Handbook of Chaos Control

angle), the spiral solution is of the form A(r, 0, *) = F(r) exp[i(-ujt + mO

(5.5)

where the (nonzero) integer m is called the topological charge or winding number, and uj is the (rigid) rotation frequency of the spiral. Since the functions F and x/j have the following asymptotic behavior \-Q2

,drtl)-+Q

for r -> oo

r m , ^ - const 4- r m + 1 for r -> 0

(5.6)

the wavefronts assume the form of an Archemedian spiral far away from the center of rotation (spiral core). Thus spirals emit asymptotic plane waves with the wavenumber Q. The spiral's wavenumber Q is an unique function of the parameters 6, c, which provides wavenumber selection mechanism (see Hagan, 1982). Encircling the spiral center on any closed curve in the counter-clockwise sense yields a phase change of 27rra. Since this cannot disappear continuously spirals have the properties of topological defects. The phase

5.3 Stability of Basic Solutions

125

Most typical are the non-charged sinks which are similar to ones in ID. However one also has localized sinks containing 'enslaved', or trapped, topological defects (Aranson et al, 1992, Weber et al, 1992, Hiiber et al, 1992) and therefore possess formally a topological charge. In contrast to spirals, which are active defects, they do not emit waves and, hence, play a minor role in the dynamics.

5.3

Stability of Basic Solutions

5.3.1 Stability of plane waves In order to test the stability of the plane-wave solutions one first considers the complex growth rate A of the modulational modes. One seeks the perturbed solution in the form A = (F + Sa+ exp[ik • r] + 8a- exp[-ik • r]) exp[i(Q • r - ut)]

(5.7)

where k is a modulation wave vector and Sa± are the small perturbations. Restricting one-selves to the most dangerous longitudinal perturbations with k || Q, one easily finds the expression for the growth rate A: \(k) = -k2 - 2iQbk - F2 ± y/(l + c 2 )F 4 - (bk2 - 2iQk + cF2)2 .

(5.8)

In the long-wavelength limit (k ->• 0) one may expand (5.8) leading to X(k)

=

ivgk - D{lk2 + O(k3)

(5.9)

Here vg = 2(c - b)Q is the group velocity and D\\ = 1 + be - 2(1 4- c2)Q2/(l - Q2) called phase diffusion coefficient. Traveling waves are long-wavelength stable for Z)|| (Q) > 0 which indicates the Eckhaus stable range:

where Q2E is the critical Eckhaus wave number. The Benjamin-Feir limit (BF) is given by the condition l + 6c = 0

(5.11)

where the solution with Q = 0 becomes unstable - all other plane-wave solutions being then unstable already.

5.3.2 Absolute versus Convective Instability of Traveling Waves Since the traveling waves have in general a nonzero group velocity the Eckhaus criterion can be taken only as a test for convective instability where a localized initial perturbation SQ(X) of the asymptotic plane wave, although amplified in time, drifts away and does not necessarily amplify at a fixed position (see Fig.

126

5 Topological Defects and Control of Spatio- Temporal Chaos

5.2). To test for absolute instability, where a localized perturbation amplifies at a fixed position and which is often relevant in finite systems, one has to consider the time evolution of a localized perturbation which is in the linear range given by S(x, t) = r

dk/{27T)SQ{k) exp(ikx + X(k)t)

(5.12)

J—oo

where 50(fc) is the Fourier transform of So(x). The integral can be deformed into the complex k-plane. In the limit t -> oo the integral is dominated by the largest saddle point ko of \(k) (steepest descent method, see e.g., Morse and Feshbach, 1953) and the test for absolute instability is Re[X(k0)} > 0 with dkX(k0) = 0

(5.13)

The long-wavelength expansion (5.9) indicates that at the Eckhaus instability, where JDJJ becomes negative, the system remains stable in the above sense. When D|l vanishes and Q ^ 0 the main contribution comes from the term linear in k that then can suppress instability.

5,3.3 Stability of topological defects in one and two dimensions The spectrum of the linear stability problem for topological defects consists of two parts: the continuous spectrum, corresponding to spatially extended modes of the emitted waves (outer stability), and the discrete spectrum of localized modes describing the destabilization of the core of the defect. The core stability is a highly nontrivial problem and we refer to the original articles (core stability of Nozakki-Bekki hole was considered by Sasa and Iwamoto, 1991, Popp et al 1995 , the core stability problem for spirals was solved by Aranson, Kramer and Weber, 1994). Since source defects emit traveling waves with a nonzero group velocity vg = DQUJ — 2(6 — c)Q directed outward, the Eckhaus criterion is not sufficient for instability. The test for absolute instability has to be applied according the criterion (5.13). The stability diagram in the 6, c-plane for the standing hole solutions (in one dimension) and spiral waves (two dimensions) is shown in Fig. 5.4 a-c. Using the general expression (5.8) together with the wavenumber emitted by the holes and the spirals one obtains both the Eckhaus limit (shown in green line) and the absolute stability limit (blue line) limit according to the conditions (5.10,5.13)4. The convectively unstable waves with the wavenumber not equal to selected by spiral or hole are restricted by the red line (called saddle-node curve according to Weber et al, 1993). The convectively unstable but absolutely stable waves (convective range) exists between green and blue lines. On the Fig. 5.4a the core stability region is shown in solid black line. As one sees, the hole solution is stable only in a very narrow belt in the parameter space (above blue line and inside the "black balloon"). In contrast, the spirals are core stable in most of the parameter 4

Only the part of actual diagram is presented. The remaining part can be reproduced the aid of the symmetry c —» —c, b —> —b.

5.3 Stability of Basic Solutions

JD

127

-1.0

Convective Range 0.05

0.10

0.15

0.20

e=l/b Figure 5.4 Stability diagrams for standing hole (a) and one-armed spiral (b,c). Black lines depict core stability limits, green and blue lines show Eckhaus limit and absolute stability limit correspondingly. The convective range, corresponding to the waves, emitted by hole or spiral lays between between blue and green lines. Red line corresponds to the saddle-node curve. No convectively unstable waves exist right of red line

128

5 Topological Defects and Control of Spatio- Temporal Chaos

Figure 5.5 The growth of spirals from a turbulent state in the convective range. The sequence of snapshots for \A\ is taken for 6 = -2 and c = 0.5.

space. The core instability of the spirals occurs only for very large b (Aranson, Kramer, Weber, 1994). Fig. 5.4c blows up the core instability region for large b. In two dimensions the spatio-temporal chaos occurs right on the absolute stability limit (blue curve). In the convective range and core-stable regime one has a very striking behavior, shown in Fig. 5.5: starting from random initial condition, the system spent quit a long time in a turbulent state with a very large number of defects (Fig. 5.5a). Then large spirals start to nucleate and sweep the fluctuations away to the system border, since no fluctuations are produced by the stable core of the spiral (Fig. 5.5b-d). Limiting state is few motionless large spirals. Inside the core-unstable convective range (see Fig. 5.4c) the behavior is even more spectacular. In the beginning one has growth of large spirals, as in a core stable range. However, the fluctuations, produced by the core instability, becomes amplified convectively and finally break the spiral down and bring the system back to the turbulent state. Then the process repeats. Thus, one has a persistent spatiotemporal defect-mediated intermittency with a number of defects in the system varying in a very large range. The behavior of the ID CGLE is more subtle. Since the hole solution is stable only in a very narrow region, it almost has no relevance

5.4 Control of Chaos in the Complex Ginzburg-Landau Equation

129

everywhere else. Two types of chaotic behavior are distinguished for the ID CGLE: phase turbulence (when the amplitude fluctuations are negligible) and amplitude turbulence otherwise (or defect turbulence, Shraiman et al, 1992, Egolf and Greenside, 1995). The phase turbulence is restricted from the left by the Benjamin-Feir line be + 1 = 0 and separated from the amplitude turbulence for larger c by the line which is very close to the saddle-node curve. There is also a region where both types of chaotic behavior coexist (near core stability line). Practically the entire convective range for ID CGLE falls into phase turbulent domain.

5.4

Control of Chaos in the Complex GinzburgLandau Equation

5.4.1 One-dimensional situation. Control of the hole solution In the one-dimensional case, the Nozakki-Bekki holes are stable only within a narrow band in 6, c plane. Otherwise there is growth of some core-localized mode, which in the convective range brings the system into the phase turbulence regime. In order to suppress turbulence in the ID CGLE we only need to stabilize one hole in the bulk of the system by suppressing the core instability. This can be achieved by adding to the r.h.s. of Eq (5.1) the term /i/(x), where /i is a complex number and f(x) is an arbitrary localized form-factor; in our simulations, we used /(#) = l/cosh(aa;),a ~ 1. The resulting equation assumes the form: dtA = A + (1 + ib)82xA - (1 + ic)\A\2A + nf{x - x0)

(5.14)

The input for control scheme is the value of complex amplitude A at the point of control, i.e. at x = x$. The control // is governed by the equation »t =7iM + 0(4O&o))

(5.15)

where 71 < 0 is a parameter and g is a function of the control scheme. The choice of the 71, g will be specified later on. Let us consider how the control works. In the linear approximation the perturbed motionless hole can be written in the form (Popp et al, 1995) - x0)) + iBW(x - x0)) x j

exp(i— logcosh(tt(£ — xo)) - iui)

(5.16)

The (real) coefficient B is the amplitude of the unstable core mode which has a functional form denoted by W (for definiteness W(0) = 1), and is characterized by the growth rate A. The growth of this mode lead to annihilation of the hole solution 5 . 5 The mode with an imaginary value of B is a quasinuetral (zero growth-rate) translation mode and is not dangerous, since it leads only to the drift of the hole

130

5 Topological Defects and Control of Spatio- Temporal Chaos

In the presence of the control scheme, the mode amplitude is governed by the equation Bt = XB- % | sin(arg/x - arg(A(x0)))

(5.17)

where 6 characterizes the response of the core mode to the control. This equation is obtained by projecting the CGLE onto the single mode subspace and using the fact that B and A are real. In order to "close the control loop", we instantaneously adjust the phase of JI to satisfy arg/i — arg A(x$) — it/2 and also specify a dynamic equation for |/i| (compare with Eq. (5.15)) H=7i|/i|+72|A(zo)|

(5.18)

The suppression of the instability in the Eq. (5.17) is possible because near the core of the hole we have |J5| = |A(a?o)|. Note that the growth rate A typically does not exceed 1; also, the value of the constant 6 can be calculated accurately using perturbation theory (see Popp et al, 1995), but for our purposes the rough estimate <5 « 1 is good enough. Therefore, one has the not very stringent conditions for the coefficients 72 < —X2/S , 71 < —A in order to make the coupled system stable. To verify this scheme, we performed numerical simulations with Eq (5.14) in one dimension. We used the high-resolution implicit quasispectral split-step method based on the fast Fourier transformation (FFT), with (typically) 1024 collocation points. The time step was chosen to be no larger than 0.05. The simulations also were checked by doubling the space discretization. The results appear to be insensitive to the resolution of the numerical scheme. We chose the length I — 300 and periodic boundary conditions. Eventually, the particular choice of boundary conditions are not relevant in the system with active sources of waves, since the stabilized defect breaks the symmetry of the system by emitting waves outward. As a result, the perturbations from boundaries do not propagate inside the integration domain. In contrast, the configuration of shocks is affected by the boundary conditions. In the periodic system the emitted waves collide approximately in the middle, whereas in other boundary conditions (e.g. no-flux) the shocks are formed near the edges. The results are presented in Figs. 5.6,5.7. We considered the range of phase turbulence, b = - 2 , c between 0.5 and 1.1, where the spatial-temporal chaos is characterized by strong phase fluctuations and almost constant amplitude (phase turbulence). A typical spatio-temporal chaotic pattern as it evolves from small amplitude noise is shown in Fig 5.6a. As we switch on the control in the middle of the system, a hole is nucleated. Waves emitted by this holes sweep away the turbulent fluctuations toward the shock, and one eventually has a perfectly synchronized steady pattern (see Fig. 5.6b). Clearly, the maximum domain which can be synchronized here exceeds the system size. The time dependence of /i and the amplitude of the minima A is given in fig. 5.7. The level of applied control in the final state is very small, |^| ~ 10~5 - 10~6, depending on the numerical precision of the simulations 6 . Let us now consider control of an already developed chaotic state. Formally speaking our considerations based on linear stability analysis fail 6

Finite numerical precision acts in the same way as a broad-band uncorrelated noise

5.4 Control of Chaos in the Complex Ginzburg-Landau Equation

131

Figure 5.6 Evolution of |A| (blue = 0, red = 1) in ID CGLE with b = -2., c = 0.8, L = 300: a - no control, b - the hole in the middle of the system is controlled since t = 20, 71 = 72 = - 5

Figure 5.7 Time dependencies of the magnitude of |A| at the control point (a) and the control level (b); dashed line - single precision in ID, solid line - double precision in ID, dotted line - single precision in 2D.

132

5 Topological Defects and Control of Spatio-Temporal

Chaos

because one has no well-established holes in the turbulent regime. However, the control itself may relatively quickly create a hole; in our simulations it took about 15 dimensionless time units. If however the level of control is restricted to be below some value /i c , the probability of "locking" decreases. Moreover, in the range of phase turbulence small values of \A\ are very unlikely, and one needs the value of IJLC to be above some threshold /io in order to force \A\ to actually vanish at the point of control. In the case of /i c < /io the system becomes fixed at some state, characterized by /x = /xc and |A(ro)| = A(0) > 0. This state also serves as a source of waves and may stabilize the system. In the region of amplitude turbulence one may expect the locking even for arbitrary small JLIC, because the probability that |A| being close to zero is finite, but in the ID CGLE the range of convectively unstable amplitude turbulence is very narrow.

5.4.2 Control of spiral in the two-dimensional complex Ginzburg-Landau equation The situation is similar in the two-dimensional CGLE. In the region of parameters (6,c) where the spiral is core-unstable and emitted waves are absolutely stable but convectively unstable, control of the entire pattern can be achieved through control at a single point near the spiral's core. The perturbed spiral solution can be represented in the form (compare Eqs. (5.5) and (5.16) ) 4(r, t) = (F(r) + BW(r, 0)) exp i[-ut + mO + i/j(r)]

(5.19)

where B is the complex amplitude of the (unstable) core mode W. When B = 0, we have an exact spiral solution at the origin. By topological reasons, the growing mode of the core instability does not destroy the spiral, but only shifts its core away from the origin. Here argjB characterizes the polar angle of the core of the shifted spiral with respect to the origin. Numerical simulations show that without control this instability does not saturate in the small amplitude meandering of the spiral, but destroys the spatial coherence completely and produces extensive spatio-temporal chaos (Aranson, Kramer and Weber, 1994). Similarly to the onedimensional case, we consider the equation governing the evolution of B: \B\t = Re\\B\ - \5\\fi\ cos(arg/x + 0 - arg(A(r0)))

(5.20)

where r 0 is the coordinate of the control site. The dynamics of control is governed by the equation identical to Eq. (5.18). The control is achieved by forcing the spiral to drift towards the point r 0 so as to diminish the value of |J3|. The values of A, 5 = \6\el<^ can be evaluated using the theory developed by Aranson, Kramer and Weber, 1993, 1994. In practice, |A| ~ .1, \6\ ~ 1, and we choose arg/i ~ argA - . The parameters of the two-dimensional simulations were b = 14.285 (or e = 1/b = 0.07) and c = —0.6, which falls in the region of spiral intermittency (Aranson, Kramer, Weber, 1994). We now used either 128x128 or 256x256 collocation points. In the absence of control one has bursts of turbulence separated by the nucleation of well defined spirals (Aranson, Kramer, Weber, 1994). It turns out that it is difficult to

5.4 Control of Chaos in the Complex Ginzburg-Landau Equation

Figure 5.8 Four snapshots of \A\ for 2D CGLE, demonstrating control of the spiral in the regime of core instability inside convective range (b = 14.285, c = —0.6, see Fig. 5.4c). The images are taken in the moments of time 20; 30; 45; 90 correspondingly. The control is imposed in the center of the domain.

achieve an effective control starting immediately from small amplitude noise. Any defect which one attempts to control attracts a neighboring defect of the opposite sign, and they form a state with zero topological charge. This state cannot be continuously transformed to a single defect state. However, we note that after some transient behavior, large spirals (unstable) are formed. These spirals emit waves which screen the core from perturbations due to other defects. Applying control in the vicinity of the core of a spontaneously nucleated spiral one can easily obtain the desired locking. Because this procedure is rather time-consuming, we shortened the expectation time of "big" spiral formation by introducing initial conditions in the form of small amplitude noise plus a small amplitude "vortex seed", chosen in order to satisfy the requisite topological condition. Once this is done, the control suppresses the core instability and eventually produces a steady synchronized pattern similar to that seen in ID. The establishment of the controlled single spiral state is demonstrated on Fig. 5.8. The temporal evolution of the field (at the control point) and of the control strength was given in Fig. 5.7.

133

134

5 Topological Defects and Control of Spatio-Temporal

Chaos

5.5

Control of Spatio-Temporal Chaos in Reaction-Diffusion Systems

An important class of systems exhibiting spatio-temporal complexity and relevant for various chemical and biological processes involves two very different scales of time (e.g. fast relaxation and slow recovery). A well-established model of such systems is a couple of reaction-diffusion equations, called usually FitzHugh-Nagumo model (Keener and Tyson, 1986): & ^ e dtv = eDvV2v + g{u) - v ,

(5.21) (5.22)

with certain nonlinear functions / and g describing reaction kinetics, 6 < 1 characterizes the relaxation rate, u is the concentration of fast activator and v of slow inhibitor. A characteristic feature of these systems is segregation of the field u into two phases (excited and quiescent) separated by a thin (of order e) interface. This interface often takes the form of a spiral wave, which is in many cases similar to the spiral wave in the CGLE considered above. The dynamics of spiral waves in reaction-diffusion systems is a subject of intensive research (Winfree, 1991, Karma et al 1994, Kessler et al, 1994, Aranson et al, 1995) It is known that spiral waves may exhibit various instabilities (Winfree, 1991, Karma et al 1994, Kessler et al, 1994), manifested as periodic, quasiperiodic and even chaotic meandering of the spiral core and the breakup of the waves emitted by the spiral (Karma, 1993, Bar and Eiswirth , 1993). Eventually, a spiral breakup leads to spatio-temporal chaos. This scenario is believed to be responsible for severe heart arithmia as the fibrillation of the cardiac tissue (Karma, 1993; Courtemanche, Glass and Keener, 1993; Karma et al, 1994). The control of such systems may appear extremely important for future medical applications. The existing methods for suppression of the fibrillation involve high-voltage electric shocks which of course badly damage cardiac tissue. A new method of suppression of initial stage of fibrillation (usually called tachycardia) through annihilation of the spiral wave by the short-time pulse perturbation near the spiral's core was proposed by Krinsky, Plaza and Voigner, 1995. However, these method fails to stabilize a well-developed turbulent state containing many spirals. The control of spatio-temporal chaos in the reaction-diffusion system can be achieved in a similar way as for the CGLE. A persistent spiral wave source can be introduced by applying a systematic perturbation at an arbitrary point of the system. The structure of the perturbation could be derived from the condition that a new high-frequency source is formed at the point of control as soon as possible. If the waves emitted by the source are not absolutely unstable, they will eventually invade the domain and sweep away the spatio-temporal fluctuations produced by other sources. Loosely speaking, in the context of cardiac applications, this method imposes "tachycardia against fibrillation". As soon as the new source has swept the turbulence, it can be gradually driven away. To illustrate the method we

5.5 Control of Spatio-Temporal Chaos in Reaction-Diffusion Systems

Figure 5.9 The snapshots of the slow field v. Blue color corresponds to minimum of v, red color corresponds to maximum. Typical pattern without (a) and with (b) control (a = 2, Do = 0.4); parameters of the simulation: a = .84, b = .07, e = .08, Dv = 0.1, system size is 30.

consider the model of the reaction diffusion system exhibiting spiral break-up instability. This model was introduced by Bar and Eiswirth, 1993 in the context of the catalysis of CO on Platinum single crystal surfaces (see also Hildebrand, Bar, and Eiswirth, 1995). Bar and Eiswirth introduced a set of modified FitzHugh-Nagumo equations to model this behavior with the functions / = —u{u — \){u - (v + 6)/a), and g = 0 for u < 1/3, 1 - 6.75u(u - I) 2 for 1/3 < u < 1 and g = 1 for u > 1. This chaotic behavior does not occur in more traditional excitable media such as the Belousov-Zhabotinsky reaction (Skinner and Swinney, 1990, Steinbock et al, 1993). As in other such reaction-diffusion systems, there exist spiral solutions; in a wide range of the parameters a, b the spiral core meanders. The chaotic state is due to the fact that this meandering exists in the same parameter range as a convective instability of the emitted waves. The meandering excites the unstable mode, the spiral arms break and the system becomes disordered. A similar behavior seems to occur in a more realistic (but far more complicated) model of wave propagation in cardiac tissue (Karma, 1993), due to the coexistence of spiral meandering and an (almost) period-doubling instability of the plane wave state (Courtemanche et al, 1993, Karma et al, 1994). In order to create a feedback to control site, we add a term —/x/(r - r 0 ) to the

135

136

5 Topological Defects and Control of Spatio-Temporal Chaos

slow field v Eq. (5.22). The control \i in the simplest case cab be governed by the equation dt\i = 7iM + 72(^(^0) ~vo)

(5.23)

where v0 corresponds to the value of slowfieldat the center of spiral. The control is achieved due to the fact that exactly at the spiral center v(ro) is constant whereas any deviation from the center results in periodic time dependence of v(ro). The resultant periodic oscillation of \x forces the spiral center to drift towards the point of control. Eventually the center is pinned at the control site, and the spiral emits the waves which suppress the turbulence throughout the system, as it can be seen on Fig. 5.9. The simulations for this case were carried out by modifying the v equation in the high-performance code EZ-Spiral written by Dwight Barkley, 1991. In contrast to the case of the CGLE, the exact value of VQ is not known, and any mismatch in VQ leads to a non-vanishing controller amplitude in the final stabilized state. In order to overcome this difficulty, we started from some arbitrary value of vo providing stable locking of the spiral (see also Zou et al, 1993, Steinbock and Miiller, 1992). Then we gradually decreased the value of VQ achieving as low as possible level of /x. In practice, the minimal value of JJL depends on a and for a — 1, /i ^ .19; changing the value of VQ past this point leads to a re-emergence of the original instability. This did not happen in the CGLE and may reflect the fact that a pinned spiral does not always smoothly return to the unpinned case as the pinning strength decreases to zero.

5.6

Conclusion

We considered the algorithm for suppression of spatio-temporal chaos in one and two dimensional isotropic convectively unstable media using a single site control technique. The control of the three-dimensional systems can be approached in the same manner. An even more challenging goal is to control chaos in an absolutely unstable range. Simulations show that in this case the core of the defect can also be successfully locked; however outgoing waves do not eliminate growing disturbances. An obvious and naive way to overcome this difficulty is to to apply an array of controlled active sources. However, it seems feasible to achieve control of extended system even in the regime where the waves emitted by the defects are absolutely unstable. The requirement could be just the existence of a convectively-unstable band of wavenumber (not necessary emitted by the defects). In the context of the CGLE, we expect that the control can be in principle achieved up to the saddle-node curve rather than in the convective range. Thus, we suggest that it is not only necessary to control the existing (exact) solutions of the system, but also to create new wavegenerating defects with desired properties of emitted waves (artificial defects). Applying systematic small (but non-vanishing) perturbations near the core of defect we can significantly change the core structure. This new defect may emit waves at a different frequency. If this frequency does not lie in the absolutely unstable

5.6 Conclusion

137

range, then the the defect would suppress the absolute instability of the system by sweeping the fluctuations away to the boundaries.

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5.6 Conclusion

139

45. Shinbrot, T., (1995), Advances in Physics, 44, 73. 46. Showalter K., (1995), Chem. Br., 31, 202 47. Shraiman B.L, Pumir A., van Saarloos W., Hohenberg P.C., Holen , Chate, (1992), PhysicaD, 57, 241 48. Skinner G.S. and Swinney H.L., (1990), Physica D 48, 1. 49. Steinbock 0., Zykov V., and Miiller S., (1993), Nature, 366, 322 50. Steinbock O. and Miiller S.C. , (1992), Phys. Rev. E 47 1506 51. van Saarloos, W. and Hohenberg, P.C., (1992), Physica D, 56, 303 52. Weber A., Aranson I., Aranson L., and Kramer L., (1992), Physica D, 1, 279. 53. Weltmann K.-D., Klinger T., and Wilke C, (1995), Phys. Rev. E., 52, 2106 54. Winfree A.T., (1991), Chaos, 1 303 55. Witkowski F.X., Kavanagh K.M., Penkoske P.A., Plonsey R., Spano M.L., Ditto W.L., and Kaplan D.T., (1995), Phys. Rev. Lett., 75, 1230 56. Zou X., Levine H., and Kessler D.A., (1993), Phys. Rev. E , 47 R800

6

Targeting in Chaotic Dynamical Systems

E. J. Kostelich Department of Mathematics, Arizona State University, POB 871804, USA

6.1

Introduction

"Targeting" refers to a process wherein one uses a suitable sequence of small controlling perturbations to steer an initial condition on an attractor to a neighborhood of a prespecified point (target) on the attractor. Here it is shown that targeting can be done in high dimensional cases, as well for one- and two-dimensional maps. The method is demonstrated with a mechanical system described by a four dimensional mapping whose attractor has two positive Lyapunov exponents and a Lyapunov dimension of 2.8. The target is reached by making very small successive changes in either one or two control parameters. Targeting often makes it possible to reach a neighborhood of a prespecified target point several orders of magnitude more quickly than would be the case without targeting. Targeting can be used to rapidly switch a chaotic process between different periodic orbits [1]. The distinguishing feature of chaotic processes is their sensitive dependence on initial conditions and highly irregular behavior that is difficult to predict except in the short term. The so-called "strange attractors" associated with chaotic processes often have a complex, fractal structure. The existence of chaotic behavior in a wide variety of mathematical, physical, and biological contexts is well known. There are many introductory texts on chaos; see for example [2]-[7]. See also the reprint collections by Cvitanovic [8] and Hao [9] for some important early papers on observations of chaotic dynamics in a variety of laboratory experiments. Reference [10], subsequent proceedings, and the references therein contain a wealth of additional applications. Ott et al. [11] introduced the idea that control of chaos could in some cases be attained by feedback stabilization of one of the infinite number of unstable periodic orbits that naturally occur in a chaotic attractor. Their method and variations thereof have been used in many experimental situations; see the other contributions in this volume and Ref. [12] for additional references. Although the ergodic nature of the dynamics on the attractor ensures that typical initial conditions eventually reach the neighborhood of almost any prespecified target point on the attractor, the time needed to reach the target might be very long. Targeting algorithms often reduce the waiting time by several orders of magnitude [13]. In this paper, we will briefly outline one approach to the targeting problem and describe some recent extensions to this work.

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6 Targeting in Chaotic Dynamical Systems

Targeting algorithms can be combined with "control of chaos" algorithms to switch a chaotic process to different periodic behaviors using only small perturbations to the system. For example, a target point can be selected that is in a small neighborhood of a periodic saddle orbit on the attractor. We apply the targeting algorithm, described below, to steer an initial condition to a neighborhood of the target. Then a control of chaos algorithm, similar to those described in Ott et al. [11] or Romeiras et al. [14], can be applied to maintain the trajectory near the periodic saddle orbit.

6.2

An Outline of Targeting Algorithms

The One-Dimensional Case Given an initial condition and a target point on a chaotic attractor, we wish to direct the resulting trajectory to a small region around the target as quickly as possible. Because of the inherent exponential sensitivity of chaotic time evolutions to perturbations, one expects that this can be accomplished using only small controlling adjustments of one or more available system parameters. This was demonstrated theoretically and in numerical experiments for the case of a two dimensional map by Shinbrot et al. [15] and also in a laboratory experiment for which the dynamics were approximately describable by a one dimensional map [16]. Consider a dynamical system given by a one-dimensional map, Xn+l = U(Xn),

(6.1)

where ji is an adjustable parameter. In the simplest case, we suppose that \i can be varied about a nominal value, say /x0. If the perturbation 8fi is small, then the change in x\ given the initial condition XQ is approximately

If we restrict \8/JL\ to be less than or equal to A/x, then 8x\ lies in the interval A/i = [xi — Aa?i, x\ + Axi], where A/i depends on #o and A/i. In a chaotic process, the intervals A/i, A/ 2 , • • • typically form an expanding set that eventually contains any target point xt in the domain of / . The precise sequence of parameter changes needed to reach a small neighborhood of xt can be found by bisection (among other means). See [15, 16] for additional details.

The Higher-Dimensional Case We next consider a targeting algorithm that can be applied in higher-dimensional settings. We illustrate its application with a four-dimensional map, the kicked double rotor [17]. For the parameters used here, typical points on the chaotic attractor have two positive Lyapunov exponents, which implies that there is a two dimensional unstable set and a two dimensional stable set associated with most

6.2 An Outline of Targeting Algorithms

143

Figure 6.1 The double rotor.

points. However, the procedure can be extended in principle to maps with any number of positive and Lyapunov exponents. The first rod, of length £i, pivots about Pi (which is fixed), and the second rod, of length 2£2, pivots about P2 (which moves). The angles #i(£), 02(t) measure the position of the two rods at time t. A point mass mi is attached at P2, and point masses ra2/2 are attached to each end of the second rod (at P3 and P4). Friction at Pi (with coefficient vi) slows the first rod at a rate proportional to its angular velocity 6\(t)', friction at P2 slows the second rod (and simultaneously accelerates the first rod) at a rate proportional to #2(t) — Oi(t). The end of the second rod, marked P 3 , receives impulse kicks at times t = T, 2T, . . . , applied to the second rod with strength p at an angle a relative to the vertical direction. Gravity and air resistance are absent. Figure 6.1 illustrates the setup. The double rotor map is the four dimensional map xn+\ = F(xn), defined by / GWi \ Xn+l =

.

/ (M0 n + 0 n ) mod =

V 6n+i J Here Qn and 0 n are 2-vectors,

ClSin v ; G(Q)=( \ c2sm92

2TT

\

.

V £0

'

G ( 0 )

J

(62)

144

6 Targeting in Chaotic Dynamical Systems

where the angles 0\ and 82 are taken to lie in [0,2?r]. The positions of the rods at the instant of the nth kick are given by 0^ = 0i{nT), and the angular velocities of the rods immediately after the nth kick are given by 0t- — 0i(nT+). L and M are constant 2 x 2 matrices. For the sake of simplicity, we assume that (mi + m2)^l = m2f% = /. Then

L = J2 Wi*XiT,

M = £ Wi(e

XiT

with "

where a = | ( 1 + i/i/A), d = |(1 — i^i/A), b = —1/2/A, A = \[v\ +4i/|? Ai^ = — \{yi + 2^2 ± A), and ci,2 = phj/I- In all the numerical work described in this paper, we fix the values of the parameters v = T = / = mi = m2 = t>2 — I?

^1 = ""F-

We use both the force p and the angle a as the control parameters, taking as the nominal values p = p = 9 and a = a = 0. A derivation of the double rotor map is given in [14]. In the remainder of the paper, we write xn = Fn(xo) to mean the n times iterated point xo, i.e., the point obtained by iterating the map n times starting from XQ. The double rotor map is invertible, so F~n(xo) refers to the nth iterate of x under the inverse map. The notation F(x) means that the map is applied with the kick set to its nominal value (here p = 9); the notation F(x,p,a) means the map applied to x with the kick set to p and its angle of application as a.

The Higher-Dimensional Targeting Algorithm Now suppose that we are given a target point XT on the attractor. The point XT must be "typical," that is, XT is not a periodic point and has two positive and two negative Lyapunov exponents associated with it. Let xo, #i> ..., XT-I be the T preimages of XT, as shown in Fig. 6.2. We call this set of points a path to the target. (As discussed below, it is often useful to choose XT in a small neighborhood of a periodic saddle point.) The stable (resp. unstable) manifold of x is the set of points y with the property that \\Fn{x) - Fn{y)\\ -* 0 (resp. ||F" n (x) - F~n(y)\\ -> 0) as n -> 00. In other words, the trajectories of points on the stable manifold of x on the average approach the trajectory of x under forward iteration of the map. (The same is true for the backward iterates of points on the unstable manifold.) If x has two positive and two negative Lyapunov exponents, then both its stable and unstable manifolds are two dimensional.

6.2 An Outline of Targeting Algorithms

145

Figure 6.2 Schematic diagram of the basic control procedure. Two successive perturbations of the kick are applied at yo to steer it to the stable manifold associated with the point #2, which is a preimage of the target point.

Suppose that the map is iterated from an arbitrary initial condition in the basin of attraction for the double rotor attractor, and that one of the iterates (call it yo) falls within a suitably small neighborhood of the point XQ in the path to the target. Without any control, the points yi, y2, • • -, rapidly diverge from the trajectory xi, x2, ..., starting at x0. The goal of the targeting procedure is to apply suitable perturbations to the map at y0 to steer the perturbed point onto the stable manifold of one of the points on the orbit of xo. Suppose for example that perturbations can be applied so that after two iterations of the map, yo is steered to a new point y2, relatively close to y2 , that lies on the stable manifold 52 of x2. The contracting nature of the dynamics on 52 implies that the orbit of y2 and the orbit of x2 approach one another. The approach of the trajectory starting from 2/2 to the one leading to the target need not be uniform. However, on the average, it should approach the target trajectory at a rate proportional to eLzt, where L3 is the largest negative Lyapunov exponent and t is the number of iterations, counting from y2. The perturbations can be applied in various ways. For example, two successive perturbations to the kick at j/o and y\ can be applied to try to steer y0 to a point y2 on 52. Note that 52 is the stable manifold of x2 for the unperturbed map at the nominal kick value p = p. If only the kick can be varied, then we try to find two values po and p\ close to the nominal value p of the kick so that y2 = F(F(yo,po),Pi) £ 52. (Recall that 52 typically is a 2-dimensional set. The intersection of 52 with a 2-plane generically is a single point in R4. Two successive small perturbations to the kick yield vectors go = dF(F(yo>po),Pi)/dpo and

146

6 Targeting in Chaotic Dynamical Systems

gi = dF(F(yo,po),pi)/dpi that typically span a 2-plane through y2 that intersects 52 at a unique point 2/2-) Alternatively, if two different parameters can be varied independently (for example, the kick and the angle at which the kick is applied), then we can attempt to find a perturbation to each to try to steer y0 to a new point 2/1 on the stable manifold of Si of x\. The difficulty in finding the intersection point y2 with 52 (or y\ with S\) is that the stable manifolds are not well approximated by a plane except in a small neighborhood of x2 (resp. x\). Although ||:ro - 2/0II may be small (perhaps of order 10~2), ||x2 — 2/2II and ||xi — 2/11| generally are not (they may be of order 1). Consider the case where we wish to make two successive perturbations to the kick at 2/0 to try to hit 52. We can approximate 52 by looking at the inverse images of the stable manifold of a point that is further down the path. Consider for example 5g, the stable manifold through x8. Under the inverse map, 5s is an expanding set. Suppose we take a point z near x$ on the tangent plane to Sg. Although z does not lie exactly on 5g, the inverse images F~ 1 (z), F~2(z),... approach the corresponding sets 57, 56,... because under the inverse map, errors damp out along the directions associated with the unstable manifolds of 2:7, XQ, ... . We approximate 5s as follows. Let A = DF{x2)DF{x^) • • • DF(xi) be the product of the Jacobian matrices of partial derivatives of the double rotor map F at the indicated points. The eigenvectors of A associated with the contracting eigenvalues typically span a 2-plane. Let SQ and s\ be two unit basis vectors for the contracting eigenspace of A. If G$ and o\ are small numbers, then z = x$ + 00 $o + 0"isi is a point that is close to 5s, and its inverse images should approach the corresponding stable sets quickly. In particular, F~6(z) should lie very close to the stable set 52 associated with x2 even though ||F~6(^) - # 2 || may not be small. The basic targeting step is a kind of shooting procedure. We attempt to determine values ao and o\, together with two values of the kick p0 and p\, such that F~6(x8 +a o $o + <7i$i,p,a) = *XF(j/o,Po),Pi) =2/2-

(6.3)

Equation (6.3) often can be solved numerically using Newton's method. Figure 6.2 is a schematic illustration of the situation. If it is possible to perturb both the strength of the kick and the angle a at which it is applied, then we try to solve an equation of the form F~6(xs

where £1 is a point near 2/1 that lies on the stable manifold S\ associated with x\. Because the negative Lyapunov exponents are relatively large in absolute value, I do I and \<TI\ typically are small, usually of order 10~6 even when \\y2 - x2\\ is of order 1. Thus, a successful solution of Eq. (6.3) determines two consecutive kicks such that the trajectory starting from the new point y2 = F(F(yo,po),pi) falls within 10~6 of the trajectory leading to the target after six iterations. (The values of ao and a\ in a typical solution of Eq. (6.3) depend on the negative Lyapunov exponents associated with the points on the path.)

6.2 An Outline of Targeting Algorithms

147

There is nothing special about the choice of Sg and the use of six inverse iterations to find an intersection point on S2. For example, if the target point is XQ, then we try to find a point on 52 whose fourth iterate lands close to XQ. If the target point is at, say, #20> then one can look at the inverse image of points on SQ or S10 instead of Sg. Going further down the path (say to Sg or S10) typically yields a point whose appropriate inverse image is closer to the stable set 52. However, it also makes the numerical solution of Eq. (6.3) more ill conditioned. (If we look at the inverse images on 5s, then it is necessary to evaluate the matrix product DF(x2)DF(xs) • • • DF{xi). If we look at S10 instead, then we must evaluate DF(x2)DF(xs) • • • DF(xg), and so on. These matrix products become more singular as more terms are added.) Thus there is a tradeoff between numerical precision and approximation errors arising from the dynamics. For the parameters of the double rotor map that we have used, six inverse iterations is a good compromise between the accuracy in finding, say, #2 and the accuracy in iterating to the target. (Of course, fewer iterations of the inverse map are used as the controlled trajectory gets within six iterates of the target.) Because of small errors in the initial approximation of S$ and numerical roundoff errors, the control described above must be repeated from time to time to keep the new trajectory close to the path leading to the target. For instance, the two kicks p0 and pi might be applied successively to y0; afterward, the system might be set to the nominal value of the kick p for the next six iterations. The resulting point will be about 10~6 away from the target trajectory. The control step can be repeated at x$ to steer the trajectory close to #16, etc. Only very small perturbations of the kick are required at x$ to accomplish this, because the controlled trajectory is usually within 10~6 or so of the path to the target at this point. Equation (6.3) may not have a solution, and sometimes Newton's method diverges because good starting values of the parameters cannot be obtained by linearizing Eq. (6.3). In such cases, it is not possible to initiate a control at the point 2/o to bring the trajectory close to the target. If the procedure fails, then one must wait until the trajectory again approaches a neighborhood of XQ and try again. For the parameter values described above, we have successfully found solutions of Eq. (6.3) about 90% of the time when ||a?o-yo|| < 0.01 and about 50% of the time when ||xo — 2/o 11 < 0.05. Moreover, the controlled trajectory rapidly approaches the trajectory leading to the target once a solution of Eq. (6.3) is determined. Typically, the distance between xi$ and yie (after two iterations of the control procedure) is close to the numerical precision of the computer (about 10~14). The basic control method described above can be applied when there is only one positive Lyapunov exponent. In that case, one must determine only a single perturbation p0 to the kick so that the new point y\ = F(yoipo) intersects the stable manifold S\ of x\ (the next point in the path) because S\ is three dimensional. The method can be extended to other maps in different dimensions in a straightforward way. For example, if the attractor sits in a six dimensional space and has three positive Lyapunov exponents, then the basic control procedure requires three successive changes to a single parameter to hit the three dimensional stable

148

6 Targeting in Chaotic Dynamical Systems

manifold of the appropriate point in the path. If three parameters can be varied independently, then one tries to hit the stable manifold of #i, and so on. The double rotor map for the parameters used here is not hyperbolic. In particular, there exist saddle fixed points in the attractor that have one unstable direction and three stable directions. This is in contrast to our determination that orbits for typical points on the attractor (i.e., almost every point with respect to the natural measure) have two positive and two negative Lyapunov exponents. In spite of this nonhyperbolic situation, we do not find any problem in our numerical experiments. As we iterate, the stable manifold of each point in the paths to the target appears to be two dimensional. (This is not surprising since the set of points for which this is not true has zero natural measure.) Finally, we remark that it is possible in principle to steer a given initial condition to the target using a sequence of four perturbations of the kick to hit a point on the trajectory leading to the target. For example, if xo,xi,X2,..., is a trajectory leading to the target and if t/o is a point near #o, then one can try to apply a sequence of kicks po,pi,p2,P3 at 2/o? 2/i?2/2,2/3 to hit x±. In practice, we have not been able to use this approach, because it is not possible to get an accurate linearization of the problem unless the corresponding trajectories are extremely close together.

6.3

The Tree-Targeting Algorithm

The procedure described above can be used to steer an orbit toward a target point, but it cannot be applied until an orbit lands near one of the points on the path. The waiting time required to approach one of the points on the path is about the same as the waiting time needed to approach a comparably-sized neighborhood of the target. A long path increases the likelihood that a given iterate lies near a point on the path, but then many control steps are required to reach the target. The tree-targeting algorithm uses a hierarchy or "tree" of paths leading to the targeting point that significantly shortens the time required to steer a typical point to the target—sometimes by several orders of magnitude compared to no control at all. The tree is constructed as follows. Suppose we have already selected a target point XT and have a path xo, # i , . . . , # T - I on the attractor leading to it; this path is the "root" path. (In the results described below, the root path has 10 points, so T = 10.) We iterate the map (possibly from an arbitrary initial condition in the basin of attraction) until we obtain a point zn that lies in a suitably small neighborhood of one of the points in the target path. We store zn, together with a path of T points leading to zn\ this path is a "level 1" path. If an iterate y falls within a suitable neighborhood of one of the points z n -T, Zn-T+i,- ••> then we can apply the control procedure described in the previous section using zn as an interim target. The idea is first to steer the orbit of y using a sequence of small perturbations to a small neighborhood of zn. The controlled orbit then lies in a neighborhood of one of the points leading to the target XT> The control procedure is continued (starting in the small neighborhood of zn) with XT

6.3 The Tree-Targeting Algorithm

149

- path at level 2

xj A'f-x

t

target Figure 6.3 Schematic illustration of the hierarchy of paths leading to the target point.

as the target. If both the root path and the level 1 path have T points, then the control steers the iterate to the target in no more than 2T steps. Figure 6.3 is a schematic illustration of this approach. Other paths are added to the tree in similar fashion. As the map is iterated, we check to see whether the current point falls suitably close to one of the points in a previously stored path. If it does, then the point is added to the tree, together with its T — 1 preimages. The tree can be made as large and as deep as required. The number of possible points in the tree grows exponentially with the number of levels, but the maximum time required to reach a neighborhood of the target grows only linearly with the number of levels. We have found good results with trees containing about 104 points in 500 to 1,000 paths of length T = 10 to T = 20 in three levels. (There are no more than 3T steps from any point in the tree to the final target.) The tree is not full; that is, not every point has a path at a lower level in the tree leading to a point in its neighborhood. These trees can be built with 106 to 107 iterations of the double rotor map (requiring only a couple of minutes of CPU time on current workstation computers) and occupies about three megabytes of computer memory. Once the tree is built, it is possible to steer points to the target very quickly, as follows. Let zo be a point on the attractor. If ZQ is not close to any of the points in the path tree, then we create a new set B of points by making n small random perturbations to the kick. Here A = {zf1 : z[^ = F(zo,po + rji), 1 < i < n} where r)i is a random variable in a small interval around 0. Typically we take rji from a uniform distribution in the interval [—0.05,0.05]. We now check whether any of the points in B lies near any of the points in the path tree. If so, then we attempt the control procedure. If it is successful, then we have steered the point ZQ to the target in no more than 3T + 1 steps (the first

150

6 Targeting in Chaotic Dynamical Systems

nil

20000

Figure 6.4 A plot of the 0\ component of the double rotor map as a function of iteration number. The orbit is controlled about five different fixed points in succession. When the control is turned off, the orbit rapidly leave the neighborhood of the fixed point, and without targeting, we must wait until the orbit again comes within a suitably small distance of another fixed point before control can be resumed.

step consists of the random kick, followed by no more than 3T steps of the control procedure). Each of the points in A can be iterated (using the nominal value of the kick) until one of them can be steered successfully to the target.

6.4

Results

Rapid Switching Between Periodic Orbits The targeting algorithm described above can be coupled with a control of chaos algorithm to allow a chaotic process to be switched rapidly between different periodic saddle orbits embedded in the attractor. At the nominal parameter values given in Sec. 6.2, the Lyapunov dimension of the attractor is about 2.8 and there are 36 fixed (period 1) points embedded in the attractor. [1] Consider the problem of stabilizing several different fixed points in succession. Figure 6.4 illustrates the results of one numerical test, where five fixed points have been stabilized in succession. The ergodicity of the dynamics on the attractor eventually brings the orbit within a suitably small neighborhood of the fixed point A so that a control algorithm (as described in [14]) can stabilize the trajectory

6.4 Results

151

near A. The control is continued for a time, then turned off. The trajectory rapidly leaves the vicinity of A, and we wait until it approaches a neighborhood of the fixed point J9, when the control is applied to keep the orbit near B for a time. Figure 6.4 displays the 0\ component of the state versus the iteration number. The switching times t follow an exponential distribution (t)"1 exp(t/(t))y which is typical of chaotic systems[l, 6]). In this example, (t) ranges from 12,000 ± 80 iterations to 252,000 ± 3,000 iterations, depending on the fixed point. The targeting algorithm can be applied as follows. Starting from a point in the basin of attraction, the map is iterated until the trajectory falls within a tolerance distance of one of the saddle fixed points. This trajectory point becomes the target, and a tree of paths leading to it is constructed as described in Sec. 6.3. One tree is constructed for each of the five fixed points considered here. Given an initial condition on the attractor, the map is iterated until the orbit falls within 0.05 of a point on the tree leading to one of the fixed points of interest. The targeting algorithm is applied as described above. When it is successful, the orbit lies close to the fixed point, and the control of chaos algorithm can be used as before. This approach allows the target to be reached very quickly, usually in an average of 17-19 iterations using one parameter control (where only the kick strength p is perturbed), and 13-15 iterations using two parameter control (where p and the angle of the kick a are varied). These switching times represent improvements of 3 to 4 orders of magnitude compared to the algorithm without targeting. Of course, there is overhead associated with building the trees. Usually about 107 iterations are required to construct the five trees used here. This price is worthwhile in a situation where repeated switching between periodic saddle orbits is required. However, if it is desired simply to stabilize one fixed point, then it is more efficient to wait for the orbit to reach a neighborhood of the fixed point without applying a targeting algorithm.

Optimal Targeting The performance objective in the example above is to minimize the switching time between different saddle fixed points in the attr actor. In practice, however, other performance objectives might be important. For example, in targeting a spacecraft, it might be necessary to minimize the total fuel consumption at the expense of a longer arrival time. Certain regions of a chaotic attr actor might correspond to an undesirable operating mode (e.g., high pollution production), so it might be necessary to avoid such regions or minimize the amount of time that an orbit spends there. Issues like these are special cases of a more general question in dynamics of the long-time average value of a smooth function G over a given orbit {#*}. In the case of a discrete dynamical system (like the double rotor map), the average is defined

152

6 Targeting in Chaotic Dynamical Systems

as (G) = lim n->oo

Hunt and Ott [18] have investigated chaotic processes where "typical" orbits (with respect to the Lebesgue measure of initial conditions in the phase space) have welldefined long-time averages. Saddle periodic orbits embedded in the attractor are not typical in this sense, and they may yield values of (G) that are different from those of the typical (aperiodic) orbits within the attractor. Hunt and Ott examined a large number of saddle periodic orbits embedded in the attractors of some low-dimensional dynamical systems and considered a large family of performance functions, G 7 , parametrized by 7. They found that for most choices of 7, the value of (G7) on periodic orbits of low period was larger than the value of (G7) on orbits of high period. Thus, if one regards G 7 as a measure of system performance, periodic orbits of low period are more likely to be the "optimal" periodic orbits in a chaotic attractor. We now address the question of whether periodic orbits of low period are less expensive to target, on the average, than those of high period. In the tree targeting applications discussed above, the targeting times fall in a limited range, because the path length and the number of levels in each tree are fixed. Consider a collection of orbits that reach a target in the same amount of time, where the target lies close to a periodic saddle orbit in the attractor. For a discrete dynamical system of the form xn+1 = F(xn), define the average targeting cost per step of an orbit of length k as !y(Fj(x0)),

(6.4)

where XQ is a point whose kth image lies on or near the target. If the time required to reach the target is held fixed, how does the average targeting cost per step depend on the performance function G 7 and the period of the saddle orbit that is targeted? Consider the tent map, .

T(xn)

.

f 2x n ,

= xn+1 = I 2 _ ^

0 < xn < 1/2 1/2 < X

<

(6.5)

l

We call y a period-p point if Tp(y) = y and Tj(y) ^ y for 0 < j < p.) We say that x0 is eventually periodic to y in k iterations if Tk(xo) = y but Tj(xo) £ orbit(y) for 0 < j < k. Let Sp(k) be the set of all points x0 such that XQ is eventually periodic to a period-p point of the map T in k steps. The average targeting cost per step to reach a period-p point in k steps is ^(*o,fc), xoesp(k)

(6.6)

6.4 Results

7 6 5 4 3 2 1 0

153

period p that minimizes

CJp) \

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Y Figure 6.5 Period of the orbit with minimal average targeting cost per step. The average is taken over all eventually periodic points in 7 steps for the tent map.

where N = cardinality Sp(k). The set Sp(k) is straightforward to generate using the symbol sequence associated with each orbit of the tent map. In this study, we consider the set Sp(7), that is, all orbits leading to a periodic point in seven steps. For each period p < 14, we evaluate C1{p) for 105 uniformly spaced values of 7 between 0 and 1, using the performance function G7(x) — COS2TT(X-7). Figure 6.5 shows, as a function of 7, the period p that minimizes the value of C7(p). The results imply that for this family of performance functions, the low-period orbits have the lowest targeting cost per step. Analogous results are found for the family of performance functions G1{x) = COS2TT(X — 7) + sin67r(x — 7). In practice, one does not have a convenient way to characterize all the periodic and eventually periodic orbits of a map using symbolic sequences alone. Instead, one uses a targeting algorithm to get within a small neighborhood of a target point in a certain number of steps. We now ask whether this case gives similar results to the tent map example, Eq. (6.5). Consider the quadratic map, Q(xn) = x n +i = axn(l — xn). If a = 3.72, then the map Q appears to have chaotic orbits, but it is not conjugate to the tent map. We can locate all the periodic orbits for p < 9 using a careful numerical search. (There are no orbits of period 3 or 5 for a = 3.72.) We then target each periodic point, starting from random initial conditions, using the algorithm of Shinbrot et at. [15], until we find a collection I of 105 initial conditions that land within 10~4 of the target in 7 steps by varying the parameter a within the interval [3.69,3.75]. We define the average targeting cost per step to reach a period-p orbit as = 10 - 5

(6.7) xoei

where c is defined as in Eq. (6.4). Figure 6.6 shows the period p that minimizes C 7 (p) as a function of 7. The family of performance functions is F7(x) = cos 2n(x — 7). As in the case of the tent

154

6 Targeting in Chaotic Dynamical Systems

6

5 - period with lowest average targeting cost per step 4 3 2 \ 1 : 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y

Figure 6.6 Period of the orbit with minimal the average targeting cost per step for the quadratic map with a = 3.72.

map, these numerical results suggest that the low-period orbits have the lowest average targeting cost per step. As a last example, we consider the targeting costs associated with the double rotor map. For each period p from 2 to 10, we choose approximately 100 different period-p orbits. We apply the tree targeting algorithm to each periodic orbit; for each orbit, we generate a set / of 2,000 different schemes to reach the target from various initial conditions around the attractor. (Both the strength and the angle of the kick are varied.) For each orbit leading to the target, we define the average targeting cost per step as c7(x0,k) -=1 k. - 1

J £ L

(6.8)

( ! - •

max

/

where Spj and 8aj are the perturbations applied to the kick and angle of application as part of the targeting scheme, and A/omax and Aa max are the maximum allowed values. We evaluate c7(xo,k) for 10,000 different values of 7 between 0 and 1 in each targeting scheme. Let C7(p) be the average value of c7(xo,k). (For each p, the mean is taken over 200,000 different paths.) Figure 6.7 shows the period p of the orbit with the lowest average targeting cost per step as a function of 7. The results suggest that, on the average, smaller perturbations are needed to steer an orbit to a neighborhood of a saddle orbit of low period than a saddle orbit of higher period.

6.5

Conclusions

Tree targeting algorithms have a variety of applications. The use of trees gives one a variety of ways to select paths to a given target point according to some criterion.

References

10 period with lowest average targeting 9 cost per step 8 7 6 5 \ 4 : 3 2 1 0

155

i

i

i

1

-

i

0

0.5

1

Figure 6.7 The period of the orbit of the double rotor map with lowest average targeting cost per step, using the family of cost functions in Eq. (6.8).

Additional examples of applications will appear in future work [19]. Trees can be combined with different targeting schemes [20] to minimize the targeting time.

Acknowledgments This work has been supported by the National Science Foundation (the Computational and Applied Mathematics Program and the Divisions of Mathematical Sciences and Physics) and by the Department of Energy (Office of Scientific Computing, Office of Energy Research, and the Mathematical, Information, and Computational Sciences Division, High Performance Computing and Communications Program). The author thanks Jim Yorke, Celso Grebogi, Ed Ott, Brian Hunt, Erik Bollt, and Dan Lathrop for helpful discussions.

References [1] E. Barreto, E. J. Kostelich, C. Grebogi, E. Ott and J. A. Yorke, Phys. Rev. E 51 (1995), 4169. [2] K. T. Alligood, T. D. Sauer, and J. A. Yorke Chaos: An Introduction to Dynamical Systems. New York: Springer-Verlag, 1997.

156

References

[3] R. L. Devaney, An Introduction to Chaotic Dynamical Systems. Redwood City, CA: Addison-Wesley, 1989. [4] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1983. [5] P. Manneville, Dissipative Structures and Weak Turbulence. Boston: Academic Press, Inc., 1990. [6] E. Ott, Chaos in Dynamical Systems. Cambridge: Cambridge University Press, 1993. [7] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, 1990. [8] P. Cvitanovic, ed. Universality in Chaos. Bristol: Adam Hilger, Ltd., 1984. [9] B.-L. Hao, ed. Chaos II. Singapore: World Scientific, 1990. [10] S. Vohra, M. Spano, M. Shlesinger, L. Pecora, and W. Ditto, eds. Proceedings of the First Experimental Chaos Conference. Singapore: World Scientific, 1992. [11] E. Ott, C. Grebogi and J. A. Yorke, Phys. Rev. Lett. 64 (1990), 1196. [12] K. Judd, A. Mees, K. L. Teo, and T. L. Vincent, eds. Control and Chaos. Boston: Birkhauser, 1997. [13] E. J. Kostelich, C. Grebogi, E. Ott and J. A. Yorke, Phys. Rev. E 47 (1993), 305; T. Shinbrot, C. Grebogi, E. Ott and J. A. Yorke, Nature 363 (1993), 411; E. M. Bollt and J. D. Meiss, Physica D 81 (1995), 280. [14] F. J. Romeiras, C. Grebogi, E. Ott and W. P. Dayawansa, Physica D 58 (1992), 165. [15] T. Shinbrot, E. Ott, C. Grebogi and J. A. Yorke, Phys. Rev. Lett. 65 (1990), 3250. [16] T. Shinbrot, W. Ditto, C. Grebogi, E. Ott, M. Spano and J. A. Yorke, Phys. Rev. Lett. 68 (1992), 2863. [17] E. J. Kostelich, C. Grebogi, E. Ott and J. A. Yorke, Physica D 25 (1987), 347; Phys. Lett. A 118 (1986), 448; erratum 120A (1987), 497. [18] B. R. Hunt and E. Ott, Phys. Rev. E 54, 328 (1996). [19] E. J. Kostelich and E. Bollt, in preparation. [20] S. Boccaletti, A. Farini, E. J. Kostelich, and T. Arecchi, Phys. Rev. E 55 (1997), 4845-4848.

7

Using Chaotic Sensitivity

Troy Shinbrot Laboratory for Chaos, Fluid Dynamics & Mixing, Northwestern University, Evanston, IL 60208

7.1 Historical Setting Prior to 1890, it was universally accepted that physical bodies were well characterized by the "clockwork universe." That is, given a set of bodies and a mathematical description of how they interact, it was believed that we could, given sufficient resources, calculate the positions and trajectories of these bodies for all future times. Bodies should interact predictably and their behaviors should be prescribed forever. This view was to be contradicted in 1890 by Henri Poincare. On January 21, 1889, in celebration of the 60th birthday of King Oscar II of Sweden, a prize was awarded to Poincare for his work on the reduced celestial 3 body problem: the problem of an infinitesimal body in the gravitational fields of two larger bodies. In a 200 page manuscript published in the prestigious Ada Mathematical Poincare demonstrated that formal, series, solutions to this problem converge, and consequently there was thought to be hope that celestial problems such as the orbit of the moon around the earth could be stable. Our story would end there, except for one thing: Poincare's proof was wrong. In a later, 270 page, paper, Poincare provided the correct proof (reproduced in Poincare, 1957), which demonstrated that celestial orbits can in fact be unstable. All copies of his earlier work were systematically sought out and destroyed (Peterson, 1993). Poincare's proof in itself was controversial (Poincare, 1896), though the controversy was moderated somewhat by Poincare's demonstration (Sternberg, 1969) that the series would nevertheless asymptotically (Poincare, 1893, Schlissel, 1977) describe the motion of the three bodies. Nevertheless his proof presented a profoundly disconcerting idea, for as Poincare pointed out1, "perhaps one day a mathematician will show by rigorous reasoning that the solar system is unstable." Thus one day, the moon could crash into the earth - or the earth could escape from the solar system altogether. More disconcerting still, given no amount of resources can we compute whether or not these catastrophes will occur. The fateful day presaged by Poincare did not for practical purposes come until quite recently - a century after Poincare's seminal proof - when careful computations showed conclusively that chaos - long term unpredictability - is readily to be found in the solar system (Laskar, 1989, Sussman, 1992). 1 "...un jour peut-etre un mathematicien fera voir, par un raisonnement rigoureux, que la systeme planetaire est instable" (Gamier, 1953)

158

7 Using Chaotic Sensitivity

100

separation between trials 1 and 3 separation between trials 2 and 3 separation between trials 1 and 4 O separation between trials 2 and 4 • separation between trials 3 and 4 — separation between numerical trials

1 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Time (seconds) Figure 7.1 Inset: Sketch of double pendulum. Main plot: Experimental separations between states of double pendulum released from, as nearly as possible, identical initial conditions during multiple trials; bold curve is numerical result. The words of Poincare, recognized at the time as the (Segre, 1980) "greatest living mathematician", a man who became the youngest ever president of the French Academy of Sciences, fell largely on deaf ears. To this day, Physics is often taught as if Poincare never lived - as if we live in a clockwork universe where everything can be predicted, everything understood. Yet we know this to be false. We know that even exceedingly simple systems can defy all attempts at prediction. We know further that the source of this unpredictability is the exponential growth of uncertainties in what we now (Li, 1975) call "chaotic" systems. One need look no further for a practical example of chaos than Goldstein's standard Classical Mechanics text (Goldstein, 1950). On page 12 of Goldstein's classic, one finds the double pendulum (sketched in inset to Fig.7.1) - one of the simplest of physical devices - described as an example of coupled oscillation. And indeed, for small oscillations one can use this device, as remarked by Professor Goldstein, to demonstrate the existence of normal modes. In addition to normal modes, however, if one studies the double pendulum for larger oscillations - where linearization is no longer appropriate - one can see a dramatic demonstration of chaos. Not only is the motion of the double pendulum unexpectedly wild (Richter, 1986), but it is straightforward to quantify the degree of unpredictability expected from this simple device. In Fig.7.1, we plot separations between trajectories for a physical double pendulum released numerous times from, as nearly as possible, identical initial conditions (Shinbrot, 1992d). From the figure we see that trajectories diverge from one another roughly exponentially in time, as is characteristic of a chaotic system. Thus in the very same system which exhibits linear, stable,

7.2 Targeting

159

and completely predictable motion, one can find alongside nonlinear, chaotic, and unpredictable behavior. The co-existence of these two antithetical behaviors is a theme that we will return to.

7.2

Targeting

Apparently very simple systems can be extremely sensitive to tiny perturbations. There are many examples; the double pendulum is just one. The question that several teams of researchers have raised over the past decade is, "can this sensitivity be used to control chaotic systems?" The mechanism which we call targeting (Shinbrot, 1990, 1992a, 1992b, 1992c) is easy to understand. As an example, consider the logistic map: Xn+1=pXn(l-Xn),

(7.1)

where the parameter, p, is chosen to equal 3.9. For any X between 0 and 1, say X = 0.3, it is easy, using only a hand calculator, to adjust p slightly so that any given target, again between 0 and 1, is quickly reached. Thus to reach X = 0.5, we set p = 3.9191..., and find that the target is reached in 5 time steps. In the following sections we examine generalizations and applications of this idea.

7.2.1 Background The earliest reference we find to the control of complex dynamical systems dates to the early 1950's, when Johann von Neumann reportedly gave a talk at Princeton on the subject of weather control. In his talk von Neumann foretold of a day when (Dyson, 1988) "as soon as we have some large computers working, the problems of meteorology will be solved. All processes that are stable we shall predict, and all processes that are unstable we shall control..." Thus Von Neumann imagined that, "a committee of computer experts and meteorologists would tell airplanes where to introduce small disturbances in order to make sure that no rain would fall on the Fourth of July picnic." The first practical contribution to the control of a chaotic system (Wisdom, 1987) came from NASA. A little over a decade ago, NASA had an interesting problem. The comet Giaccobini-Zinner was due to travel through our solar system, and NASA scientists wanted to send a spacecraft through its tail. At the time, this had never been done. But NASA couldn't budget a spacecraft to do this; it would have cost billions of dollars. What NASA did have was a craft that had previously been used to study the solar wind. This craft, the International-Sun-Earth-Explorer-3 (ISEE-3), was parked in a "halo orbit" - a controlled, elliptic trajectory - encircling a so- called Lagrange point, where the gravitational attractions of the earth and sun cancel out. Because of the essential nonlinearity in celestial mechanics remarked upon by Poincare a century earlier, NASA was able to send the craft - renamed the International-Cometary-Explorer-3 (ICE-3) - in an complex, unstable trajectory

160

7 Using Chaotic Sensitivity

halfway across the solar system to meet the comet, using only small amounts of residual hydrazine fuel left over from its solar wind mission. It is important to stress that this was made possible only by the nonlinearity of the 3 body problem; NASA's maneuver would not have been possible in a linear system. In a linear system, a large effect requires a proportionately large control. Only in a nonlinear system can one achieve disproportionately large effects using small controls. NASA's method was customized to their task and is exceedingly interesting (Efron, 1985, Dunham, 1985, Farquhar, 1985, Muhonen, 1985a, 1985b). What they did in the ISEE-3/ICE-3 mission (and what they do today in planning new missions) was to compile a library of maneuvers that would take them from one kind of trajectory to another. For example, they would calculate a single slingshot trajectory around the moon or a multiple slingshot with an orbit around the earth intervening, and so forth. Then they looked for ways to piece maneuvers from this library end-to-end in such a way that they could get from their starting point to their target. Fuel was reserved to redirect the spacecraft from one maneuver to another and for periodic mid-course corrections.

7.2.2 Using Chaotic Sensitivity Clearly the spirit, if not the precise implementation, of the control of chaos has a long history. Let us turn then to more generic techniques which have been developed to use chaotic sensitivity to direct trajectories. As we have seen, chaotic systems are extremely sensitive to tiny perturbations. So if at time t=0 we can adjust some parameter in a chaotic system, we expect generically that the future trajectories of the system will diverge exponentially in time as indicated on the sketch shown in Fig.7.2 Here we see that if a particular chaotic trajectory follows the solid curve at the nominal parameter value, p = po, then by perturbing the system a little i.e. changing the parameter to p = p0 + Sp - we expect the trajectory to veer away exponentially, as shown in the dashed curve. This means that by choosing a suitable perturbation between 0 and 6p, we can reach any point on some grey curve shown in the figure. Thus if we want to reach a particular target, our job will be to sweep the grey curve forward in time until it strikes the target, and then identify the parameter which leads to the target region.

7.2.3 Implementations: Lorenz Attractor Let's see how the idea works. In Fig.7.3 we show everyone's favorite attractor, originally published by Ed Lorenz in 1963 (Lorenz, 1963). We imagine that we want to direct the trajectory from any arbitrary starting point on the attractor to, say, a neighborhood of the unstable stationary point of the flow, the origin (encircled region in figure). This is an interesting region because it contains exceedingly low measure: only 1 in 1010 orbits around either lobe of the attractor falls within the target region shown. To put this in perspective, the probability that a flipped coin will end up resting on its edge is a factor of a million greater than this (Murray,

7.2 Targeting

X(t=to)

161

X(t=to) p=po+8p

target Figure 7.2 Exponential growth of disturbance in generic chaotic system. Solid line indicates trajectory at nominal parameter value, po\ dashed line at perturbed value, po 4- Sp. Thus any target on grey curve - which typically grows exponentially in length with time - can be struck using a perturbation between zero and 5p.

1993). As in any real physical problem, we imagine that we have control over some parameter, p - some knob or flow control, say. The Lorenz problem can be described in terms of a simple heated convective fluid loop (Yorke, 1987), and we imagine that the parameter represents an asymmetric heating rate. To direct a trajectory to this target, in a computer model of the Lorenz system, we perturb the system by adjusting this parameter by some amount p(t) = ±Sp. Analytically, the Lorenz system with this parametric adjustment is written: X = a(Y-X) Y = -XZ + rX-Y" Z = XY-bZ

(7.2)

where we use Lorenz' values for the parameters: a — 10, r = 28, and 6 = 8/3. We trace these two trajectories forward in time until they eventually fall on opposite sides of the target. We can identify these events by placing planes through the origin, labelled F< and Y> in the figure, and noting when a trajectory passes through one or the other plane. Once the trajectory associated with p = po — Sp strikes one plane and the trajectory associated with p = p0 -f Sp strikes the other, if our system is sufficiently low dimensional and well behaved, then we expect that some perturbation in the range between po—Sp and po+Sp will strike the target. All that remains is to refine the estimate of the correct perturbation. This can easily be done, for example by successively halving the parameter range and repeatedly

162

7 Using Chaotic Sensitivity

Figure 7.3 Lorenz attractor including target point, the origin. choosing the halves which contain the target. Readers may recognize this as a simple shooting method. What is new here is that this simple method has been applied to a chaotic system, and far from linearizing the problem - as is frequently done for control calculations - we include the full nonlinearities of the system. In so doing we can turn the exponential sensitivity of the system to our advantage. Through this procedure, we can choose the perturbation to any accuracy we desire which should get us from any point on the attractor to a neighborhood of the origin - or any other target on the attractor - in a short time. How short? We mentioned that only 1 in 1010 orbits around the lobes of the attractor will fall in a small neighborhood of the origin. By contrast, this technique typically brings us to the origin in 5 to 10 orbits, again using only small perturbations. This represents a factor of a billion improvement in the time required to reach the target. This improvement is accomplished in this example using controls which amount to less than a tenth of a percent of the nominal heating rate, p0. In Fig.7.4 we show a typical trajectory brought to the origin in a numerical experiment including additive noise. The same trajectory without control leads nowhere near the origin in any sensible amount of time (cf. Fig.7.3). Again, our recipe is to calculate a perturbation in a 'model' Lorenz system, and then to apply the perturbation in a separate 'real' system including additive noise 2 . To deal with noise, we re-apply the targeting algorithm periodically to obtain mid-course corrections (this is discussed in greater detail shortly). This is the same thing that NASA did in their ISEE3/ICE-3 mission and that any robust controller does, and it can easily be shown 2

For experiments done on this system, see (Singer, 1991).

7.2 Targeting

163

that by doing so, we can reduce the exponential growth of errors to a growth that is no worse than linear. Notice that we are playing both sides of the street 50 Starting point

-20

20

Figure 7.4 Targeting of origin in Lorenz system.

here: because this system is chaotic, we can make use of exponential sensitivity to rapidly steer trajectories to highly improbable states. But because we correct the trajectory regularly, we can prevent the exponential build-up of errors in our trajectory. There is, of course, nothing magic about the origin as a target; any other accessible state would work as well. For example, in Fig.7.5 we show a chaotic trajectory which is brought to a figure-eight unstable orbit (unlike targeting of the origin, in this case we only need one plane, and we identify trajectories which strike one or the other side of the target point on this single plane). Once near this orbit we can apply the same targeting algorithm to bring the state back to the same orbit in the shortest possible time. Thus the targeting technique can be applied to stabilization of unstable periodic orbits (UPOs) as well, with the added advantage that targeting, requiring a global model, includes all nonlinear effects and requires no linearization (cf. (Ott, 1990), now a classic for stabilization in chaotic systems). A further feature of targeting is that it can be used to send signals (Hayes, 1993, 1994a, 1994b). In one implementation which has been developed, one UPO of a nonlinear transmitter is associated with a zero, and a second UPO is assigned to a one. By targeting one orbit and then the other, one can switch the system state in such a way as to send digital information. Indeed, since there are typically an infinite number of periodic orbits embedded in a chaotic attractor (Grebogi, 1988), one is not restricted to base-2; one can efficiently (Barretto, 1995) send information in any base that might be desired.

164

7 Using Chaotic Sensitivity

Figure 7.5 Targeting of figure-eight orbit in Lorenz system.

7.2.4 Implementations: Higher Dimensionality We have so far shown that straightforward techniques can be used to steer trajectories to a desired state in simple, temporally chaotic systems like a heated fluid loop. For problems of this kind, it should not be surprising that control is possible. Indeed recent research has proven that there is an intimate relation between ergodicity - a hallmark of dissipative chaos - and controllability (Colonius, 1993). For the Lorenz case, this relation is facilitated by the low dimensionality of the problem: the correlation dimension (Schuster, 1989) of the Lorenz attractor for the parameters used in Fig.7.3 is 2.06. The technique which we have described works by detecting intersections between the flow and surfaces of section (e.g. the plane shown in Fig.7.5), on which the dimension is 1.06. So we have a nearly one-dimensional problem which we are controlling using one parameter (Shinbrot, 1992b). It comes as no surprise that control is readily achievable. What about higher dimensional problems? Strategies for dealing with high dimensional targeting are discussed in a later section (after Kostelich, 1993), but generically we can play the following game, which we illustrate using the Henon system:

= xn

(7.3)

where we use a nominal value for the parameter p of p0 = 1.4. Equations [3] define the discrete time evolution of points (Xn,Yn) at times n. The chaotic attractor (Ruelle, 1980) resulting from these equations is shown in Fig.7.6(a). We imagine that the current system state is defined by A= (Xo, Yo), and we want to reach a

7.2 Targeting

165

new system state, B = (Xt,Yt). If we continuously vary a control parameter, we can sweep out a short3 line segment, identified by Fl{A) in Fig.7.6(b), By iterating this segment using the map [3], we obtain F 2 , F 3 , etc., and in a chaotic system, we expect the lengths of these curves to typically grow exponentially in time. We repeat these iterations until the length of the curve is on the order of the size of the attractor. Next, we surround the target point with a small4 circle (for higher dimensional problems a hyper-surface), and iterate this circle backward in time, using the inverse map, f"1. For the Hnon map [3], this is unique; for other problems multiple branches may result, but the same idea still holds for any given branch (Kostelich, 1993). We repeat this process until, once again, the size of the circle is on the order of the attractor size. We then search for intersections between the forward iterates emanating from A and the backward iterates from the circle surrounding B. These intersections represent trajectories that start at A and end near B. Intersections typically exist due to the hyperbolic structure of stable (Fig.7.6(c)) and unstable (Fig. 7.6(a)) manifolds found in characteristic chaotic systems. Moreover, because of the exponential growth characteristic of chaotic systems, the time required to reach a target using this technique is exceedingly short, and goes only as the logarithm of the target size (Shinbrot, 1990); we discuss this next.

7.2.5 Time to Reach Target If the process under study is ergodic, then in the absence of controlling perturbations, the time required to travel from an initial point A to a small neighborhood of radius C-Q of a target point B in the ergodic set is typically r 0 « l///(eg), where fi denotes the natural probability measure of the chaotic set. The measure /i(eg) typically scales with the information dimension5 , £>/; so the time required obeys:

for small eg. Thus in the absence of perturbations, the amount of time required to reach a desired target increases according to a power law as the size of the target region decreases. Now suppose that perturbations to a parameter p can be applied. After one iteration of the return map, f(X,p), the change of the state, SX, relative to the point, f(A,p 0 ), due to a small perturbation, Sp, about the nominal parameter value, p 0 , is given by the Taylor expansion: (7.5) 3 For the plots shown in Fig.7.6, we use a total parameter variation p = 0.001po, where again po = 1.4. In this example, the starting point is: (0.94228972,0.8512537). 4 In the example shown, we use the target point, (0.5999968, -1.58540), and surround this point by a circle of radius 0.01 (in units defined by Eq. [3]). 5 See, for example, (Farmer, 1983).

166

7 Using Chaotic Sensitivity F5(A)

4

Yn

o-

Yn

0-

Yn

o-

\*-F
Figure 7.6 Targeting in Henon system.

where: D f _a/(x,p)

(7.6)

9 Letting Jp vary through a small interval, |(5p| < p, Eq. [5] defines a small line segment through the point f(A,p 0 ). We denote this line segment F1(A) and we denote its length CA- Since our system is chaotic, the length of the image of this line segment will grow roughly geometrically with each successive iteration of the map f. Let UA denote the number of iterates required for the small line segment to stretch to a length of order one. This typically happens when CA exp(n^Ai) « 1 if €A is small, where Ai is the largest positive Lyapunov exponent obtained for typical initial conditions on the attractor. Defining T ^ = A^ 1 ln(l/e^), the length of the line segment becomes of order one after about TA iterates (i.e., n^ r ^ ) if CA is small. Without loss of generality, we take the size of the relevant ergodic region to be of the order of one so that TA is approximately the number of iterates required for the line segment to span the ergodic region. Likewise, if we map the region defined by eg backward in time, we find that its pre-image spans the ergodic region after a number of pre- iterates which is

7.2 Targeting

167

typically of the order of rB = |A2| 1 ln(l/eg), where again eB is assumed to be small, and 12 is the largest negative Lyapunov exponent for the map f for typical initial conditions on the attractor. Thus the total amount of time to reach from A to an e# neighborhood of B goes as (Shinbrot, 1990): r = TA+TB =

ln(l/eA)

ln(l/eB) |A2

i

Ai

(7.7)

where we can write that CA — |Df \p. Consequently the amount of time required to reach from A to a neighborhood of B goes as the logarithms of the sizes of the maximum parameter perturbation allowed and the target neighborhood size. This is confirmed in Fig. 7.7, where we plot the time required to reach neighborhoods of points chosen at random on the Hnon attractor described in the previous section as a function of eg. For simplicity in this plot we choose p so as to set ej± = eg.

25

r

j -

20 " •

U

3v

is r

at 10 -• ©

g

i

-

jc^i j.

...O

i~ __.

.**?(*..

K ^

<

•

10

100

i

•

i

1 1 1 1

1000

Figure 7.7 Average time required to reach typical target neighborhood from typical initial state with control versus neighborhood size, eB (eA = eB). Solid line has slope predicted from Eq. [7]; broken line indicates expected behavior without control. Error bars are standard error for 25 point means.

7.2.6 Why Search for Intersections? In practice, we cannot actually iterate either the line denoted FX(A) or the surface surrounding the target, B. Rather, we iterate discrete approximations to them and make successive refinements until a sufficiently accurate intersection is obtained. We do this by putting a fixed number NA > 1 of equally spaced points on F 1 (A),

168

7 Using Chaotic Sensitivity

iterating these points and joining their images with straight line segments. Similarly, we iterate NB > 1 points on the perimeter of B backward in time and join their images with straight line segments. Once an intersection is detected, say at nj± forward and n g backward iterates, we refine its accuracy by repeatedly halving the intersecting forward and backward line segments and determining which of the halves actually contain the intersection. We must achieve accuracy sufficient to strike the n g backward iterate of the target, which is a long, thin region with width of order eg exp(-Aing), where eg is the radius of the surface surrounding the target, B, and Ai is again the positive Lyapunov exponent. If we choose the size of the attractor to be of order one, the curvature of the line F n A (A) and the long sides of the n g backward iterates of the surface surrounding B will also be of order one. Thus, taking account of the curvature, to resolve the intersection within a distance of egexp(-Aing), we require that the distance between points on F n A(A) and on the n g backward iterate of the surface surrounding B be of order ^/egexp(Aing). The square root results because the maximum distance between the curve and its discrete straight line approximation is quadratic in the length of the straight line segment. Each time we halve the line segments, we increase the resolution by a factor of two at the expense of including three additional points (one on FnA (A) and one each on the two segments bounding the backward iterate of the surface surrounding B near the intersection). Thus to resolve the intersection, we require a number of points N' additional to the original N = Nj± -f N g points, where N' obeys: 2N'I\

(7.8)

Using the relation, e^exp(|A2|ng), this can be rewritten as: N' > |jD L ln(l/e B ),

(7.9)

where DL = 1 + Ai/|A2| is the Lyapunov dimension of the attractor, A2 is the maximum Lyapunov exponent of the time-reversed map, and e ^ is the length of F 1 1(A). It should be stressed that N is fixed (typically N « 100) and does not depend on eg or e^. Consequently, as eg is reduced, the required computational effort increases logarithmically in eg as shown in Eq. [9]. In order to show why the method of using forward and backward iterations is useful, let us contrast it with another conceivable procedure. If one iterated the line segment F1(A) forward until it first intersected the region surrounding B, it would do so on iterate nj^ -f n g . One could then choose a point in this intersection, iterate the point backward n ^ + n g steps to find the corresponding point on the original line segment F 1 (A). While this would work in principle, the numerical requirements of this pure forward scheme are needlessly more severe than determining an intersection by iterating F1(A) forward n g steps and the surface surrounding B backward n g steps. In the pure forward method, to detect an intersection between the target and the nj± + n g iterate of the initial state, we require that the approximation of F n A + n B (A) obtained by joining the

7.2 Targeting

169

points with straight line segments intersect the surface surrounding B. Since the curvature of FnA n B (A) is typically of order one, we thus require: l

[

< V^

(7.10)

The curve emanating from A will have length unity after n ^ iterates, and will then expand by roughly exp(n^A) during the next n-Q iterates, where A is the topological entropy6.So: ^ A > ^ 1 / 2 exp(n B A)

(7.11)

or: /

-, \ Ai/|A2|+i/2

NA > ( - J

(7.12)

Thus the number of points required by the pure forward method increases exponentially with l/e-g, but only increases logarithmically with I/eg in the forwardbackward method .

7.2.7 Effects of Noise and Modeling Errors The preceding discussion demonstrates that targeting can be achieved for chaotic systems using only small controls. It remains to be shown, however, that these ideas can remain effective in the presence of noise or modeling errors. Thus we suppose that the real system obeys X n + i = g(X n ,p) -f n . Here we imagine that the model map, f(X n ,p), is slightly in error, and that, unknown to us, the correct form is g(X n ,p). We further allow small amplitude random noise to disturb the system at each iteration as indicated by the term n - To investigate the effect of noise alone, we take f = g. The following test is performed. Initial and target locations are chosen at random on the Henon attractor, and a trajectory between the initial state and the target neighborhood is found for the case without noise as previously described. As an example, if the target neighborhood size, eg, is chosen to be 0.01, the time required to hit the target in the absence of noise and with only Sp 0 is ten iterations. Then if for each of the ten iterations, a random amount of noise is applied with n distributed uniformly in the interval |nI <max, the results are as follows. As shown in Fig.7.8 for the case max = 0.01, the noise displaces the tenth iteration to a point (denoted £10 in the figure) far away from the target point B. Since the noise is applied at every iteration, we next compensate by recomputing the trajectory at every iteration and adjusting the applied perturbation correspondingly. That is, at each iterate we use the map 6

The length of a small line segment typically grows at the exponential rate Ai. After the length becomes of order one, however, it grows at the exponential rate A (Newhouse, 1987). 7 We note that some improvement can be obtained by using higher order fitting (e.g. parabolic rather than linear) of the curves to the iterated points. The exponential and logarithmic dependencies of the pure forward and the forward-backward methods remain, however.

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f to determine dp by calculating the intersection of the forward iteration of the line determined from <SXn+i = 5pn+idf/dp with the backward iteration of the region surrounding B. The result of this procedure is shown in the inset to Fig.7.8. The tenth iteration (denoted XiO in the figure) now lies within the target region. Thus the method can be effective in the presence of small amplitude noise provided that we apply a correction Sp at each iterate. We can similarly show that targeting can

-2

Figure 7.8 Initial state, A, and target region, eg on Hnon attractor. Inset: When the targeting procedure is applied at every iteration, the noise or modeling error can be compensated for, and resulting states both lie within the target region (Xio and X'1O). Without error compensation, trajectories would lie far from the target region (£io and £'1O) also be achieved even when the system is imperfectly modeled, i.e. when f differs slightly from the true map, g. After each iteration, we have to compensate for the difference g - f. For example, let us consider the Henon map for the case without noise, where f is the Henon map with po = 1.4, and g - f = 0.014. eg is still 0.01, and we use the same initial state and target as in the noise example (cf. Fig.7.8). If we apply the targeting procedure only on the first iterate from A (as we would if f = g, then the trajectory again ends at a point (denoted £io in Fig.7.8) far from the target. As before, however, if the targeting algorithm is applied at every iteration, the tenth iterate (denoted £io in Fig.7.8) arrives in the target neighborhood despite the modeling error.

7.2.8 Experimental verification In this section we describe the first confirmation of the targeting method in an experimental chaotic system. For our purposes, we consider the case in which the

7.2 Targeting

171

system at hand can be approximately described by a one- dimensional map, Xn+1 =f(XN,p).

(7.13)

We assume that some parameter p can be varied by some small amount about its nominal value po p = po + Sp, and we seek a value for the small perturbation Sp in some allowed limited range, — p Sp p, which will take us from a current state, X = A, to a desired state, X — B. As before, the variation in the state after one iterate of the map due to the variation in p is given by:

8X1 * (?£) I \dpj

\(A,PO)

Sp

(7.14) '

Since \8p\ is restricted to be less than or equal to p, this defines an interval, X. This interval will typically grow with each successive iteration of the map until it encompasses the desired point, B. Once B is contained within the interval, we know that some parameter value, pt, between pmin = po —p and pmax = Po+ P will lead to B. All that remains is to estimate pt, which can be done by a variety of means. This procedure will give us a value of p which would, in the absence of noise or modeling errors, lead us along some idealized trajectory directly to the target after some small number of iterations. In a real physical system, noise and modeling errors will cause the actual trajectory to wander off the idealized trajectory, however. Therefore we make periodic corrections by reapplying the targeting algorithm after every iteration. Thus we have a different value of the parameter on each iterate, n, and we denote this value pn. If the system truly were described by a one dimensional map, then the replacement of p in Eq. [13] by pn would be exact. For the system we deal with, this turns out to be a useful approximation. In general, however, the validity of such an approximation has to be examined on a case by case basis. We discuss this issue at the end of this section. To experimentally evaluate the effectiveness of this method, we use a vertically oriented, magnetoelastic ribbon, which is known (Ditto, 1989) to vibrate chaotically in response to an external applied magnetic field of the form, H = Hf)c + HAccos(wt). As sketched in Fig.7.9, the ribbon is clamped at its base but is otherwise free to move. The elastic modulus of the ribbon is nonlinearly dependent on the applied field, so that as the field oscillates, the ribbon alternately buckles and stiffens under the influence of gravity. The position, X, of the ribbon is measured at a point near its base with an optical sensor. In order to apply the targeting algorithm, we must construct a map from the experimental system8. The DC field , HDC, is chosen as the control parameter, p, and a map is constructed as follows. First, nominal values of HAC, HAC, and w, are selected, and an experimental delay plot of 500 points is formed by sampling the position of the ribbon once per driving period, 27r/w. This is shown9in Fig.7.10. These points are then 8

The other parameters, HAC and w have been tried as well. The targeting method works about equally well with either AC field, HAC, or DC field, Hoc- Changes to the frequency are to be avoided since the ribbon position is sampled once per frequency cycle. 9 It is worth mentioning that the position of the ribbon is sampled at zero phase of the sinusoidal driving function. Since the experimental system is continuous, one could instead sample the

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Vdc +

sin(cot)

Figure 7.9 Magnetoelastic ribbon

fit with a robust spline curve. This gives us a map, /(X n ,p o ), for a nominal value, po, of the parameter. Next, the map is changed by decreasing the parameter to po - p, obtaining 500 new data points which again are fit with a second spline. Finally, the parameter is increased to po + p, again obtaining 500 points, which once more are fit with a third spline. The spline fits are used to estimate the value of the map function f for the three parameter values at any given value of X. By interpolation, f(Xn,p) can then be estimated for any value of the parameter, p, in the range, \po — p,po + p]- Given this model for the map, the targeting method can be applied in a straightforward way. To illustrate the procedure, the point, X = 2.5, is targeted in the experimental system described. Other accessible points can also be targeted; it is interesting to discuss targeting of X — 2.5 because it is in a region of relatively low measure and therefore in the absence of targeting its vicinity typically takes a long time to reach. In Fig.7.11, we show a time series of the ribbon position with fixed, nominal, applied field, showing that the vicinity of X = 2.5 is seldom visited10 . Indeed, for each of the three fixed parameter values, Po —P,Po and Po+P, the ribbon position reaches the neighborhood, [2.49, 2.51], only about 1 time in 500 iterates. By contrast, Fig.7.12 shows the results of targeting for several representative trajectories. For each trajectory shown, the ribbon is allowed to vibrate chaotically for 100 iterates, and then on the 101st iterate, denoted n — 1, the targeting algorithm is initiated. The position of the ribbon at n = 1 is position at any other phase. The delay plot would then have looked different, but the technique would work equally well. 10 We remark that successive points, Xn, are indicated in the figure by open circles, and that the lines connecting the points are provided solely to aid the eye. Thus in the figure, the point, Xn = 2.5 is never visited.

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173

Figure 7.10 Experimental delay plot of magnetoelastic ribbon at nominal parameter value, p = po.

shown as an open circle, and the position when the ribbon reaches the neighborhood, [2.49, 2.51], is shown as a solid circle in the figure. Every time that targeting is turned on, the ribbon is brought rapidly to the desired target neighborhood, [2.49, 2.51]. Due to noise and modeling errors, it is necessary to repeat the targeting algorithm at every iteration as described previously. This procedure is carried out for 100 randomly chosen initial conditions on the attractor, and each time the trajectory is rapidly brought to the targeted point. On average, the neighborhood can be reached in 20 iterates11 , as compared again with the 500 iterates typically required without targeting. Despite the apparent effectiveness of the targeting algorithm, two complications have been observed which may be of importance in future applications. First, as we mentioned earlier, the technique used relies on the map [13] being approximately one dimensional. Yet higher dimensional behavior is manifestly present in the experimental system described, as can be seen in Fig.7.13 (in particular, note the structure on the right half of the map, which is suggestive of fractal behavior). The targeting algorithm is nevertheless successful. It is worthwhile, however, to consider the limits of the targeting algorithm due to the use of a one dimensional approximation. To evaluate the effect of approximating behavior whose dimension is slightly above one with a one dimensional map, we can implement the targeting algorithm numerically on a projection of the Henon system defined by the map: 11

In 100 realizations, a few outlying cases are observed which require abnormally long targeting times ( 50 iterates). Deletion of these few cases from the record results in a significantly shorter average targeting time ( 10 iterates).

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o

50

100 150 Iteration, n

200

250

Figure 7.11 Ribbon displacement versus iteration for fixed, nominal parameter value.

Xn-i - Xl

(7.15)

for various values of b. The parameter, p, is used as a control, and p is varied by up to p — 0.05 about the nominal value, po — 1.5. The system is modeled as a one-dimensional map in precisely the same way as in the experimental system. That is, three data sets are generated by iterating Eq. [15] using p = 1.45, p = 1.5, and p = 1.55, and, using only this data, f(Xn,p) is estimated by interpolating between quadratic fits to these data sets. Using this model, initial points on the attractor are chosen at random12 , and a particular point, X — 1.1, is targeted in 25 realizations at each of several values of the parameter b. The results of this process are shown in Fig.7.13. The RMS error of the quadratic fit is shown in the abscissa of the figure13 . We see from the figure that the targeting procedure can tolerate higher dimensionality only until the RMS error due to higher dimensionality reaches within an order of magnitude of the change (±0.05) which we can produce by parametrically varying the position on the attractor. In the physical experiment, the maximum change14 in the position on the attractor produced by varying the parameter amounts on average to 5% of the position itself. By contrast, the RMS error in the spline fits used is about 1% of the position. According to the Henon simulation of the effect of higher dimensionality, the experiment is apparently operating very near the margin of targeting effectiveness. Indeed, the 12 To choose random initial points, Eq. [15] is iterated 25 times using p — 1.5, starting from Xn — 1 = 0 and using Xn randomly distributed on [-0.5, 0.5]. 13 To aid the eye, a best fit hyperbolic tangent obtained by nonlinear regression is also shown in the figure. 14 This is the average of the difference between robust spline positions from the three curves divided by the nominal position. We normalize the position of the ribbon to a scale from 0 to 1.

7.2 Targeting

10

15

20

175

25

Iteration, n Figure 7.12 Typical ribbon displacements versus iteration with targeting on. The point, X = 2.5 is targeted starting at iteration, n = 1.

targeting algorithm is observed in the experiment to be substantially less effective for smaller parameter variations. A second complication which may occur has to do with the fact that the analysis presented assumes that the system dependence on the parameter is given by xn -f 1 = f(xn,pn) with / a one dimensional map. We deal in the experiment, however, with a surface of section for a continuous time, infinite dimensional system: a spatially extended vibrating beam. Under these circumstances, even if a one dimensional map applies at each fixed parameter value, and even if contraction to the attractor is very rapid compared to the surface of section map iteration time, t, then the switching of pn-\ to pn in principle leads to a dependence of x n +i on both pn-1 and pn. To see this, note that the attractor position in the full infinite dimensional phase space depends on p. A point with scalar coordinate xn at time n is on the p = pn - 1 attractor at time t = tn-. At a time shortly after the switch from pn-\ to pn the state is essentially on the p = pn attractor (assuming rapid contraction), and in general has now moved to a new location that has an x-coordinate, x'n = g(xn,pn,pn-i), which depends on the location of the attractors for p = pn-\ and for p — pn. The location of a?n+i is determined by iterating the point x'n via the map that applies for p fixed for all time at pn. Thus xn+i depends on both pn and pn-i> An added dependence of £n_|_i on pn-i is also implied by the use of delay coordinates (Dressier, 1992, Nitsche, 1992). To examine the dependence of rrn+i on pm for m n — 1, the following experiment has been performed. The parameter is fixed at the minimum value, pmin — po — p, for 10 iterates. Then, the parameter is switched to the maximum value, Pmax — Po+P, for 10 more iterates. At this time, the parameter is switched

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CO

o

3

0.001 RMS Error in 1 -D Fit Figure 7.13 Effect on targeting of RMS error due to approximating higher dimensional attractor by a one dimensional map

back to Pmin. This process is repeated for 50 cycles, producing a delay plot of the position of the ribbon, shown in Fig.7.14. For comparison, Fig.7.15 shows plots of xn+i versus xn both for the case in which p is kept fixed at p = pmax and for the case where p is kept fixed at p = pminFor the data in Fig.7.14, following most of the switches, the subsequent x-point does indeed land on the appropriate approximate curve in Fig.7.15. However, occasionally, when p is switched from p = pmin to p — pmax, and the p = pmin x-coordinate is in the region labelled 0 in Fig. 7.14 , then the subsequent x-value occurs in the region labelled 1. Region 1 is not on the p = pmax curve (the lower curve of Fig. 7.15). Region 1 in turn iterates to region 2 (which also is appreciably far from the appropriate curve) and thereafter rapidly approaches the p = pmax curve. While this effect degrades the targeting performance, it occurs relatively rarely. Presumably, a more elaborate procedure could correct for this kind of effect.

7.3

Outlook

For many years it has been accepted as axiomatic that the exponential sensitivity which makes chaotic systems unpredictable also makes them uncontrollable. It is remarkable that in less than a decade the paradigm has shifted to the point that it has been proven numerically (Shinbrot 1990, Kostelich, 1993), experimentally (Shinbrot, 1992c, Hayes, 1994b), and now mathematically (Colonius, 1993), that in fact the very properties that make chaos difficult to predict make chaotic systems ideally suited for control.

7.3 Outlook

177

1.4 1.4

1.6

Figure 7.14 Delay plot for parameter alternately switched between pmin and pmax. The nominal parameters, HDc, HAC, and w used in this experiment are slightly different from those in Fig. 7.10 Thus the exponential sensitivity of the three body problem permitted NASA to achieve the first ever scientific cometary encounter using only the residual fuel left in a nearly spent spacecraft. This sensitivity has enabled the use of chaotic circuits to efficiently send signals by rapidly switching between unstable periodic states. Likewise the presence of chaos permits the time required to reach a desired target in a chaotic system to be reduced by as much as a factor of a billion, and allows the dependence of targeting time to be transformed from exponential in the target size to only logarithmic. Experiments and simulations have confirmed that targeting algorithms can be made robust against small amounts of noise, modeling errors, and contamination due to higher dimensional influences. The challenge for the future will be to extend the frontiers of this research to include more real, practical applications. Indeed, new applications using a variety of control techniques have already been proposed in diverse applications ranging from communications (e.g. Hayes, 1994a, Barretto, 1995) and electronics (e.g Roy, 1992, Hunt, 1991, Carroll, 1992) to physiology (e.g. Garfinkel, 1992, Ding, 1991), fluid mechanics (e.g. Singer, 1991, Gad-el-Hak, 1994) and chemistry (Peng, 1991, Qammar, 1991). The growth of applications such as these lead us to look forward to an exciting and fruitful future for the control of chaotic systems.

Acknowledgments I wish to thank Professors C. Grebogi, E. Ott, and J.A. Yorke for their support and direction. I also wish to acknowledge the support of the U.S. National Science Foundation and the Department of Energy.

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1.4 1.4

1.6

1.8

2.D

2.2

2.4

2.6

IS

3.0

3.2

Figure 7.15 Delay plots for parameter fixed at pmin, and for parameter fixed at pmax.

References Barreto, E., Kostelich, E.J. Grebogi, C. Ott, E. Yorke, J.A. (1995): Efficient Switching Between Controlled Unstable Periodic Orbits in Higher Dimensional Chaotic Systems. Phys. Rev. E 51 4169-72. Carroll, T., Triandaf, I., Schwartz, I.B., Pecora, L. (1992): Tracking Unstable Orbits in an Experiment. Phys. Rev. A 46 6189-92. Colonius, F., Kliemann, W. (1993): Some aspects of control systems as dynamical systems. J. Dyn. & Diff. Eq. 5 469-494. Ding, M., Kelso, J.A.S. (1991): Controlling Chaos: A Selection Mechanism for Neural Information Processing?, in Measuring Chaos in the Human Brain, ed. by D. Duke, W. Pritchard (World Scientific, Singapore) 17-31. Ditto, W. L.,Rauseo, S., Cawley, R., Grebogi, C, Hsu, G.-H.,Kostelich, E., Ott, E., Savage, H. T.,Segnan, R.,Spano, M. L., Yorke, J. A. (1989): Experimental Observation of Crisis-induced Intermittency and its Critical Exponent. Phys. Rev. Lett. 63 923-6. Dressier, U., Nitsche, G. (1992): Controlling Chaos Using Time Delay Coordinates. Phys. Rev. Lett. 68 1-4 Dunham, D.W., Davis, S.A.(1985): Optimization of a Multiple Lunar-Swingby Trajectory Sequence. J. Astronautical Sci. 33 275-88. Dyson, F.J. (1988)infinite in all dimensions (Harper & Row, NY), 182-3. Efron, L., Yeomans, D.K., Schanzle, A.F. (1985): ISEE-3/ICE Navigation Analysis. J. Astronautical Sci. 33 301-23. Farmer, J.D., Ott, E., Yorke, J.A. (1983): The Dimension of Chaotic Attractors. Physica 7D 153- 80. Farquhar,R., Muhonen, D., Church, L.C. (1985): Trajectories and Orbital Maneuvers for the ISEE-3/ICE Comet Mission. J. Astronautical Sci. 33 235-54. Gad-el-Hak, M. (1994): Interactive Control of Turbulent Boundary Layers: A Fu-

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turistic Overview. AIAA J. 32 1753-65. Garfinkel, A., Spano, M.L., Ditto, W.L., Weiss, J.N. (1992): Controlling Cardiac Chaos. Science 257 1230-1235. Gamier, R., Leray J. ed's, (1953): Oeuvres de Henri Poincare, 8, (Gauthier-Villars, Paris) 538. Goldstein, H. (1950): Classical Mechanics (Addison Wesley, Cambridge, MA), 12. Grebogi, C , Ott, E. Yorke, J.A. (1988): Unstable periodic orbits and the dimensions of multifractal chaotic attractors. Phys. Rev. A 37 1711-24. Hayes, S., Grebogi, C , Ott, E. (1994a): Communicating with Chaos. Phys. Rev. Lett. 70 3031-4. Hayes, S., Grebogi, C , Ott, E., Mark, A. (1994b): Experimental Control of Chaos for Communication. Phys. Rev. Lett. 73 1781-4. Hunt, E.R. (1991): Stabilizing High-period Orbits in a Chaotic System: the Diode Resonator. Phys. Rev. Lett. 67 (1991) 1953-5. Kostelich, E., Grebogi, C, Ott, E., Yorke, J.A. (1993): Higher-dimensional Targeting. Phys. Rev. E 47 305-10. Laskar, J.A. (1989): A numerical experiment on the chaotic behavior of the solar system. Nature 338 237-8. Li, T-Y, Yorke, J.A. (1975): Period three implies chaos. Amer. Math. Monthly 82 985-92. Lorenz, E.N. (1963): Deterministic Nonperiodic Flow. J. Atmos. Sci. 20 130-41. Muhonen, D., Davis, S., Dunham, D. (1985):a Alternative Gravity-Assist Sequences for the ISEE- 3 Escape Trajectory. J. Astronautical Sci. 33 255-73. Muhonen, D., Folta, D. (1985b): Accelerometer-Enhanced Trajectory Control for the ISEE-3 Halo Orbit. J. Astronautical Sci. 33 289-300. Murray, D.B., Teare. S.W. (1993): Probability of a tossed coin landing on edge. Phys. Rev. E 48 2547-51. Newhouse, S. (1987) in Physics of Phase Space, YS Kim & WW Zachary, ed's (Springer, Berlin), 2. Nitsche, G., Dressier, U. (1992): Controlling Chaotic Dynamical Systems Using Time Delay Coordinates. Physica D 58 153-64. Ott, E., Grebogi, C , Yorke, J.A. (1990): Controlling Chaos. Phys. Rev. Lett. 64 1196-9. Peng, B., Petrov, V., Showalter, K. (1991): Controlling Chemical Chaos. J. Phys. Chem. 4957-9. Peterson, I. (1993): Newton's Clock: Chaos in the Solar System (Freeman, NY) 157-9. Poincare, H. (1896): Comptes Rendus des Seances de L'Academie des Sciences, 179 497ff. & 557ff. Poincare, H. (1957): La Mcanique Cleste, Tome 1 (Dover Publications, NY) 350. Poincare, H.(1893): Solutions Asymptotiques, Mthodes Nouvelles de la Mcanique Cleste, II, (Gauthier-Villars et Fils, Paris), 335ff. Qammar, H.K., Mossayebi, F. Hartley, T.T. (1991): Indirect adaptive control of a Chaotic System. Chem. Eng. Comm. 110 99-110.

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Richter, P.H., Scholz, H.J. (1986): The Planar Double Pendulum. Film C1574 (Naturwiss. Series 9, Number 7/C1574) Publ. Wiss. Film, Sekt. Techn. Wiss., Gttingen. Roy, R., Murphy, T.W., Maier, T.D., Gills, A., Hunt, E.R. (1992): Dynamical Control of a Chaotic Laser: Experimental Stabilization of a Globally Coupled System. Phys. Rev. Lett. 68 1259-62. Ruelle, D. (1980): Strange Attractors. Mathematical Intelligencer 2 126-37. Schlissel, A. (1977): Poincare's Contribution to the Theory of Asymptotic Solutions. Archive for History of the Exact Sciences, 16 325ff. Schuster, H.G. (1989): Deterministic Chaos (VCH, Weinheim) 127-9. Segr, E. (1980): From X-Rays to Quarks. (WH Freeman k Co, NY ) 9. Shinbrot, T., Ditto, W., Grebogi, C, Ott, E., Spano M., Yorke, J.A. (1992c): Using the Sensitive Dependence of Chaos (the 'Butterfly Effect') to Direct Orbits to Targets in an Experimental Chaotic System. Phys. Rev. Lett. 68 2863-6 Shinbrot, T., Grebogi, C, Ott, E., Yorke, J.A. (1992a): Using Chaos to Target Stationary States of Flows. Phys. Lett. A, 169 349-54. Shinbrot, T., Grebogi, C, Wisdom J., Yorke, J.A. (1992d):a Chaos in a Double Pendulum. Am. J. Phys. 60 491-9. Shinbrot, T., Ott, E., Grebogi, C , Yorke, J.A. (1990): Using Chaos to Direct Trajectories to Targets. Phys. Rev. Lett. 65 3215-8 Shinbrot, T., Ott, E., Grebogi, C, Yorke, JA (1992b): Using Chaos to Direct Orbits to Targets in Systems Describable by a One-dimensional Map. Phys. Rev. A 45 4165-8 Singer, J., Wang, Y-Z, Bau, H.H. (1991): Controlling a Chaotic System. Phys. Rev. Lett 66 1123- 5. Singer, J., Wang, Y.Z., Bau, H.H., (1991): Controlling a Chaotic System. Phys. Rev. Lett. 66 1123-5. Sternberg, S. (1969): Celestial Mechanics, Part II, (WA Benjamin, NY). Sussman, G.J., Wisdom, J. (1992): Chaotic evolution of the solar system. Science 257 56-62. Wisdom, J. (1987): Urey Prize lecture: chaotic dynamics in the solar system. Icarus 72 241-75. Yorke, J.A., Yorke, E.D., Mallet-Paret, J. (1987): Lorenz-like Chaos in a Partial Differential Equation for a Heated Fluid Loop. Physica D 24 279-91.

8

Controlling Transient Chaos on Chaotic Saddles

T. Tel 1 , Y.-C- Lai2, and C. Grebogi3 1. Institute for Theoretical Physics, Eotvos University, H-1088 Budapest, Puskin u. 5-7, Hungary 2. Departments of Physics and Astronomy and of Mathematics, University of Kansas, Lawrence, Kansas 66045, USA. 3. Institute for Plasma Research, Department of Mathematics, Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA

8.1

Introduction

Chaotic saddles are invariant nonattracting chaotic sets in the state space of dynamical systems [l]-[6]. They are the fractal analog of saddle points characterizing isolated unstable behaviors. In contrast to saddle points, chaotic saddles are responsible for a globally unstable behavior known as transient chaos (for reviews see [4, 5]) which provides an example of a kind of "dynamical nonequilibrium state" that cannot be understood as an asymptotic state. In such cases one observes chaotic-like behavior and then, rather suddenly, a settling down to another state which is either periodic or chaotic, but different from the transient. In other words, chaotic saddles are state space objects responsible for temporally chaotic behavior with finite life time. Sole investigation of the asymptotic behavior of such systems would miss the interesting chaotic part contained in the transients. Chaotic transients and chaotic saddles are common in dynamical systems. They are ubiquitous in periodic windows of bifurcation diagrams [7] in which chaos may be present in the sense that there exists an infinity of unstable periodic orbits but their union is not attractive. Transient chaos can also be a sign for the birth of permanent chaos [8]. More generally, all types of crisis configurations [1] are accompanied by long chaotic transients. Large attractors born at crises incorporate into themselves chaotic saddles existing before [9, 10, 11]. Chaotic saddles can also give rise to a plethora of dynamical phenomena. For example, if two or more periodic or chaotic attractors coexist, trajectories may hesitate, possibly for a long time, to which of the attractors they asymptote. The attractors are typically separated by fractal basin boundaries [12] which are stable manifolds of chaotic saddles [13]. For scattering processes in open Hamiltonian systems the only way that chaos can appear is in the form of transients. Chaotic scattering is then also governed by underlying chaotic saddles that now have a symplectic character [14]. The passive advection of tracer particles (e.g., small

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8 Controlling Transient Chaos on Chaotic Saddles

dye droplets) in open hydrodynamical flows with uniform inflow and outflow velocities is one of the most appealing applications of chaotic scattering [15] - [17]. In systems subjected to external random forces, the form of attractor observed might depend on the noise intensity. The phenomenon, when a system with simple periodic attractors turns to be chaotic at sufficiently strong noise, is called noise induced chaos [18]. In such systems there is always a chaotic saddle coexisting with periodic attractors so that at increasing noise intensity all becomes embedded into a noisy chaotic attractor. There is also a growing experimental evidence of motion on chaotic saddles [19] -[29]. The most detailed ones are convection loop experiments [20, 25], the investigation of chemical reactions preceeding thermal equilibrium [26], NMR lasers [27],

and a pendulum experiment [28]. Recently, the first experimental investigation of chaotic scattering has been carried out in the form of chaotic tracer dynamics in the wake of a cylinder [29]. There are also every-day observations of transient chaos on finite time scale. The majority of the battery-driven chaos machines settles down to periodic motion after a typical time of no longer than one hour. The wheels of the supermarket trolleys often exhibit, even at a constant speed of towing, a shimmying motion which goes over into a smooth rolling after a while [30]. This list clearly shows the pervasiveness of transient chaos which is as a general phenomenon as stationary or permanent chaos. In fact, chaotic saddles are as robust invariant state space objects as chaotic attractors. The rest of the Chapter is organized as follows. In Sec. 8.2, we review the basic dynamical properties of chaotic saddles. In Sec. 8.3, we present the basic idea and methodology for controlling transient chaos. In Sec. 8.4, we compare the control of transient chaos and that of sustained chaos (on attractors). In Sec. 8.5, we discuss characteristic changes in the control when a chaotic saddle is converted into a chaotic attractor, or vice versa, at a crisis. In Sec. 8.6, we present an example of controlling fractal basin boundaries. In Sec. 8.7, we describe applications to chaotic scattering. In Sec. 8.8, we discuss the idea of maintaining sustained chaotic motion in a transient chaotic regime and outline a procedure to achieve this, which can yield a high percentage of initial conditions to be controlled. The details of the algorithm are relegated to the Appendix. In Sec. 8.9, we discuss recent developments of controlling transient chaos and applications.

8.2

Properties of chaotic saddles

Chaotic saddles, similarly to chaotic attractors, contain an infinite number of unstable periodic orbits. They are also fractal objects. These saddles are globally not attracting but have a basin of attraction of zero volume in the full phase space, which is itself a fractal foliation. As an example, Fig. 8.1 shows the chaotic saddle of the Henon map (at parameter values different from the usual ones a = 1.4, b = 0.3 that lead to the standard Henon chaotic attractor.) A chaotic saddle is the chaotic analog of a saddle point which repels trajectories from its neighborhood, but has nevertheless an invariant subspace of volume zero along which it is attractive. This

8.2 Properties of chaotic saddles

1 -

183

r *

*

0-

1

-

t 1

1 t

-1-

/

W x%

u

Figure 8.1 The chaotic saddle of the Henon map xn+i = 1 — ax% + byn,yn+i = xn at parameters a = 2,6 = 0.3 where no other finite invariant sets of the map exist. Trajectories escaping from this set go to minus infinity. subspace is called the stable manifold of the chaotic saddle. The stable manifold provides a fractal foliation of the phase space (see Fig. 8.2). Similarly, there is an invariant subspace along which the chaotic set repels out those exceptional trajectories that have hit it infinitely long time ago, the set's unstable manifold. More precisely, this can also be defined as the stable manifold of the time reversed dynamics. It is worth sometimes considering the chaotic saddle as the common part of its invariant manifolds. A chaotic saddle has more pronounced fractal properties than a chaotic attractor. Due to the escape, the chaotic saddle has holes on all scales along its unstable manifold^ too. The saddle of the Henon map shown in Fig. 1 appears to be the direct product of two Cantor sets. Chaotic saddles typically coexist in dissipative cases with an attractor, and in Hamiltonian systems with an exit channel leading to an asymptotic motion. Trajectories starting from randomly chosen initial points then approach the attractor or the asymptotic state with probability one. Because of the saddle's stable foliation, however, they might come close to the saddle and stay in its vicinity for a long time. What is observable numerically or experimentally is not the chaotic saddle but rather a small neighborhood of it. This results in the appearance of a chaotic motion on finite time scales. The actual time spent around the saddle depends sensitively on initial conditions but the average transient lifetime is typically well defined. When distributing a large number of initial points in a region containing the chaotic saddle (but not the attractor(s)), trajectories emanating from these points will leave the region with certainty. Those with initial points close enough to a branch of the stable manifold have a long time before escaping. Thus the number of points N(n) staying still in the preselected neighborhood after a discrete time

184

8 Controlling Transient Chaos on Chaotic Saddles

1 -

-1-

-1

Figure 8.2 The stable (dotted lines) and unstable (full lines) manifolds of the Henon chaotic saddle at a = 2, b = 0.3. if+ represents an unstable fixed point on the saddle. n is a function that monotonously decreases to zero. How rapidly it decreases is an important characteristics of the saddle. Often one finds that the decay is exponential for long times [2], i.e., N(n)

(8.1) for n —> oo. The positive number K is called the escape rate of the saddle and turns out to be independent of the shape of the region containing the intial points. The escape rate measures the saddle's strength of repulsion: a large escape rate marks a rather unstable saddle. Conversely, 1/K is considered to be the average lifetime around the chaotic saddle. The case of permanent chaos is formally recovered in the limit K -» 0. Just as for chaotic attractors, there exist invariant distributions on chaotic saddles, too. The so-called natural distribution [2, 3, 4] on a saddle is obtained by distributing an ensemble of points around the set and following those with long lifetimes. The frequency of visiting different regions of the saddle by these trajectories defines the natural probability measure. One can then speak about characteristic numbers taken with respect to this measure. The average Lyapunov exponent A of the saddle is of special importance since it characterizes the typical divergence of nearby trajectories staying for long times around this set. The Lyapunov exponent describes the local instability of the saddle, while the escape rate is a global measure of instability. Numerically, chaotic saddles can be constructed with arbitrary accuracy by a technique known as the Proper-Interior-Maximum triple (PIM-triple) method [31], which is specially designed for selecting a long trajectory which is never further away from the saddle to a predefined small threshold distance value. A recent generalization of the method is able to create a trajectory which visits different parts

8.3 The basic idea for controlling chaotic saddles

185

of the saddle according to the natural distribution [32], and thus to reconstruct the natural measure with arbitrary accuracy, too.

8.3

The basic idea for controlling chaotic saddles

The preceeding discussion implies that controlling the motion on chaotic saddles is a generic possibility to convert even transiently chaotic dynamics into periodic one, i.e. stabilizing one of the saddle's periodic orbits. The novel feature of this type of control is that it stabilizes an orbit which is not on the actual attractor. One selects thus an atypical behavior which cannot be seen by a long time observation of the unperturbed motion. One can say that this control means stabilizing a metastable state. These general features of controlling chaotic saddles hold for any particular control method applied to transient chaos. To be specific, we shall treat in the following the Ott-Grebogi-Yorke (OGY) method [33] because this is an algorithm that is capable of carrying out the finest possible selection of the target orbit to be stabilized and of applying the weakest possible perturbation. Other methods, e.g. the delayed feedback control of Pyragas [34] or the geometric control method of Toroczkai [35], are also applicable. To achieve control by means of the OGY method, one has to use an ensemble of TVo >> 1 trajectories [36] because any randomly chosen single initial point belongs with high probability to a trajectory which escapes any neighborhood of the saddle rather quickly. This ensemble is typically chosen to start in a compact region having an intersection with the saddle's stable manifold. One also selects a target region around a predetermined hyperbolic periodic orbit on the chaotic saddle. Then the ensemble of trajectories is started and one waits until any trajectory enters the target region where and when the control algorithm is applied. The controlling perturbation is adjusted so that the predetermined periodic orbit is stabilized. Only small local perturbations are allowed, smaller in size than some value S that we call the maximum allowed perturbation. It is supposed that S is proportional to the linear extension of the target region [33]. The actual algorithm selecting the proper value of the perturbation parameter pn (| pn |< 6) is exactly the same as in the OGY control of the motion on chaotic attractors, and has been given in other chapters of this book. Fig. 8.3 shows a succesful control of the fixed point on the Henon saddle in comparison with the uncontrolled trajectory [36]. The striking feature is that the controlled motion is not a part of the asymptotic dynamics which is a period-5 attractor in this case. Next we summarize the scaling behavior characterizing the ensemble in the limit of small allowed perturbations 6 « 1. Many of the trajectories escape this region before they can enter the target region selected around a periodic orbit on the chaotic saddle. Short transients are therefore irrelevant for the controlling process, while very long ones are unlikely to find. It is thus qualitatively clear that the average time r needed to achieve control is independent of S and is limited from above by the chaotic lifetime \/K when the

186

8 Controlling Transient Chaos on Chaotic Saddles

Figure 8.3 (a) Transient chaotic signal xn versus n starting from the point XQ = yo = 1.014782 in the Henon map at parameters a = 1.45, b = 0.3 where the attractor is a period-5 cycle. The trajectory ceases to be chaotic at about the 38th time step where it comes to the neighbourhood of the periodic attractor. (The average lifetime of chaotic transients is 1/K = 22 at these parameters.) (b) Controlled signal started from the same initial point. The Henon map was taken in the form given in the caption to Fig. 0.1 with a = 1.45 -f pn where the maximum allowed perturbation in | pn | is 6 = 0.1. The fixed point is at XF = 2/F = 0.868858. Control sets in at the 26th step and the fixed point on the saddle becomes stabilized. maximum allowed control 6 goes to zero: T(S) = independent of S < 1/K.

(8.2)

Because of escape, only a small portion of all trajectories can be controlled. Moreover, the number of controlled trajectories N(S) has been found [36] to decrease with decreasing S according to a power law: N{8)

(8.3)

where the exponent J(K) depends on the escape rate, too. In the particular case when the target region is a ball around a fixed point of a two-dimensional map, the exponent 7(/c) characterizing the decay of the number

8.4 Comparison with controlling permanent

chaos

187

of controlled trajectories is [36] ~T»

(8-4)

with A^ and A ^ as the eigenvalues of the fixed point to be stabilized.

8.4

Comparison chaos

with

controlling

permanent

When applying the OGY method to controlling permanent chaos, the scaling properties of an ensemble of No trajectories are different. The average time r needed to achieve control is a function of the linear size of the target region which is supposed to be proportional to the maximum allowed perturbation S during control. It has been pointed out [33] that r(6) increases as a power of S when 6 tends toward zero: T(S)

~
(8.5)

where 7 > 0 is a characteristic exponent. This scaling law, which shows that the dynamics of reaching the target region is itself a kind of transient chaos, has recently been verified experimentally [37]. On the other hand, the number N(S) of controlled trajectories does not depend on 6: N(S) = No

(8.6)

since all of the iV0 trajectories of the ensemble are controlled sooner or later. Thus, the shown scaling behavior in the control of permanent and transient chaos appear to be the two extremes of a general process (in the first case N(S) is constant, while in the second r(S)). Interestingly, there exists a unifying relation between N(S) and r(S) which holds in both cases [38]. The key observation is that the number of controlled trajectories in the entire process is proportional to the average number of trajectories controlled per unit time multiplied by the average time needed to achieve control. The average number of trajectories controlled per time step is proportional to the probability to fall in the target region, i.e., to some measure fi(S) of the target region. The measure /i to be taken for permanent and transient chaos is the natural measure of the attractor and the conditionally invariant measure of the nonattracting chaotic set, respectively. The latter is the probability that trajectories which have not yet escaped a given neighbourhood of the chaotic set fall into the target region [39, 4]. Thus we can write N(6)

~ »(6)T(6).

(8.7)

This relation contains as special cases Eqs. (8.3,8.6) (for r =const and for iV =const, respectively), and shows that the exponents 7 of (8.5) and J(K) of (8.3) also govern the scaling law of /J>(8):


(8.8)

188

8 Controlling Transient Chaos on Chaotic Saddles

Since for permanent chaos K — 0, 7 = 7(/c = 0)

(8.9)

must also hold. As an important consequence of Eq. (8.8), the number g(S) of trajectories controlled per time steps follows a simple scaling law: F<*\

(8.10)

and this is valid in both permanent (K, = 0) and transient (/c > 0) chaotic cases.

8.5

Crossover around crises

The essential difference between the scaling of N{5) and r(S) of the controlling process for permanent and transient chaos can only be explained by the fact that the limits S -» 0 and K -* 0 are not interchangable. There must, therefore, exist a crossover from Eq. (8.2) to Eq. (8.5) (and from Eq. (8.3) to Eq. (8.6)) as K goes to zero. The phenomenon of diminishing escape rate, i.e., diverging chaotic life-time, can always be observed when a nonlinearity parameter a decreases in the transient chaotic region and tends to a critical value ac marking the crisis value where permanent chaos first sets in. The dependence of the escape rate on a has been found [9] to follow a power-law in (a — a c ): K(a)=C(a-acy

(8.11)

with a positive 7 as a -> ac. In the range of small values of 8 and a — ac > 0, there are two time-scales which at suitable choices of a and 6 might be of the same order of magnitude: the average lifetime of chaos at parameter a, 1/K(CL), and the average time needed to achieve control T((5, ac) ~ 6~y^ at the crisis point. Thus one expects that the average time r(61a) needed to achieve control for (a — ac)/ac « 1 follows a scaling law: r((J, a) = T(S, ac)/[«(o)r(<S, ac)]

(8.12)

where the scaling function f(x) depends on the dimensionless variable product K{a)T(8,ac) only. In fact, f(x) must tend to one as x -» 0 and must be inversely proportional to x as x -» 00. It is easy to derive [38] that the scaling function has the explicit form f{x) = (l + x)-1

(8.13)

which is a mean-field-type result. ^From this the rules Eqs. (8.2,8.3) and (8.5,8.6) are recovered for 6 -> 0 at a fixed (when r(S,ac) -> 00, and f(x) -» 1/x) and for a -> ac at S fixed (when

8.6 Controlling motion on fractal basin boundaries

189

K,(a) -> 0 and f(x) -> 1), respectively. A crossover between these behaviors takes place when the two characteristic times are of the same order of magnitude, i.e., if r(<J,a c )«(a)«l.

(8.14)

The crossover region is typically rather narrow because of the small values of S used [38]. Therefore, it is unlikely to numerically find the crossover behavior, one either sees the scaling (8.2,8.3) valid for chaotic saddles or (8.5,8.6) valid for attractors. It is, however, important to know that between these drastically different behaviors there is a smooth interpolation.

8.6

Controlling motion on fractal basin boundaries

An immediate application of the control of chaotic saddles is the control of the motion on fractal basin boundaries [40], which in physical systems typically contain a chaotic saddle whose stable manifold is the boundary [12]. One can then select lowperiodic cycles of the saddle to control. By applying the method sketched above, a hyperbolic orbit hesitating to go to any of the attractors, (a Balam's donkey) is converted into an attracting orbit by applying weak perturbations. Figure 8.4 shows a continuous-time example, the case of a driven damped pendulum. At the parameter set investigated there are two attractors, each corresponds to a periodic winding motion (interrupted by a swinging), whose basin of attraction is bounded by a fractal curve in the phase space. A period-1 swinging motion is selected as the motion to be stabilized on the chaotic saddle sitting on the basin boundary. The OGY control algorithm is applied at integer multiples of the driving period, i.e. on the stroboscopic map of the system, by making the driving amplitude slightly time-dependent. The upper part of Fig. 8.4 exhibits a successful control process leading to the stabilization of a periodic motion which is basically different from any of the two periodic attractors. This method is potentially important in applications where periodic driving can result in a catastrophic failure of the system. A particular example can be ship capsizing. The method of controlling motion on fractal basin boundaries has succesfully been extended to preventing chaos-induced ship capsizing even in cases when the driving due to environmental influences (e.g. waves) is not periodic but has substantial irregular (chaotic) component [41].

8.7

Controlling chaotic scattering

A novel feature of chaotic saddles in Hamiltonian systems is that they typically also have a nonhyperbolic component where the local Lyapunov exponents might be arbitrarily close to zero. An interesting question is therefore to investigate the influence of the nonhyperbolic component on the control process. The hyperbolic component has a similar direct product Cantor set structure as shown in Fig.

190

8 Controlling Transient Chaos on Chaotic Saddles

Phil

3 2 1

0 -1

i

I

\\ :: :

i

:#

-

H !i ft ft

*:

:

•

3 2 1 0 -1 -2 -3

20

40

60

ij il! f"

80

100 120 140 160

80

100 120 140 160

\ ' II,"

LJU 111 0

Figure 8.4

:

• H N il il il n 0

Ch

:

til Inlinmu r| niiniyiyyi

-2 -3

•H

—

20

40

60

(a) Time dependence of the angle $(t) of the driven pendulum $ 4- /33> + sin$ — p(t) cos art with p(t) = p 0 + g(£) started from <3>0 = 0.5806, $o = 2.7524 at t = 0. The parameters are p = 0.2, a; = 1, p 0 = 2. The trajectory enters the target region, a disk of radius 0.05 around the unstable fixed point (-2.6313,0.6516) on the stroboscopic map talen at t = 0 mod 2n/u>. After 10 driving cycles control sets in, g(t) is chosen to cause the pendulum to keep swinging without rotation, i.e., to remain on the basin boundary (upper panel). Without control, g(t) = 0 (lower panel), the same trajectory is attracted to a stable winding motion (interrupted by swinging).

8.1 just more symmetric since, because of the time reversal invariance, the stable and unstable manifolds are equivalent. The stabilization of periodic orbits on the hyperbolic component is therefore similar to that discussed in Sec. 8.3. The nonhyerpbolic component is close to the KAM surfaces surrrounding quasiperiodic islands. If one selects a periodic orbit close to such a KAM surface, the time to achieve control is obviously long due to the stickiness of these surfaces. Numerical investigation shows [42] that the average time to achieve control could be an order of magnitude longer than the average chaotic lifetime on the hyperbolic component. The details of the actual behavior might depend on the inital condition of the ensemble investigated. Anyhow, Eq. (8.2) is no longer valid, but r seems to be still limited from above when the maximum allowed perturbation goes to zero. This is illustrated by the numerically obtained distribution of the average times

8.8 An improved control of chaotic saddles

191

e o

bo o

o.o-4.0

-3.0

-2.0

-1.0

Figure 8.5 The average time r(e) to achieve control in a chaotic scattering system [42] for an ensemble of trajectories visiting the nonhyperbolic component of the saddle before being captured on a peridic orbit. It is given as a function of the linear size e of the target region which is proportional to the maximum allowed perturbation, 8. The straight line marks the reciprocal value of the escape rate, the average control time on the hyperbolic compoenent. Note that r(e) greatly exceeds this for small sized target regions.

to achieve control shown in Fig. 8.5 which holds for trajectories coming in the vicinity of a KAM torus, i.e., moving on the nonhyperbolic component, before being controlled. The number of controlled trajectories scales according to Eq. (8.3) since this number depends only on the probability to hit the target region, no matter how long it takes for the individual trajectories to enter this region. Generally speaking, controlling a coUisional scattering process means stabilizing the intermediate complexes of a reaction which would otherwise be of finite life-time. The findings mentioned above show that although the effect of KAM surfaces can be important for the controlling process, the qualitative behavior of the controlled ensemble is similar to that of fully hyperbolic systems.

8.8

An improved control of chaotic saddles

As we discussed in Sec. 8.4, a major difference between stabilizing unstable periodic orbits embedded in a chaotic attractor and a chaotic saddle is that for the chaotic attractor, the probability that a chaotic trajectory enters the neighborhood of the

192

8 Controlling Transient Chaos on Chaotic Saddles

CO CM

c\i

CD

-3

-2

-1

Figure 8.6 A chaotic saddle for the Henon map xn+i = a—x^+6yn, yn+i = xn at a = 1.5, b = 0.3. The crosses denote the locations of a period-8 orbit embedded in the chaotic saddle.

desired unstable periodic orbit is one [33]. Hence, trajectories originating from almost any initial condition in the basin of the chaotic attract or can eventually be stabilized. While for the case of transient chaos, only a small set of initial conditions can be controlled. The reason is that most trajectories will have already left the chaotic saddle before entering the neighborhood of the target periodic orbit. A possible remedy is to simultaneously launch an ensemble of initial conditions and to control any one of the trajectories that come close to the target periodic orbit as demonstrated in Sec. 8.3. Clearly, for transient chaos we can only talk about the probability that a randomly chosen initial condition can be controlled. This probability is usually very small [36]. The main objective of this Sec. is to describe a scheme that can be utilized to maximize this probability. The key observation is that there exists a dense chaotic orbit on the chaotic saddle (the complement of the set of all unstable periodic orbits on the saddle) that comes arbitrarily close to any target unstable periodic orbit, as shown in Fig.

8.6 for a chaotic saddle of the Henon map. By using the PIM-triple method [31], one can compute such a long reference orbit on the chaotic saddle. The probability that a trajectory approaches this reference orbit is greater than the probability that this same trajectory enters the neighborhood of the target unstable periodic orbit before it escapes, if the length of the reference orbit is long enough. By stabilizing a trajectory around the reference orbit first, and then switching to stabilize it around the target periodic orbit after the trajectory comes close to the periodic orbit, we

8.8 An improved control of chaotic saddles

193

can substantially increase the probability that a trajectory can be controlled. This can indeed be achieved since there exist stable and unstable directions at each point of the reference orbit on the chaotic saddle. Hence, in principle, controlling a trajectory near the reference orbit is equivalent to stabilizing a long unstable periodic orbit as in Ref. [33]. The longer the length of the reference orbit is, the larger the probability for controlling periodic orbits can be. Our idea of control is as follows [43]. Let x n +i = F(x n ,p) be a two-dimensional invertible map that exhibits transient chaos. Let {y n } (n = 0,1,2,..., N) denote a long reference orbit on the chaotic saddle obtained by the PIM-triple method [31]. Now generate the orbit {x n } to be stabilized around the reference orbit. Randomly pick an initial condition x 0 , assume that the orbit point x n (n > 0) falls in a small neighborhood of the point y^ of the reference orbit on the chaotic saddle at time step n. Without loss of generality, we set k = n on the reference orbit. In this small neighborhood, the linearization of F is applicable. We have, thus, x n +i(p n ) - yn+i(po) = J • [xn(po) - Ynipo)] + KAp n ,

(8.15)

where Apn = pn — p0, Apn < 6, J is the 2 x 2 Jacobian matrix and K is a two-dimensional column vector, J = DxF(x,p)|x=v n ,p = P 0 ,

K = DpF(x,p)|x=yn,p=P0.

(8.16)

Without control, i.e., Apn = 0 , the orbit x^ (i = n + 1,...) diverges from the reference orbit yi (i = n - f l , . . . ) exponentially. Our task is to program the parameter perturbations Apn in such a way that the trajectory x stays near the reference orbit on the chaotic saddle (or equivalently, |x^ — yj| —> 0|) for subsequent iterates

i > n+ 1. For each reference orbit point on the chaotic saddle, there exist both a stable and an unstable direction. (For higher dimensional maps, there may be several stable and unstable directions. The algorithm to control chaos in such cases is more complicated [44] and will not be discussed here.) The existence of the stable and unstable directions at each reference orbit point can be seen as follows. Let us choose a small circle of radius e around some orbit point y n and map this circle to y n - i by the inverse map. In a Cartesian coordinate system with the origin at y n _i, the deformed circle can be expressed as A{dx')2 + B{dx')(dy') + C{dy')2 — 1 which is typically an ellipse. Here A, B and C are functions of the entries of the inverse Jacobian matrix at y n . This deformation from a circle to an ellipse means that distance along the major axis of the ellipse at y n _i contracts as a result of the map. Similarly, the image of a circle at y n - i under F is typically an ellipse at y n , which means that distance along the inverse image of the major axis of the ellipse at y n expands under F. Thus the major axis of the ellipse at y n _i and the inverse image of the major axis of the ellipse at y n approximate the stable and unstable directions at y n -i- It should be noted that, typically, the stable and unstable directions are not orthogonal to each other. In nonhyperbolic chaotic systems they can even coincide [45]. To achieve control, it is neccessary to calculate the stable and unstable directions along the reference orbit. We use an algorithm developed in Ref. [45]. This

194

8 Controlling Transient Chaos on Chaotic Saddles

numerical method, however, requires that the Jacobian matrix of the map be explicitly known. The stable and unstable directions are then stored together with the reference orbit, and they are used to compute the parameter perturbations applied at each time step. The details of the algorithm are given in the Appendix. We now present a numerical example. Figures 8.7(a-b) show an example of applying our algorithm to the chaotic saddle of the Henon map at a = 1.5 and b = 0.3. We use a reference orbit on the chaotic saddle of length N = 10000. The maximally allowed parameter perturbation is 5 = 0.01 and the size of the small neighborhood around each point on the reference orbit is chosen to be e = 0.005. We can choose both 8 and e arbitrarily, as long as they are small. We start the trajectory to be stabilized with initial condition: (#o,yo) = (0.5,-0.1). After 4 initial iterates, the trajectory falls into the neighborhood of a point of the reference orbit [(x,y) « (-1.8393,1.8387)]. When this occurs, parameter control based on Eq. (8.18) is turned on to stabilize the trajectory around the reference orbit. At the time step n = 846, the controlled chaotic trajectory comes into the vicinity of the period-8 orbit, at which time we immediately turn on a new set of parameter perturbations calculated with respect to the period-8 orbit. The trajectory stays in the neighborhood of the period-8 orbit in subsequent iterations as long as the parameter perturbation is present. Figure 8.7(b) shows values of the parameter perturbations applied. Numerically, the controlled trajectory rapidly converges to the reference orbit both after n = 4 (stabilized around the reference orbit) and after n = 846 (stabilized around the period-8 orbit). After a few iterates, the parameter perturbations required become extremely small (around 10~10). The probability that a randomly chosen initial condition can be controlled, P(N,e), depends both on the length of the reference orbit N and the size e of the small region around each reference point. Figure 8.8(a) shows the P(N,e) versus N curve, where e — 0.005. This curve is calculated by varying N systematically and randomly choosing 104 initial conditions with uniform probability distribution in the square region of Fig.. 8.6 for each fixed N value. The probability is given by the ratio between the number of initial conditions that approach the reference orbit before escaping to infinity and the total number of initial conditions chosen (104). For small N values, say N < 800, P(iV, e) increases approximately linearly. The reason is that the probability that a trajectory enters the neighborhood of the chaotic saddle is approximately proportional to the total area of the small circles surrounding all the reference orbit points. This area is approximately ire2N when overlaps between neighboring circles are small. As N increases further, the overlaps between neighboring circles become significant, thereby causing P(AT, e) to saturate. In fact, when N > 1000, P(N,e) increases very slowly. For N = 1000, P(N,e) = 0.546. For N > 10000, we have P{N, e) > 0.66. We expect the optimal length of the reference orbit to be N ~ 1/e, the point when overlap starts to become significant. If a trajectory is directly stabilized around the period-8 orbit without being stabilized around the reference orbit, the probability that an initial condition can be controlled is only 0.04 as shown by the lower straight line in Fig. 8.8(a). Thus, by using a reference orbit of length about 1000, a factor of more than 10 improvement in this probability

8.8 An improved control of chaotic saddles

195

(a)

./..-'. ' \:- ;• ••• •••• ••••. -V..^% •'•. CM control on to stabilize around chaotic saddle

control on to stabilize around period-8 orbit

500

1000

1500

time step n

(b) g *
in

£ C

I*)

I ^ CO

o

nfJpHmWr^^WT|ff^MTr^'^^TT'

i control on to stabilize around period-8 orbit

control on to stabilize around chaotic saddle

500

1000

1500

time step n

Figure 8.7

(a) An example of stabilizing the unstable period-8 orbit shown in Fig. 8.6 by means of the improved control method. A trajectory starts with the initial condition (#o, yo) = (0.5, —0.1). At time step n = 4, it falls in a neighborhood of one point on the reference orbit. Parameter control is turned on to stabilize the trajectory around the reference orbit for n > 4. At n = 846, the controlled chaotic trajectory gets close to the period-8 orbit. A new set of parameter perturbations calculated with respect to the period-8 orbit is turned on to stabilize the trajectory. For n > 846, the controlled motion is period-8. (b) The time-dependent parameter perturbations applied at each time step.

196

8 Controlling Transient Chaos on Chaotic Saddles

can be achieved. The relation between P(N, e) and e for fixed N = 8000 is shown by the upper curve in Fig. 8.8(b). As a contrast, the lower curve in Fig. 8.8(b) shows the same probability when no reference orbit is used to stabilize the period-8 orbit. Remarks: (a) The above method can also be applied to convert transient chaos into sustained chaos [46]. By constructing an arbitrarily long reference orbit on the chaotic saddle, we can make other trajectories stay in the neighborhood of this reference orbit for as long as we wish by applying small parameter control. In this sense, nonattracting trajectories in the neighborhood of the chaotic saddle are transformed into stable chaotic trajectories. (b) In a similar way, a method for stabilizing chaotic orbits on the attractor has been proposed and applied to the synchronization of two almost identical chaotic systems [47], and a method of creating desired chaotic orbits on a chaotic attractor has been implemented [48].

8.9

Discussions

In this Chapter we review the basic problem of controlling transient chaos on chaotic saddles. Through the use of scaling relations, we compare controlling transient chaos with the more extensively studied problem of controlling permanent chaos on chaotic attractors. Below we discuss several applications. Controlling transient chaos may have applications to engineering (e.g., the voltage-collapse problem [49]) and ecology (e.g., the species extinction problem [50]). In electrical engineering, in a particular type of voltage collapse, the power supply system suddenly breaks down after exhibiting complicated dynamical bahavior resembling that of transient chaos. Theoretical models for this type of voltage collapse suggest that transient chaos may be the culprit [49]. Therefore, the conversion of a transient chaotic trajectory into a sustained chaotic or periodic trajectory would prevent the voltage collapse. In ecology, in certain situations the problem of species extinction can be addressed using the idea of transient chaos [50]. Specifically, it was recently suggested by McCann and Yodzis [50] that transient chaos in very simple but biologically reasonable ecosystem models, mathematically described by coupled ordinary differential equations, can provide a hint as to how local species extinction can arise without the necessity to consider temporal or spatial variations and external factors. Usually, the population size of some species can behave chaotically for a (long) period of time and then decreases to zero in a relatively short period of time. Our idea is that if species extinction is caused by transient chaos, then it is possible for human being to intervene externally by applying perturbations so as to effectively prevent species from becoming extinct. The magnitude of the applied perturbation can be made arbitrarily small, and the perturbations need to be applied only occasionally. As such, the natural dynamics of the species population is hardly influenced, and yet, the population, though still exhibiting chaotic behavior, will never become zero. The implication is that in a

8.9 Discussions

197

(a) radius of control neighborhood = 0.005

oo

2

2 p d 0

2000

4000

6000

8000

10000

Length of reference orbit

(b) length of reference orbit = 8000 00

d

1

i

° q d -3.5

-3.0

-2.5

-2.0

Iog10(radius of control neighborhood)

Figure 8.8

(a) For fixed e = 0.005 (the radius that defines the controlling neighborhood), the probability P(N,e) that a randomly chosen initial condition can be controlled versus N, the length of the reference orbit (the upper curve). This probability increases initially with N and saturates for large N. The asymptotic value of P(N, e) is approximately 0.66. The lower straight line represents the probability that the trajectory is directly stabilized around the period-8 orbit. The value of this probability is only 0.04. Therefore, by using a reference orbit of length about 1000, a factor of more than 10 improvement in this probability can be achieved, (b) For fixed N = 8000, P(N, e) versus e curve (upper curve). The lower curve is the same probability versus e when the trajectory is directly stabilized around the period-8 orbit.

198

8 Controlling Transient Chaos on Chaotic Saddles

realistic ecological environment, a very small amount of artificially imposed change to population sizes or some small disturbance to the environment, only very rarely applied, can prevent species extinction over long time scales. Potentially, this can be of paramount interest to the significant and growing environmental problem of species preservation. A strategy for controlling transient chaos based on time series with applications to the voltage-collapse and species-extinction problems was recently studied [51]. Controlling transient chaos also has applications to nonlinear digital communication. Previous work demonstrated that symbolic representations of controlled chaotic orbits could be utilized for digital communication [52]. A central issue in any digital communication device concerns with the channel capacity, a quantity that measures the amount of information that the device can encode. It is thus highly desirable to have the channel capacity as large as possible to maximize the amount of information that can be encoded. For a chaotic system, channel capacity is equivalent to the topological entropy because it defines the rate at which information is generated by the system. Recently, it was pointed out that it is generally more advantageous to use transient chaos naturally arising in wide parameter regimes of nonlinear systems as information sources from the standpoint of channel capacity [53]. This is based on the observation that, typically, the orbital complexity associated with trajectories on a chaotic saddle can be greater than that of trajectories on a chaotic attractor. Thus, transient chaos can yield larger channel capacity. Thus, it is highly desirable to design a chaotic system operating in a transient chaotic regime for digital communication. In Ref. [53], a procedure was proposed for encoding digital messages into trajectories that live on chaotic saddles and, it was also argued that digital encoding with chaotic saddles can be robust against environmental noise, thereby significantly reducing the probability of bit error in communication. Another promising application is based on controlling point vortices in ideal fluids around an obstacle by means of an OGY-type control. In open flows this corresponds to the class of problems we have investigated. The surprising outcome is, however, that the viscous problem, in which one can speak of smooth vorticity ditributions instead of singular point vortices, can also be controlled practically with the same strategy as the point vortex problem. This has been poined out by Kadtke, Pentek, and Pedrizzetti [54] who developed a numerical method to stabilize a finite, concentrated vortex around a rotating cylindrical body embedded in an open fluid flow. As control parameter they used both the rotation frequency of the cylinder and the background flow velocity. The control algorithm is based on a low-dimensional Hamiltonian point vortex model, which is practically a scattering problem. This is the first example to control transient dynamics of spatially extended systems based on low-dimensional approximate dynamics. In a practical sense, the method has important consequences for any problem with vortex-body interactions, like e.g., the interaction of atmospheric vortex rings with aircraft wings where it can reduce the chance for a sudden decrease of the lift.

8.9 Discussions

199

A further practical application of controlling motion on chaotic saddles can be a semiconductor double heterostructure in which the electric currents were shown to exhibit transiently chaotic dynamics in a wide range of parameters. In a fourdimensional model is was demonstrated by Reznik and Scholl [55] that the continuous delayed feedback method [34] can succesfully be used to stabilize different periodic motions on the saddle. This opens the possibility to create this way widely tunable self oscillators which might be of practical relevance. Very recently even experimental realizations of the control of transient chaos have been reported by Kopp [56]. The motion on the chaotic saddle of the double scroll circuit as well as of an electrochemical reaction have been controlled based on time series observation and the delayed feedback method. This illustrates that controlling transient chaos in laboratories is feasible, and it will hopefully be followed be several other experiments.

Acknowledgments We would like to thank the enjoyable and fruitful collaboration with I. Janosi, C. Jung, Z. Kovacs, A. Pentek, K. G. Szabo, and Z. Toroczkai. This work was supported by the Hungarian National Science Foundation grants TO 19483, and the US-Hungarian Joint Fund No.286, 501.

Appendix: The improved control algorithm To find the stable direction at a point y, we first iterate this point forward N times under the map F and get a trajectory F 1 (y), F 2 (y), ..., FAr(y). Now imagine we put a circle of arbitrarily small radius e at the point FAr(y). If we iterate this circle backward once, the circle becomes an ellipse at the point FiV~1(y) with the major axis along the stable direction of the point F7V~1(y). We continue iterating this ellipse backwards, while at the same time normalizing the ellipse's major axis to be of the order e. When we iterate the ellipse all the way back to the point y, the ellipse becomes very thin with its major axis along the stable direction at point y if N is large enough. It should be mentioned that this same method can also be used to compute the stable and unstable directions along unstable periodic orbits. For an unstable period-k orbit, we choose TV = mk so that N is large, where m is an integer. In practice, instead of using a small circle, we take a unit vector at the point F iv (y) since the Jacobian matrix of the inverse map F " 1 rotates a vector in the tangent space of F towards the stable direction. Thus, we iterate a unit vector backward to the point y by multiplying by the Jacobian matrix of the inverse map at each point on the already existing orbit. We normalize the vector after each multiplication to the unit length. For sufficiently large N, the unit vector so obtained at y is a good approximation of the stable direction at y. A key point in the calculation is that we do not actually calculate the inverse Jacobian matrix along the trajectory by iterating the point F N (y) backwards using the

200

References

inverse map F l. The reason is that if we do so, the trajectory will usually diverge from the original trajectory F Ar (y), F iV ~ 1 (y), ..., F*(y) after only a few backward interations. What we do is to store the inverse Jacobian matrix at every point of the orbit F*(y) (i = 1,..., N) when we iterate forward the point y beforehand. Similarly, to find the unstable direction at point y, we first iterate y backward under the inverse map N times to get a backward orbit F~*7(y) with j = N,..., 1. We then choose a unit vector at point F~Ar(y) and iterate this unit vector forward by multiplying by the Jacobian matrices. The final vector at point y is a good approximation of the unstable direction at that point if N is large enough. Again, to avoid divergence from the original trajectory, we do not actually iterate the inverse map. What we do in this case is to choose y to be the end point of a forward orbit, all the points before y are the inverse images of y and we store the Jacobian matrix of forward map at those points. The method so described is efficient. For instance, the error between the calculated and real stable or unstable directions is on the order of 10~10 for chaotic saddles in the Henon map if TV = 20 [45]. Let es(n) and eu^ be the stable and unstable unit vectors at y n and, fs(n) and fu(n) be the corresponding unit contravariant vectors that satisfy fu(n) • eu(n) = fs(n) • e s(n) = 1 and fu(n) • e s(n) = f,(n) • eu{n) = 0. To stabilize {x n } around {y n }, we require the next iteration of x n , after falling into a small neighborhood around y n , to lie on the stable direction at y(n+i)(Po)> *-e-> [x n+ i - y(n+i)(po)] • f«(n+i) = 0(8.17) Substituting Eq. (8.15) into Eq. (8.17), we obtain the following expression for the parameter perturbation,

It is understood in Eq. (8.18) that if Apn > 5, we set Apn = 0. After a trajectory is stabilized around the reference orbit, we monitor the trajectory to see if it gets close to the target periodic orbit. To guarantee that the trajectory will always approach the target periodic orbit at later times, a possible strategy is to let the end point of the long reference orbit be in the neighborhood of the target periodic orbit. As soon as the controlled chaotic trajectory is in the vicinity of the target periodic orbit, a new set of parameter perturbations computed with respect to the periodic orbit is turned on to stabilize the trajectory around it. The new parameter perturbations can be computed similarly [Eq. (8.18)], except that the stable, unstable and their corresponding contravariant vectors are now associated with the target periodic orbit. These directions can be calculated using the same method discussed above [57].

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Periodic Orbit Theory for Classical Chaotic Systems R. Artuso Istituto di Scienze Matematiche, Fisiche e Chimiche Via Lucini 3, 1-22100 Como, Italy, Istituto Nazionale di Fisica della Materia, Unita di Milano and I.N.F.N. Sezione di Milano

9.1

Introduction

In the last decades we have been exposed to a wide variety of systems exhibiting chaotic behaviour: the physical import of this comes from the fact that the main features hold true at any conceptual level, as chaos is manifested in idealized, analytically tractable examples, in numerical simulations of approximated models as wella as in real laboratory experiments. Our theoretical intuition is however modelled after the paradigmatic idea that complicated phenomena might be investigated by means of theoretical procedures in which the lowest order approximation is built upon integrable (ordered) cases: the successes of this approach to the analysis of physically relevant systems are too numerous and too illustrious to bore the reader with any example, but my claim, in this introductory review, is that a perturbative approach to chaotic behaviour is most naturally performed by selecting as zeroth order approximation fully chaotic systems, or, to say it in other words, a generic chaotic system might be more easily approximated by idealized systems behaving as randomly as coin tossing than in terms of uncoupled harmonic oscillators. This program might sound like a phylosophic proclaim, if just stated in general terms: we will argue that instead this is the backbone of a perturbative approach to complexity. The key ingredient of the recipe is the recognizion that unstable periodic orbits are the skeleton underlying the structure of chaotic phase space: we will show this using a simple one-dimensional repeller as a prototype example, which will allow us to formulate the general structure of periodic orbit expansions. Then we will describe how various chaotic averages are computed in terms of cycles, and then discuss in some detail the physically relevant problem of deterministic diffusion: in this context we will also point out which problems may arise in cycle expansions when the system has a very complicated symbolic dynamics, or marginally stable regions of the phase space rule the asymptotic dynamics. We will also try to provide the reader with references to other applications of the theory, and parts of it not dealed with in this contribution.

206

9 Periodic orbit theory for classical chaotic systems

0.2

0.4

0.6

0.8

X

Figure 9.1 An overshooting parabola map, Tx = 6#(1 — x). Only the branches originating the strange repeller are shown.

9.2

Strange repellers and cycle expansions

In this section we will provide an example on how a dynamical calculation may be turned into a perturbative procedure, based on the ordering of periodic orbits according to their period. We consider a map T with two full branches, like the one in fig. 1. Whenever an iterate hits the roof the particle escapes from the interval. A natural quantitative measure of the decaying process is the escape rate, giving the exponential decay rate of particles initially spread at random along the unit interval. We will see in the final part of this contribution that deviations from exponential decay may arise from lack of hyperbolicity, but for the moment we ignore this problem. So we define the decay rate 7 through N8urv(n) Nsurv(0)

(9.1)

where Nsurv(n) is the number of initial conditions survived up to the n-th iteration. We may rewrite the ratio appearing in the left hand side of (9.1) by Nsurv(n) N8urv(O)

= I dxdyS{x-Tny)

= f dxdy£n(x\y)

(9.2)

where we have introduced the kernel of the transfer operator C C{x\y) = 6(x - Ty) The action on functions of the transfer operator is thus written as 1

(£g)(x)= J

dy£(x\y)g(y)=

(9.3)

9.2 Strange repellers and cycle expansions

207

and, as we may easily check, this operator enjoys the semigroup property. While in successive sections we will introduce and study generalized transfer operators, we remark that the operator in (9.3) is the so called Perron-Probenius operator which admits as an eigenfunction the invariant density under the dynamics (if it exists), as it is easily seen by considering the identity

dxf(x)g(Tx) = J dx (£/) (x)g(x) If we consider (9.2) we notice that the asymptotic behaviour is determined by the dominant eigenvalue of C (with exponentially small corrections in n, provided there is a gap for the spectrum of £, separating the dominant eigenvalue from the rest of the spectrum). Thus we may rephrase our problem in terms of spectral properties of £, by writing 7 = -log|A 0 | Ao being the dominant eigenvalue of C: if the system admits an invariant density Ao = 1 and the escape rate is zero. Before illustrating how periodic orbits come into play we have to remark, as it is clear since the very beginning of this contribution, that our treatment skips all mathematical problems and subleties: we refer to (Ruelle, 1978) and (Mayer, 1980), for the foundations of the mathematical theory of transfer operators, and warmly recommend (Baladi, 1995), for an overview on rigorous approach to dynamical zeta functions, and updated mathematical references. We now turn our attention to the evaluation of the dominant eigenvalue (and follow (Artuso, Aurell and Cvitanovic, 1990a) and (Cvitanovic, 1991)), by putting z0 = A^1, where z0 is the smallest solution of the secular equation det(l-z£) = 0

(9.4)

How is an expression like (9.4) treated in practise? One first goes to traces by rewriting the Predholm determinant in the following way OO

-

F(z) = det (1 - zC) = exp - ] T -zntrCn

(9.5)

n=l

where trCn = f dxCn(x\x)

(9.6)

These traces are naturally connected to periodic orbits as trCn = f dx8{x-Tnx) where n-l

3=0

=

^

——j

r

(9.7)

208

9 Periodic orbit theory for classical chaotic systems

is the stability of the periodic orbit to which x belongs (note that Ax(n) has the same value for all cycle points). We remark that the sum (9.7) is over all periodic points of period n (while the contribution is the same for points belonging to the same periodi orbit). Moreover periodic points of period n include all cycles of prime period dividing n: by taking into account the repetitions we may thus rewrite (9.5) as

where {p} indicates the set of all prime periodic orbits (each counted once), nv being the prime period of p and Ap its stability. Now we consider uniformly hyperbolic maps, for which |T"(x)| > l,Vx, so that |AP| > lVp, and

The sum over repetitions r may now be performed explicitely, yielding

or

det(l-z£) = q=0

with

These functions are called dynamical zeta functions (Ruelle, 1978). Before going on and showing how the Fredholm determinant or dynamical zeta functions are used in actual calculations, we briefly comment on some virtues of an approach to dynamical systems based on periodic orbits properties (Cvitanovic, 1988): first of all cycles are invariant under smooth conjugacies, and this property is also shared by the stability factors, so the expressions (9.5) or (9.8) are independent of the particular representation of the dynamics: moreover they provide, as we will see, a perturbative approach to chaotic properties in which the order of the expansion is determined by the longest prime period taken into account. In practise the full power of this approach is exploited in situations in which we have a complete control over the topology of the dynamical system, so that we can efficiently encode the hierarchy of periodic orbits, and the system itself is fully hyperbolic (marginally stable orbits require a special care, as it is apparent from the expression (9.7)).

9.2 Strange repellers and cycle expansions

209

As long as we are interested only in the leading eigenvalue we may equivalently use the Predholm determinant (9.5) or the dynamical zeta function (Q1(Z) and look for their smallest zero (though though these objects enjoy different analytic properties, see (Baladi, 1995), or the considerations in (Eckhardt and Russberg, 1993)): in these notes we will consider only dynamical zeta functions, and refer the reader to (Cvitanovic, 1991) for a nice introduction to the alternative use of Fredholm determinants (examples are also discussed in (Cvitanovic et al, 1996)). For small z we can expand the infinite product defining Co"1^) a s a formal power series

n=l

and examine how the coefficients /3n are related to cycle properties. The first observation is that contributions to /3m come only from cycles whose period does not exceed m: the detailed structure of coefficients requires a knowledge on which cycles are present, their number and so on. To understand the general features we refer to our example (fig. 1), where a symbolic dynamics is easily introduced: we call / 0 and h the supports of the left and right branch respectively: surviving trajectories will be coded by a binary alphabet with a sequence eo,.. . en ... such that if Tnx e /o then en = 0 and if Tnx eh cn = 1. Each symbol sequence labels uniquely an existing trajectory: this is due to the fact that the inverse branches ipo and ipi of the map we are considering are defined over the whole interval (each branch of the map is onto [0,1]) so ip€l ... tp€n (y) may be evaluated for any possible combination t\ ... en. So for each n we have 2 n periodic points, whose infinite code is given by repetitions of any n-long subsequence: the number of prime periodic cycles is then evaluated by an easy application of Mobius inversion formula (see for example (Hardy and Wright (1979)): if we denote by Mp the number of prime periodic cycles of period p we have Mp = -

_

where /x(l) = 1, //(pi.. .pk) = (—)* (pi.. .pk prime factors) and fjt(q) = 0 if by decomposing q we get squared factors. So for instance we have M3 = | ( 8 - 2 ) = 2 with corresponding labels 001 and Oil (the label 000 corresponds to the left fixed point, which contributes to periodic points of period 3, but not to the counting of prime cycles of period 3). We will not derive the most general expressions (the interested reader may consult (Artuso, Aurell and Cvitanovic, 1990a), or (Cvitanovic, et al 1996)) and just see what happens by looking at the first few coefficients: if we denote by tp = 1/|AP| we have

210

9 Periodic orbit theory for classical chaotic systems

so that each coefficient but the first is built upon 2n l contributions (half of which with a minus sign in front): if we just have a single scale (piecewise linear maps with a single slope |AP| = A) all the coefficients n > 2 would vanish and we would have

yielding Ao = 2/A (unique zero). In general higher order coefficients do not vanish (we do not expect a single scale being present in the system): this, for instance, does not happen for the map of fig. 1: from the viewpoint of analytic properties this results in a C^1 admitting meromorphic extensions (instead of the analytic form of the uniform slope example), see (Ruelle, 1978), (Mayer, 1980) and (Baladi, 1995)): we just give an elementary argument (inspired by (Eckhardt and Russberg, 1993)) that this analytic behaviour is compatible with the observation (Artuso, Aurell and Cvitanovic, 1990a, 1990b), that for fully hyperbolic, topologically regular dynamical systems, finite order estimates, obtained by truncations of (9.9) to finite orders, converge exponentially well. Consider the simplest possible example of a function having a zero and a simple pole p{z)

~ (T^te)

where we take a, b positive, and a > 6, so that the zero is nearer to the origin. If z is sufficiently small the truncations will have the form M

PM(Z) = l - ( a - f t ) ^

bm-lzm

and 2(M)> solution of PM{Z(M)) — 0> converges to I/a exponentially fast. More direct arguments justifying the numerically observed exponential convergence of finite order estimates are given in (Artuso, Aurell and Cvitanovic 1990a) and (Cvitanovic et al. 1996). While, as we already mentioned, in this review we will always deal with dynamical zeta functions we must remark that the use of Fredholm determinants significantly improves convergence; in the latter case finite order estimates for "good" systems do in fact converge faster than exponentially (see (Cvitanovic, 1991) and (Christiansen, Cvitanovic and Rugh, 1990)). We summarize now the procedure by recalling the main steps one has to go through to perform periodic orbit calculations. • encode the dynamics by introducing a symbolic dynamics: our prototype example was characterized by a complete bynary grammar • find all periodic orbits up to a certain period, order then hierarchically according to prime period, calculate weights of interest. In the escape rate example the only index that enters the calculations is the stability of each orbit

9.2 Strange repellers and cycle expansions

211

• build up finite order truncations to the dynamical zeta function (9.9) and obtain a sequence of finite order approximations to the asymptotic exponent you are interested in (the escape rate in the present case). If the sequence is regular (as it is expected for nice hyperbolic systems) you may use acceleration convergence algorithms to improve your results (we refer to (Guttmann, 1989) for a comprehensive review). In practice detailed knowledge on the topology of the system allows one to write the dynamical zeta function (9.9) is such a way that the role of fundamental cycles is highlined:

where we have incorporated z in the definition of cycle weights £, and we factored away the contribution of cycles which are not shadowed by combination of lower order orbits: in the case of a complete binary grammar the fundamental cycles are just the fixed points 0 and 1. The fundamental cycles thus provide the lowest order approximation in the perturbative scheme: a general chaotic systems is approximated at the lowest level with its simplest poligonalization, non uniformity is incorporated perturbatively by considering curvature corrections (cn) of higher and higher order. The whole scheme relies on a symbolic encoding of the dynamics, and while we remark that finding a proper code for a given system is a highly nontrivial task in general, we have to emphasize that this cannot be considered as a shortcut on the theory proposed here, as the topological complexity cannot be eluded in any sensible treatment of general properties of chaotic systems. For unimodal maps the natural code is in terms of a binary alphabet, and a deep mathematical theory of allowable sequences is known (see (Milnor and Thurston, 1988)). A binary partition may also be introduced for two dimensional maps of Henon type (Grassberger and Kantz, 1985): the analysis of allowable sequences in this context lead to the introduction of pruning fronts (Cvitanovic, Gunaratne and Procaccia, 1988). When we consider hamiltonian systems even in the simplest cases it is known that introducing a partitioning of the phase space is a hard problem (see (Bunimovich and Sinai, 1981), as regards billiard systems), or (Hansen 1993a, 1993b)). On the other side when there is a finite set of pruning rules it is possible to redefine the alphabet such that in the new set of symbols the grammar is unrestricted: take for instance the case in which we have a binary coding {0,1}, with the pruning rule that forbids every _00_ substring: by introducing the new symbol 2 = 01 we easily see that every possible allowed sequence is built through unrestricted composition of the symbols 2 and 1, and the fundamental cycles are thus the fixed point 1 and the period two cycle 01. Other examples are discussed in (Artuso, Aurell and Cvitanovic, 1990a): in the last section we will present another example, in which prohibition of a fixed point leads to the introduction of an infinite alphabet.

212

9 Periodic orbit theory for classical chaotic systems

9.3

Recycling measures of chaos

We now want to extend the formalism by introducing generalized zeta functions and Fredholm determinants, whose zeroes yield various averages relevant to the characterization of chaotic properties of the system. We first observe that we may introduce kernels for generalized transfer operators by putting C9(x\y) = 5(x-Ty)g(y)

(9.10)

and generalized transfer operators will enjoy the semigroup property if g is multiplicative along the orbit m-l

g(T™y) = [ I g(Tiy) i=o In this way we may define a generalized Fredholm determinant (cfr. (9.5)) Fg(z) = det(l-zCg)

(9.11)

or generalized dynamical zeta functions (cfr. (9.8))

{p}

where 9P = x\.. .xnp being the cycle points. We remark that the discussion on the convergence properties of finite order estimates, contained in the former section for the escape rate case, may be carried over to the present framework: we may still take expansions like (9.9) once we redefine the weights tp = gp/\Ap\ For a one dimensional map a multiplicative weight is the stability of a trajectory: the same property is obviously also shared by gs(x) = exp(/?log|T'(:r)|) and this lead us to a cycle expansion formula for the Lyapunov exponent: ds{P)

(9.13) 0=0

where s(0) is the leading eigenvalue of the generalized transfer operator C9a, which might be evaluated, for instance, from a cycle expansion of the dynamical zeta function C[* Az). We have to remark that cycle expansions for the Lyapunov

9.3 Recycling measures of chaos

213

exponent present some problems when we try to extend them in higher dimensions, as the largest eigenvalue of the jacobian for a n-th dimensional map is no more a multiplicative quantity along the trajectory: this might be dealt with by introducing extended zeta functions (Cvitanovic and Vattay, 1993) or consistent approximations might be devised in specific applications (Dahlqvist, 1995b) We have to comment briefly on a particular feature that is emerging from our treatment: in practise we use periodic orbits to mimic phase averages, and from a naive point of view this might seem a very dangerous approach: if you think of an ergodic hamiltonian system periodic orbits are just contained in the zero measure set that is excluded from all theorem formulations. Nethertheless cycles allows for a hierarchy of finer and finer "poligonalization" of the dynamics (Cvitanovic, 1988, Artuso, Aurell and Cvitanovic, 1990a), and in this sense they provide a systematic way of approaching the ergodic measure: for nice hyperbolic systems convinging arguments that periodic points together with their expanding factors (products of unstable eigenvalues) concurr to approximate the invariant measure were provided in (Parry, 1986, and Grebogi, Ott and Yorke, 1988)). As a matter of fact this is equivalent to substitute ergodic averages with the n»4oo limit of trace formulas (see (9.7))

[A being the ergodic measure. Though trace formulas might be useful by themselves is dealing with particular applications (see (Vance, 1992) or (Morriss and Rondoni, 1994)), their convergence properties are a priori more dubious, so in principle one should undertake such an approach with some care. We will see in next sections how trace formulas may be functionally related to integrals in which Fredholm determinants appear, and these expressions will turn out to be particularly useful in the case of anomalous diffusion (induced by marginally stable orbits). Now we briefly review how other frequently used indices of chaotic behaviour may be recycled: first of all we consider the topological entropy (for a careful definition of all the quantities we will mention and their significance in the description of chaotic properties of dynamical systems we refer to (Eckmann and Ruelle, 1980)): it coincides with the growth rate of the number of periodic point of period n with respect to the period: h =

lim

The topological entropy thus does not keep into account metric properties of the system and it is easy to convince that h = —\ogztOp, where ztop is the smallest zero of the topological zeta function:

For a complete binary grammar we have

214

9 Periodic orbit theory for classical chaotic systems

and h = log 2. In this example we see how (9.15) defines an entire function on the complex plane: it is interesting to observe that recently it has been realized that complicated pruning rules might induce dramatic changes in the analytic structure of (9.15): see (Mainieri, 1995) and (Dahlqvist, 1996b). As regards other indices, we have in general just to understand the correct weigth we have to insert in tp: let pi be a probabilistic weight associated to cycle i. Let np be the asymptotic scaling of this weight:

In practice the most relevant weight is the natural measure built up as the visitation frequence of region i (at least for dynamically generated strange objects: when studying for example strange sets in parameter space the choice of weights is a priori completely arbitrary). For hyperbolic systems this weight is asymptotically equal to the absolute value of the product over unstable eigenvalues of the derivatives matrix. The generalized dimensions of Grassberger and Procaccia (see (Grassberger, 1983), and (Hentschel and Procaccia, 1983)) are given by 1 where

By taking q(r) = 0 we get in particular a cycle expansion formula for the HausdorfT dimension Do'-

We mention that in (Artuso, Aurell and Cvitanovic, 1990b) this formula has been used to evaluate the dimension of the period doubling repeller (using the universal presentation function of (Feigenbaum, 1988)), yielding Do = 0.5380451435... by using cycles up to period eight (and the number of significant digits might be pushed much farther by using Fredholm determinants as in (Christiansen, Cvitanovic and Rugh , 1990). The generalized metric entropies (Grassberger and Procaccia, 1985) are given by 9-1 where

0 = JJ(1M

9.4 Periodic orbit theory of deterministic diffusion

215

The generalized Lyapunov exponents ((Fujisaka, 1983), (Kantz and Grassberger, 1985)) are \T — where

Of special interest are the Hausdorff dimension, Do, the information dimension, Z}i,the topological entropy, Ko,the metric entropy , Ki,and the largest Lyapunov exponent. By using the preceedings cycle expansions we may rederive the following identities (Kantz and Grassberger, 1985), (Eckmann and Ruelle, 1980): K\ = AM • D\

(any measure)

K\ = X^ — 7

(natural measure)

where 7 again denotes the escape rate. Finally it is easily seen that all the infinite products are but special cases of

az^r)-1

=H(l-z***D;)9

(9.17)

{p}

so that indeed cycles provide a suitable input for every kind of average computation. Such formulas have been used for a number of thermodynamic calculation on one and two-dimensional maps in (Artuso, Aurell and Cvitanovic, 1990b), and also (once generalized to> continuous flows) to the Lorenz system (see (Franceschini, Giberti and Zheng, 1993)).

9.4

Periodic orbit theory of deterministic diffusion

We now report on some progress that have been done in the last few years as regards the problem of deterministic diffusion. While in the realm of the present contribution this might be viewed as just another context in which cycle expansions might be usefully applied, we believe that some emphasis is rightly put upon this problem, as it is both physically motivated (see for instance (Huberman, Crutchfield and Packard, 1980), (Geisel and Nietwerberg, 1987), or (Aurell and Gilbert, 1993)), and the same time presents a major theoretical challenge, as the issue is to motivate how purely deterministic systems may enjoy properties which are typical of genuine stochastic systems, as random walks. The first approach to deterministic diffusion in which the dramatis persons where assumed to be periodic orbits was put forward in (Dana, 1989): the intuition is that deterministic diffusion emerges like a balance between localized orbits

216

9 Periodic orbit theory for classical chaotic systems

(fixed periodic points of a periodic extended systems) and accelerator modes (orbits leading to a linear translation with time): the diffusion coefficient D is then thought to emerge as a statistical balance between these two classes of orbits. We consider the simplest possible framework in which the phenomenon appears: a one dimensional map, which we define over the unit interval by extending its definition over a unit cell, in the following way: r(x + n ) = T ( x ) + n ,

x G R , neZ

(9.18)

together with the property T(x) = -T(-x)

(9.19)

The property (9.19) is just imposed to prevent any drift being present in the problem. It was early recognized that maps of this form may lead to diffusive type behavious, and we refer to (Geisel and Nietwerberg, 1982) and (Schell, Fraser and Kapral, 1982)) for the possible phenomenology of such maps. We consider the simplest possible example of a map leading to diffusive behaviour, by taking a piecewise linear map yielding a simple symbolic dynamics once restricted to unit-cell torus: Ax x6 [0,1/4 4- 1/4A] T(x) = { -Ax + (A + l ) / 2 x € [1/4 + 1/4A, 3/4 - 1/4A]

(x) = i

Ax + (1 - A)

(9.20)

x € [3/4 - 1/4A, 1]

see fig. 2 Together with this map it is useful to consider the associated circle map t{x) = T(x)\modl (9.21) We can then classify peridodic orbits of the torus map (9.21) according to their behaviour with respect to the lift (9.20): we will have standing periodic orbits if Tn(x) = x and running periodic orbits (accelerator modes) if Tn(x) = x + p

p e Z

so that a; is is a periodic point of the torus map T, while corresponding to a runaway mode for the map on the real axis. If we refer to the map on the real axis the proper operator to deal with will be the Cn(x\y) = e^fn^S(y - Tnx)

(9.22)

so that the appropriate cycle weight will be tp = where ap is the "jumping" factor of each torus periodic point in the sense that whenever x is a periodic point of the map on the torus T, fn(x) = x we have that Tn(x) =x + ap.

9.4 Periodic orbit theory of deterministic diffusion

217

-1.6 x

Figure 9.2 (9.20), for A = 4. The piecewise linear map, leading to diffusive behaviour (9.20), for A = 4.

The factors ap are additive along the trajectory, so the correct quantity to take into account with the purpose of introducing the appropriate generalized transfer operator is the generating function nn(/?) = (e^*"-*0))

(9.23)

where the average (• • •) is over all xo in an elementary cell Jo- As in the first section we may rewrite the generating function by means of the S propagator as ttn(P) =

dx

(9.24)

But we now use the symmetry of the map which allows to associate to every orbit of the map T an orbit of the torus map T, and rewrite (9.24) in such a way that the integration over y is also carried out in the elementary cell /Q:

n n (/j) = f

dx f

(9.25)

so that once again the generating function will be asymptotically dominated by the leading eigenvalue of the (generalized) transfer operator with kernel £0(x\y) = e^Ty-y^S(x-Ty) By taking the same steps as in the derivation of the escape rate cycle expansion of the first section we are thus lead to the zeta function whose weigths tp are of the form _ exp(/3ap) ~ |A|A$

P

(9.26)

218

9 Periodic orbit theory for classical chaotic systems

T(X)

0.0

0.2

0.4

0.6

0.8

Figure 9.3 The torus map corresponding to A = 7 in (9.20). if we call z((3) the smallest zero of the generalized dynamical zeta function £0 1 (z, / (built upon the weigths (9.26)), we then obtain, for large n

By Taylor expanding around /3 = 0, and keeping in mind that odd moments of the relative position vanish (due to the symmetry property (9.19)), we get the expression for the diffusion constant as

D=

lim r-((x n -x 0 ) 2 ) = - -

(9.27) (3=0

This approach to deterministic diffusion was introduced in (Artuso, 1991) and (Cvitanovic, Eckmann and Gaspard, 1995); we now illustrate the procedure by calculating D for a set of particular values of A in (9.20) which induce a particularly simple symbolic dynamics in the corresponding torus map (9.21). Call a the maximum of Tx in the unit interval: this value is connected to the slope A by A = 4a — 1: whenever a is an integer the torus map consists of m = 4a — 1 full branches (see fig. 3 for the case a — 2) and thus the symbolic dynamics is complete in 4a — 1 symbols. Each of the subintervals in which the unit interval is accordingly partitioned has the same value of the absolute value of the slope |A^fc| = 4a — 1 (rjk = 1,2,...4a— 1), while we have 3 branches yielding am = 0, and the remaining 4(a — 1) branches are such that 2 branches have aVk = + 1 , 2 yield a^k = —1 up to the maximum jumping number a — 1, for which again two branches get a positive value, with another pair giving am = - ( a - 1). Both stability and exponentials of jumping factors for any orbit are just products of factors of the visited subintervals, so curvature correction vanish (like in the piecewise linear example of the

9.5 The inclusion of marginalfixedpoints

219

first section) and we get Q

C~l(z,(5) = 1 — a

4

z

""

a

~

a

~1

z y ^ cosh(fc/?)

(9.28)

A;=l

yielding the only zero 4a~l where we notice that indeed z(0) — 1, as for /3 = 0 the diffusion zeta function coincides with the escape rate one, and as no particle escapes in the torus dynamics we have to recover a zero escape rate (probability conservation). By performing the double derivative we finally get the result q(q-

D{a) D { a )

==

3(4a-l)

Periodic orbit formulas for diffusion have been applied to derive a number of exact results for one dimensional maps in (Artuso, 1991) and (Artuso, Casati and Lombardi, 1994), to investigate two dimensional area preserving maps of the standard type (Eckhardt, 1993), sawtooth and cat maps (Artuso and Strepparava, 1996) and finite horizon Lorentz gas (Cvitanovic, Gaspard and Schreiber, 1992), (Cvitanovic, Eckmann and Gaspard, 1995), (Vance, 1992) and (Morriss and Rondoni, 1994). We end the section by remarking that though diffusion for piecewise linear maps might seem a trivial example, yet it hides a number of subtle features: for instance the diffusion coefficient is in general a highly nontrivial (fractal) function of the slope (Klages, 1996), (Klages and Dorfman, 1995).

9.5

The inclusion of marginal fixed points

We have already remarked how hyperbolicity assumption is fundamental in deriving cycle expansions: in this section we want to sketch a few results that indicate how to proceed when marginally stable cycles appear, and we keep on considering deterministic diffusion as the physical context of our considerations: cycle formulas for anomalous diffusion were introduced in (Artuso, Casati and Lombardi, 1993). First of all it is convenient to introduce an alternative expression for the diffusion coefficient, by means of an inverse Laplace transform (see for instance (Dahlqvist, 1996a)): as the leading behaviour in the asymptotic limit for the generating function is induced by the dominant eigenvalue of the Fredholm determinant we can write lim nn(/?) = lim — /

dse°n-£j-t

(9.29)

220

9 Periodic orbit theory for classical chaotic systems

where the Predholm determinant which appears may be expressed through dynamical zeta functions via the expression

k=0 {p}

(see (Dahlqvist, 1996a) and references therein for further details) and we notice that, in order to catch the leading contribution, we may substitute Ql{e~s,(3) to Fp(s) in the expression (9.29). Once the generating function is Taylor expanded around 0 = 0 we get D = lim

1£ ( 1 /

dsegnCo(e,m

where anomalous diffusion is signalled by (9.30) having a nontrivial dependence on n in the asymptotic limit: in this case the proper asymptotic behaviour for the variance will be <(z n -z 0 ) 2 > *

\D{n)-n

The expression (9.30) thus allows us to draw the following observation: the presence of a polar singularity (and the fact that z{0) = 1 for 0 == 0, to guarantee probability conservation) leads to a well defined diffusion coefficient, and anomalous behaviour will be connected to different kind of singularities (typically branch points) appearind in the dynamical zeta function. Though here the focus is on diffusive properties, we observe that such a feature lies at the heart of many other dynamical manifestations expressed though periodic orbit expansions, as the development of long time tails in correlation functions, see (Dahlqvist, 1995a), (Dahlqvist and Artuso, 1996), and (Isola, 1995). The prototype example of an intermittent map, in which the dynamics is influenced in an essential way by the presence of a marginal fixed point was introduced in (Pomeau and Manneville , 1980) Xn+i = f(xn) = xn + cxzn {modi)

(9.31)

(with z > 1, c > 0): x = 0 is the marginal fixed point. By fixing c = 1 the map consists of two full branches, with support on [0,p) (Jo) and [p, 1] (Ji), where p + pz = 1. We will denote by cpi the inverse branches of /|/.. As fii is onto the unit interval, the symbolic dynamics is a priori given by an unresticted grammar in two symbols: however the 0 fixed point has stability one and cannot be included directly in cycle expansions: the class of orbits 0*1 will probe the region around 0 arbitrarily close to the fixed point and will be particularly relevant in the analysis of dynamics. We accordingly will exclude the fixed point 0 from the allowed symbol sequence: this rule may be implemented by redefining and (infinite) alphabet in which the symbolic dynamics is unrestricted: the new alphabet is {1,0*1, k = 1,2...}. We remark that from a mathematical

9.5 The inclusion of marginalfixedpoints

221

point of view this procedure may be viewed as a consequence of applying inducing techniques: see (Prellberg, 1991), (Prellberg and Slawny, 1992), (Isola, 1995) and (Rugh, 1996). The unit interval is accordingly partitioned into a sequence of subsets /i,/Ofcl5 where I\ = [p, 1] and IOki = (PQ(II). We denote by £e the widths of these intervals and approximate the map with a piecewise linear version, introduced in (Gaspard and Wang, 1988) (see also (Wang, 1989)): the corresponding slopes will be s(Ii) = t[l = p/(l — p) = A, while s(/Ofci) = ^Ofc-1i/^ofci: t n e corresponding cycles stabilies will be Ai = A, A ^ = t~^v The asymptotic behaviour of £Ok1 is determined by the intermittency exponent z, in fact if we put yo = 1, y\ = p, y\ = ipo(yi-i) for / = 2,3,... we have that y m _i = ym + yzm and S \~l/ym~l)j ~ Sm so that

This implies that the widths will scale as £Ok1 ™ q/ka+1. We now assume the former as an exact expression, so that t\ = A" 1 a^nd £Ofci = ^/fc a+1 and choose q in such a way that probability conservation is respected: oo

°nl = 1 = A"1 + g • C(l + a) n=l

so q = (A — 1)/(A • C(l + a:)) and the stabilities are finally written as Ai = A, while AOki = A • £(1 + OL) - ka+1 /(A — 1). The dynamical zeta function now picks only contributions from the fundamental cycles (labellel by alphabet letters): (932)

where F(z,s) is known as the Jonquiere function:

n=l

while

Now 2? = 1 is a branch point and we have an example of a zeta function which cannot be extended as a meromorphic function over all the complex plane (the first example of this fact was introduced in (Gallavotti, 1976), see also (Artuso, 1988)). The development of a branch point is connected with the absence of a gap in the spectrum of the Perron Probenius operator, and the analysis of correlations requires a special care (see, in the context of Sinai billiards (Dahlqvist and Artuso, 1996)), as we have no more Ruelle resonances (Ruelle, 1986) dictating exponential decay.

222

9 Periodic orbit theory for classical chaotic systems

Further calculations require some control over F: First of all observe that F(z,l) — -log(l - z), which diverges as z 4 1". Now if 0 < s < 1 again F(z,s) diverges as z »-> 1~~, with a behaviour like F(z,s) ~ F(l - s) • (1 — z)s~l: this may be seen for instance by employing Tauberian theorems for power series, see (Feller, 1966). Thus far we have remarked a few fact on simple maps with a marginal fixed point: we now want to see how the appearance of such points might modify the properties of deterministic transport: to this purpose we consider a map with the same "symbolic properties" of (9.20) with a = 2 in which the central (standing) region is however modelled after the piecewise linear approximation of the intermittent map (9.31): so the map consists of 5 full branches whose support we denote by Ii i = 1,2,3,4 (right and left branches) with uniform slope (absolute value A), while f\j0 (central branch) is of (piecewise linear) intermittent form. We again prune the 0 fixed point away so that the symbolic dynamics is determined by the countable alphabet {1,2,3,4,0*1,0^2,0*3,0*4} z, j , k, I = 1,2,.... The corresponding weights are

o fc :*,0* 4 ol 1, 012

Ap Ap

-

= ££ , = ±A , = ±A

3, 4 2, 1

1

g/2

op = - 1 op = 1 dp = - 1

(9.34)

where q is to be determined by probability conservation for / :

+ 2a* + 1) = 1 so that q = (A — 4)/2AC(a +1). The fundamental cycles coincide with the alphabet letters and we have Ql{f3)

1

and its first zero z(/3) is determined by

4

A-4

„,

n

By using implicit function derivation we see that D vanishes (i.e. z"(/3)\g_0 = 0) when a < 1. This is easily interpreted from a physical point of view, as marginal stability implies that a typical orbit will be sticked up for long times near the 0 indifferent fixed point, and the 'trapping time' will be larger for higher values of the intermittency parameter z (recall a = (z — I)" 1 ). A closer look at diffusive behaviour must take (9.30) into account: to get asymptotic estimates we again rely on Tauberian theorems (Feller, 1966): and in particular we will use the following version: take oo

Xxn dxe~-Xx u{x)

9.5 The inclusion of marginal fixed points

223

(with u(x) monotone in some neighborhood of infinity): then, as A »-> 0 and x \-¥ 0 respectively (and p £ (0, oo)),

±L (I) if and only if u(x) ~ where L denotes any showly varying function (i.e. such that lim L(ty)/L(t) = 1). t—KX)

Now, if we denote the second derivative of the dynamical zeta function by ga(s) we may estimate the behaviour near s = 0 through the estimates of the function F near z — 1, and then find the asymptotic law for the growth of the variance through Tauberian theorems. We thus get

{

s~2

far a > 1

5 -(a+i)

for

l/(s 2 logs)

for a = l

a e (0)

-g

from which we get the estimates

{

t

for a > 1

t° for a G (0,1) (9.36) t/\nt for a = 1 which coincide with the results formerly obtained via other ways in (Geisel and Thomae, 1984) and (Geisel, Nierwetberg and Zacherl, 1985). We conclude this section by giving an overview on a recently developped method to build systematically approximate zeta functions, which is particularly useful to deal with marginally unstable systems: this approach relies heavily on a method introduced in (Baladi, Eckmann and Ruelle, 1989) to calculate in a probabilistic framework Ruelle resonances for systems close to intermittency, and was extended in (Dahlqvist, 1994) to the construction of approximate zeta functions. The physical idea underlying such approaches is to suppose dynamical evolution may be mimicked by a sequence of time intervals determined by t\ < £2 < • • • < tn < • • • such that the time laps Aj = tj —tj-x form a sequence of random variables with common distribution t/>(A) dA and the orbit properties before and after tn are independent of n. A typical choice for {tj} in the case of intermittent maps consists in collecting the reinjection times in the laminar region: the second property mentioned above is thus related to the "randomization" operated by the chaotic phase. We refer the reader to (Geisel and Nietwerberg, 1984) for a determination of if>(A) in the case of Pomeau-Manneville intermittent map (9.31), and to (Dahlqvist, 1995a) for the construction of the same quantity in the case of infinite horizon Lorentz gas.

224

9 Periodic orbit theory for classical chaotic systems

The starting point in the construction of the approximation is to consider the generalized transfer operator (9.10): we may write -i

ra+ioo ra

pi / \

= 27± /

™ Ja (where we recall that w is a generic multiplicative weight along the trajectory). If the system is ergodic (and the phase space volume is normalized to one) we can rewrite trClw as a time average trClw = < W(X(T), t) 6(X(T) - x(t + r))

»T

Now we write this average as a series over contributions conditioned that r E A n and r + te A n + m :

m=0

2

dz

J

*

dz

J

m 1}

"

*

d

Jo

u

Jo

dvzlP™(z -

{t-u-v)

(9.37)

where *(n) denotes n-fold involution, and where p+(z,u) is the probability that w(xtiu) = z and u + t is the next exit time, p™(z,ii) is the probability that w(xt-v,v) = z and the present interval has been entered since time v, while W(t) is the weight associated with A = t (and it is assumed to depend only on the size of the time interval). Now we neglect trClw.Q (which takes care of events happening within the same interval A^) and take the Laplace transform of the right hand side of (9.37): by taking (9.29) into account we obtain the identity

F'w(s) Fw(s)

JZodte-tfidUfdz1JdzazlI%{zl,u)zip
where we have used the convolution theorem for Laplace transforms. Now, since

JJ du J

dz2 zlP%(Zl, u)z2puL(z2,t-

u)

may be identified with til)(t)W(t), we can consistently put FZrob(s) = 1 - / dte-stW(t)ip(t) (9.39) ./o which gives the probabilistic approximation to the zeta function (Predholm determinant). We may use once again Tauberian theorems to deduce from the leading behaviour of i^(T) as T !->• oo, how F behaves near its first zero. This approach has been used for a series of investigations on the Lorentz gas with infinite horizon (see for instance (Dahlqvist, 1996a) and (Dahlqvist and Artuso, 1996)), and we believe that a number of physically challenging problems may be tackled by using it.

9.6 Conclusions

9.6

225

Conclusions

We have reviewed the foundations of the approach to chaotic dynamics based on periodic orbit expansions: we showed how the analysis of generalized evolution operators naturally leads to focus on properties of unstable periodic orbits, which represent a conjugacy-invariant skeleton over which asymptotic motion is suspended: moreover they lead to a hierarchical presentation of dynamics, induced by order with respect to their prime period. Sensible applications of this approach require a detailed understanding of the symbolic dynamics: this may hardly be considered a drawback of the method, as this seems an obliged step in any serious analysis of chaotic systems. The various thermodynamic functions, which have been so popular in the last few years, served us mainly to show how generalized zeta functions may be tailored to one's favourite average, while we discussed in some detail applications of cycle expansions to the problem of deterministic diffusion. This framework also provided a way to discuss how marginally stable orbits deeply influence the dynamics. Of course a great number of important applications could not be mentioned, and notably all applications of periodic orbits to semiclassical quantum mechanics. Also within the realm of classical chaos fundamental contributions have not been mentioned: we here provide a very partial list, which does not pretend to be complete in any respect, but just suggests further reading for anyone seriously interested in using cycle expansions as a working tool. In this review we always dealed with discrete maps: the extension of the formalism in the case of flows is described in (Cvitanovic and Eckhardt, 1991). The factorization of zeta functions when the system has discrete simmetries is discussed in (Cvitanovic and Eckhardt, 1993). On the basis of the statistical analysis analogue it is easy to realize that correlation decay is ruled by next to leading eigenvalue of the evolution operator: this is discussed in (Christiansen, Paladin and Rugh, 1990), the role of further eigenvalues is analyzed in (Christiansen, Isola, Paladin and Rugh, 1990), an alternative treatment was introduced in (Eckhardt and Grossmann, 1994). Finally it is interesting to observe that the these techniques may be projected back to statistical mechanics with success: see (Mainieri, 1992).

Acknowledgements I thank Predrag Cvitanovic for continuous interchange of ideas over the years, since early times in Copenhagen. I enjoyed very much collaborating with Erik Aurell in the first stages of cycle expansion theory, and I learned many things in the friendly environment of the Niels Bohr Institute, to which I am most grateful: I thank all the people I met and discusses with there. In the last few years Per Dahlqvist teached me a great number of things. Let me finally mention the discussions I had with Giovanni Paladin: his frank and profound attitude will be always remembered by all the people that knew him.

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9 Periodic orbit theory for classical chaotic systems

References Artuso. R. (1988), J.Phys. A21, L923 Artuso, R., Aurell, E. and Cvitanovic, P. (1990a), Nonlinearity 3, 325 Artuso, R., Aurell, E. and Cvitanovic, P. (1990b), Nonlinearity 3, 361 Artuso, R. (1991), Phys.Lett. A160, 528 Artuso, R., Casati, G. and Lombardi, R. (1993), Phys.Rev.Lett. 71, 62 Artuso, R., Casati, G. and Lombardi, R. (1994), Physica A205, 412 Artuso, R. and Strepparava, R. (1996), Phys.Lett.A (submitted) Aurell, E. and Gilbert, A.D. (1993), Geophys.Astrophys.Fluid Dyn. 73, 5 Baladi, V., Eckmann, J.-P. and Ruelle, D. (1989), Nonlinearity 2, 119 Baladi, V. (1995), in Proceedings of the NATO ASI "Real and Complex Dynamical Systems", Kluwer, Dordrecht Bunimovich, L.A. and Sinai, Ya.G. (1981), Commun.Math.Phys. 78, 479 Christiansen, F., Cvitanovic, P. and Rugh, H.H. (1990), J.Phys. A23, L713 Christiansen, F., Paladin, G. and Rugh, H.H. (1990), Phys.Rev.Lett. 65, 2087 Christiansen, F., Isola, S., Paladin, G. and Rugh, H.H. (1990), J.Phys. A23, L1301 Cvitanovic, P., Gunaratne, G. and Procaccia, I. (1988), Phys.Rev. A38, 1503 Cvitanovic, P. (1988), Phys.Rev.Lett. 61, 2729 Cvitanovic, P. (1991), Physica D51, 138 Cvitanovic, P. and Eckhardt, B. (1991), J.Phys. A24, L237 Cvitanovic, P., Gaspard, P. and Schreiber, T. (1992), Chaos 2, 85 Cvitanovic, P. and Eckhardt, B. (1993), Nonlinearity 6, 277 Cvitanovic, P. and Vattay, G. (1993), Phys.Rev.Lett. 71, 4138 Cvitanovic, P., Eckmann, J.-P. and Gaspard, P. (1995), Chaos, Solitons and Fractals , Cvitanovic, P. et al. (1996), "Classical and Quantum Chaos: a Cyclist Treatise, on http://www.nbi.dk/ predrag/QCcourse/ Dahlqvist, P. (1994), J.Phys. A27, 763 Dahlqvist, P. (1995a), Nonlinearity 8, 11 Dahlqvist, P. (1995b), "The Lyapunov Exponent in the Sinai Billiard in the Small Scatterer Limit", preprint, Royal Institute of Technology, Stockholm Dahlqvist, P. and Artuso, R. (1996), Phys.Lett. A (1996) Dahlqvist, P. (1996a), J.Stat.Phys. 84, 773 Dahlqvist, P. (1996b), "On the Effect of Pruning on the Singularity Structure of Zeta Functions", preprint, Royal Institute of Technology, Stockholm

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Dana, I. (1989), Physica D39, 205 Eckhardt, B. (1993), Phys.Lett. A172, 411 Eckhardt, B. and Russberg, G. (1993) Phys.Rev. E47, 1578 Eckhardt, B. and Grossmann, S. (1994) Phys.Rev. E50, 4571 Eckmann, J.-P. and Ruelle, D. (1980), Rev.Mod.Phys. 57, 617 Feigenbaum, M.J. (1988), J.Stat.Phys. 52, 527 Feller, W. (1966), "An Introduction to Probability Theory and Applications, Vol. IF, Wiley, New York Franceschini, V., Giberti, C. and Zheng, Z. (1993), Nonlinearity 6, 251 Fujisaka, H. (1983), Progr.Theor.Phys. 70, 1264 Gallavotti, G. (1976) Accad.Lincei Rend.Sc.ns.mat. e nat. 61, 309 Gaspard, P. and Wang, X.-J. (1988), Proc.Natl Acad.Sci. U.S.A. 85, 4591 Geisel, T. and Nierwetberg, J. (1982), Phys.Rev.Lett. 48, 7 Geisel, T. and Nierwetberg, J. (1984), Z.Physik B56, 59 Geisel, T. and Thomae, S. (1984), Phys.Rev.Lett. 52, 1936 Geisel, T. Nierwetberg, J. and Zacherl, A. (1985), Phys.Rev.Lett. 54, 616 Grassberger, P. (1983), Phys.Lett. A97, 227 Grassberger, P. and Procaccia, I. (1985), Phys.Rev. A31, 1872 Grassberger, P. and Kantz, H. (1985), Phys.Lett. A113, 235 Grebogi, C, Ott, E. and Yorke, J.A. (1988) Phys.Rev. A37, 1711 Guttmann, A.J. (1989), in C. Domb and J.L. Lebowitz (eds), "Phase Transitions and Critical Phenomena, Vol. 13", Academic Press, London Hansen, K.T. (1993a), "Symbolic Dynamics in Chaotic Systems", Ph.D. Thesis, University of Oslo Hansen, K.T. (1993b), Nonlinearity 6, 753 Hardy, G.H. and Wright, E.M. (1978), "An Introduction to the Theory of Numbers", Clarendon Press, Oxford Hentschel, H.G.E. and Procaccia, I. (1983), Physica D8, 435 Huberman, B.A., Crutchfield, J.P. and Packard, N.H. (1980), Appl.Phys.Lett. 37, 750 Isola, S. (1995), "Dynamical Zeta Functions and Correlation Functions for Nonuniformly Hyperbolic Transformations", Preprint, Bologna Kantz, H. and Grassberger, P. (1985), Physica D17, 75 Klages, R. and Dorfman, J.R. (1995), Phys.Rev.Lett. 74, 387 Klages, R. (1996), "Deterministic Diffusion in One-Dimensional Chaotic Dynamical Systems", Wissenshaft und Technik Verlag, Berlin

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Mainieri, R. (1992), Phys.Rev. A45, 3580 Mainieri, R. (1995), Physica D83, 206 Mayer, D. (1980), "The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics", Lecture Notes in Physics, Vol. 123, Springer, Berlin Milnor, J. and Thurston, W. (1988), in Lecture Notes in Mathematics No. 1342, Springer, Berlin Morriss, G. and Rondoni, L. (1994), J.Stat.Phys. 75, 553 Parry, W. (1986), Commun.Math.Phys. 106, 267 Pomeau, Y. and Manneville, P. (1980), Commun.Math.Phys. 74, 189 Prellberg, T. (1991), "Maps of the Interval with indifferent fixed points:thermodynamic formalism and phase transitions", Ph.D. Thesis, Virginia Polytechnic Institute Prellberg, T. and Slawny, J. (1992), J.Stat.Phys. 66, 503 Ruelle, D. (1978), " Thermodynamic Formalism", Addison-Wesley, Reading MA Ruelle, D. (1986), J.Stat.Phys. 44, 281 Rugh, H.H. (1996), "Intermittency and Regularized Fredholm Determinants", Preprint, Warwick Schell, M., Eraser, S. and Kapral, R. (1982), Phys.Rev. A26, 504 Vance, W.N. (1992), Phys.Rev.Lett. 69, 1356 Wang, X.-J. (1989), Phys.Rev. A40, 6647

Applications of Chaos Control

10 Synchronization in Chaotic Systems, Concepts and Applications L. M. Pecora1, T. L. Carroll1, J. F. Heagy2 1. Code 6343, Naval Research Laboratory, Washington, DC 20375, USA 2. Computer Sciences Corporation, System Sciences Division, GT II, 10110 Aerospace Road, Lanham, MD 20706

10.1 Introduction and Motivation Descriptions of chaos as unpredictable, noise-like behavior suggested early on that such behavior might be useful in some type of secure communications. One glance at the Fourier spectrum from a chaotic system will suggest the same. There are typically no dominant peaks, no special frequencies. The spectrum is broad-band. How then would one use a chaotic signal in communications? A first approach would be to hide a message in a chaotic carrier and then extract it by some nonlinear, dynamical means at the receiver. If we are to do this in real time, we are immediately led to the requirement that somehow the receiver must have a duplicate of the transmitter's chaotic signal or, better yet, synchronize with the transmitter. In fact, synchronization is a requirement of many types of communication, not only chaotic possibilities. If we look at how other signals are synchronized we will get very little insight as to how to do it with chaos. Thus we are forced to find new synchronization methods. There have been suggestions to use chaos in robotics or biological implants so as to achieve more nature movement or system behavior than one would find in periodic mechanical motion. Inevitably, this also often leads to the problem of synchronizing chaotic systems when we desire separate parts of such devices to move chaotically, but the same way at the same time. For simplicity we would like to be able to achieve such synchronization using a minimal number of signals between the synchronous parts; one signal passed between them would be best. Even when we're not synthesizing chaotic systems we have spatiotemporal systems in which we are often faced with the study of the transition from spatially uniform motion to spatially varying motion, perhaps even spatially chaotic. For example the Belousov-Zhabotinskii chemical reaction can be chaotic, but spatially uniform in a well-stirred experiment [1], This means that all spatial sites are synchronized with each other - they are all doing the same thing at the same time, even if it is chaotic motion. But in other circumstances the uniformity can become unstable and spatial variations can surface. Such uniform to non-uniform bifurcations are common in spatiotemporal systems. How do such transitions occur? What are the charakteristics of these bifurcations? We are asking physical and dynamical questions regarding synchronized, chaotic states. All

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three of the previous examples strongly suggest that we need a way to view the synchronous state and analyze its properties. Early work on synchronous, coupled chaotic systems was done by [2,3]. In that work, some sense of how the dynamics might change was brought out by a study of the Lypunov exponents of synchronized, coupled systems. Although, Yamada et al were the first to exploit local analysis for the study of synchronized chaos, their papers went relatively unnoticed. Later, a now-famous paper by Afraimovich, Verichev, and Rabinovich [4] exposed many of the concepts necessary foe analyzing synchronous chaos, although it wasn't until many years later that wide-spread study of synchronized chaos [5-10] to develop a geometric view of this behavior.

10.2 The Geometry of Synchronization 10.2.1 Simple Examples Let's look at two simple examples. The first is a nonautonomous system made from two similar subsystems. This is very similar to the system studied first by Afraimovich et al. It consists of two driven Duffing oscillators [11],

-j£ = Asint - ky{ - ax{ + x? + Cu

(10.2)

for i=l,2, plus a coupling term Ci which equals c(y2 — yi) for i=l and c(yi — 2/2) for i=2, where c is the scalar coupling strength. The coupling acts like negative feedback. If we set c=0, and A=7.5, a=0.2,fc=0.05we get two independent chaotic Duffing systems. The dynamical variables (x and y) in each system remain uncorrelated with each other. The attractor of the system occupies the full 4-dimensional space (plus an extra dimension for the time coordinate). If we set c=0, we see a new behavior set in. As t -> 00, | Xi — x 0 and | 2/1 — 2/2 |-> 0. We now have a set of synchronized, chaotic systems. The dynamical variables in one system are equal to their counterparts in the other. More importantly, we can get an idea of what the geometry of the synchronous attractor looks like in phase space. Typical figures displaying synchronous systems usually look like Fig. 10.1, just a 45° line showing that a variable from one system equals its counterpart in the other system at all times. A better way to display the geometry of the synchronous attractor in the total phase space is suggested by Fig. 10.2. There the plot of x\, y\, y2 suggests that the trajectory or attractor is constrained to a plane in the phase space. This is correct and is easily generalized. Two identical systems are in identical synchronization (IS) if the attractor lies on a hyperplane whose dimension is strictly less than the full phase space dimension. This holds even if we do not have the same number of variables in each system. For example in our original work we used the following approach to synchronizing chaotic systems [5,6,8,9,12- 14]. We started

10.2 The Geometry of Synchronization

231

yi Figure 10.1 Typical display of synchronization between two systems using only one variable from each.

with two identical systems and completely replaced one of the variables in one system (called the response) by its counterpart in the other system (called the drive). As the names suggest the coupling is one-way, as opposed to the two-way coupling in the Duffing systems above and we will call this particular one-way approach complete replacement (CR) drive-reponse. If we do this with the Lorenz system [15,16] and let the x variable in the reponse be replaced by the x in the drive we have the following: = on (2/1 -

xi =x2 y2 = -x2z2 z2 =x2y2

+ r2x2 - 2/2 -b2z2,

(10.3)

which is a 5-dimensional (5D) dynamical system. When we run the system we find that \yi — y2\ —> 0 and \z\ — z2\ —> 0. Hence, we end up with the constraints 2/i — 2/2 and %\ — %2 with (in a sense) x arbitrary. The motion takes place on a 3D hyperplane in a 5D space.

10.2.2 Some Generalizations and a Definition of Identical Synchronization We can immediately make another generalization about the synchronization manifold. There is synchronization in any system, chaotic or not, if the motion is continually confined to a hyperplane in phase space. To see this note that we can change coordinates with a constant linear transformation and keep the same

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geometry. These transformations just represent changes of variables in the equations of motion. Likewise, we can assume that the hyperplane contains the origin of the coordinates since this is just a simple translation which also maintains the geometry. The result of these observations is that the space orthonogal to the synchronization manifold, which we will call the transverse space, has coordinates which will be zero when the motion is on the synchronization manifold. Simple rotations between pairs of synchronization manifold coordinates and transverse manifold coordinates will then suffice to give us sets of paird coordinates which are equal when the motion is on the synchronization manifold as in the examples above. There is one other general property that we will note, since it can eliminate some confusion. The property of having a synchronization manifold is independent of whether the system moves onto that manifold when started off of it. The latter property is related to stability and we take that up below. In our Duffing example, if we start the systems on the synchronization manifold (yi = y
10.3 The Dynamics of Synchronization 10.3.1 Stability and the Transverse Manifold Stability for One-Way Coupling or Driving In our example of two synchronized Lorenz systems we noted that the differences |2/x — 2/21 —^ 0 and \z\ — 221 —> 0 went to zero in the limit of t —> 00. Why does this happen? The answer is to be found in the stability of the synchronization manifold. Let's transform to a new set of coordinates: x\ stays the same and we let y±=(yi - 2/2), y\\=(yi + 2/2), z±=(zi - z2), and z\\=(zx + z2). What we have

10.3 The Dynamics of Synchronization

233

done here is to transform to a new set of coordinates in which three coordinates are on the synchronization manifold (x\,y\\, and z\\) and two are on the transverse manifold (y± and z±_). We see that, at the very least, we need to have y± and z± go to zero as t ->• oo, that is, we want to guarantee that small differences will die out in time and the systems will synchronize. Thus, the zero point (0,0) in the transverse manifold must be a fixed point within that manifold (or equivalently, within the transverse dynamical subsystem). We reach our first stability requirement: perturbations in the direction of the transverse manifold must die out. This leads to requiring that the dynamical subsystems dy±/dt and dz± be stable at the (0,0) point. In the limit of small perturbations we end up with typical variational equations for the response, where we approximate the differences in the vector fields F by the Jacobian DF = dFi/dxj :

( ; ) (

1

^ ) ( l ) . ao.4)

where y± and z± are considered small. Solutions of these equations will tell us about the stability. The most general and, it appears the minimal condition for stability is to have the Lypunov exponents of Eq. (10.4) be negative for the transverse subsystem. We easily see that this is the same as requiring the response subsystem 2/2 and z
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10 Synchronization in Chaotic Systems, Concepts and Applications

System Lorenz

Driving variable X

y z

Rossler

X

y z

Response System (y,z) (x,z) (x,y) (y,z) (x,z) (x,y)

Cond.Lypunov exponents (-2.5, -2.5) (-3.95, -16.0) (+0.0, -17.0) (+0.2, -8.89) (-0.056, -8.81) (+0.0108, -11.01)

Table 10.1 Conditional Lyapunov exponents for various CR drive-response setups. Lorenz parameters a = 16,6 = 4, r = 45.92 and Rossler parameters a = 02fe = 0 2 c = 90

before let x±=(xi - x 2 ), x\\-(xi + x 2 ), y±=(yi - 2/2), y\\=(yi + 2/2) and examine the equations for x± and y± in the limit where these variables are very small. This leads to a variational equation as before, but one which now includes the coupling. This is best seen when writing the equations in a general form first, since this form will be generic. z± |= (x±,y±) and (F* = (yi, Asint-kyi — axi+x?),i = 1,2). As before DF is the Jacobian. Then we have for small pereturbations in the transverse manifold, (10.5) Note that the coupling matrix C is included in the equations for stability. The hope in such coupling schemes is that for positive scalar coupling the coupling term will decrease the Lypunov exponents, perhaps to the point in which they are all negative and the transverse manifold is stable at the origin, implying the synchronous state is stable. As we shall see the effect of coupling is not always so straightforward. In the end we must still solve Eq. (10.5) to determine the real exponents' values. Eq. (10.5) is actually quite general. It occurs in almost all mutual, diffusive coupling schemes. In larger arrays of diffusively coupled, chaotic, oscillators which have a shift invariance (we can translate the oscillators in the array and still have the same, equivalent array), we have shown that one is led to solve several equations like Eq. (10.5) to determine the stability of the synchronous state. An example of this would be a set of chaotic systems diffusively coupled in a ring as in Fig.10.3. If there are n identical (or nearly so) systems we will get one variational equation on the synchronization manifold which will give the usual Lypunov exponents of a single, chaotic subsystem, and n-1 equations like Eq.(10.3) with different expressions for C depending on the coupling and the discrete, spatial Fourier transform applied to the system. For more details and proofs, see [10]. Another interesting paper that examines parameter details and their effects on stability of synchronous, chaotic arrays is Ref. [20]. Also, consult the papers [2,3,21-30] for various viewpoints on stability of synchronous states. For all these cases on the synchronization manifold the variational equation is just the same as for the individual subsystems before

10.3 The Dynamics of Synchronization

235

Figure 10.2 More realistic view of synchronization between two dynamical systems showing that the motion is confined to a (flat) hyperplane.

they were coupled. That is, generically, dx/dt = DF.x and we find the usual single system spectrum of exponents. The interesting thing that has emerged in the last several years of research is that the two general methods we have shown so far for linking chaotic systems to obtain synchronous behavior are far from the only approaches. In the next section we show how one can design several versions of synchronized, chaotic systems.

10.3.2 Synchronizing Chaotic Systems, Variations on Themes Cascaded Drive-Response Synchronization Once one views the creation of synchronous, chaotic systems as simply "linking" various systems together a "building block" approach can be taken to producing other types of synchronous systems. For example, our drive-response approach for 3D subsystems, above, can be viewed as in Fig. 10.4. We can quickly build on this and produce and interesting variation which we call a cascaded drive-response system (see Fig. 10.5). Now, provided each response subsystem is stable (has neg-

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10 Synchronization in Chaotic Systems, Concepts and Applications

Equation of motion for each node, i=l,...,6 Figure 10.3 Schematic of a ring of diffusively-coupled, chaotic systems. The typical equation of motion for one subsystem or node is shown with the coupling to nearest neighbors. Such a system is shift invariant when synchronized: if we translate all systems around the ring the same amount of steps, we get the same system again. This means we can analyze the stability of the synchronous state using Fourier methods (see Ref. [10]) and get transverse manifold equations like Eq. 10.4.

ative conditional Lypunov exponents), both responses will synchronize with the drive and with each other. A potentially useful outcome is that we have reproduced the drive signal, x\ by the synchronized x$. Of course, we have xi=x% only if all systems have the same parameters. If we vary a parameter in the drive, the difference x\ — x 3 will become non-zero. However, if we vary the responses' parameters in the same way as the drive, we will keep the null difference. Thus, by varying the response to null the difference, we can follow the internal parameter changes in the drive. If we envision the drive as a transmitter and the response as a receiver, we have a way to communicate changes in internal parameters. We have shown how this will work in specific systems (e.g. Lorenz) and implemented parameter variation and following in a real set of synchronized, chaotic circuits [6].

10.3 The Dynamics of Synchronization

yi

y2

z

z2

i

Drive

237

Response

Figure 10.4 Building block view of a synchronous system using a drive-response approach.

Nonautonomous Synchronization Nonautonomous synchronization has been accomplished in several nonautonomous systems and circuits [11,16,31-35], but the more difficult problem of synchronizing two nonautonomous systems with separate, but idential forcing functions has not been treated except for the work by Carroll and Pecora [7]. In this system we start out with a cascaded version of a three-variable, nonautonomous system so as to reproduce the incoming driving signal when the systems are in synchronization (see ??). As in the cascaded, parameter variation scheme when the phases of the limit-cycle forcing functions are not the same we will see a deviation from the null in the difference x\ —x$. We can use this deviation to adaptively correct the phase of the response forcing to bring it into agreement with the drive. A good way to do this is to use a Poincare section consisting of x\ and xs which is "strobed" by the response forcing cycle. If the drive and response are in sync, the section will center around the origin. If the phase is shifted with respect to the drive, the points will cluster in the first or third quadrants depending on whether the response phase lags or leads the drive phase, respectively. The shift in Poincare points will be roughly linear and, hence, we know the magnitude and the sign of the phase correction. This has been done in a real circuit. See Ref. [7] for details.

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Partial Replacement Synchronization In the drive-response scenario thus far we have replaced one of the dynamical variables in the response completely with its counterpart from the drive (CR driveresponse). We can also do this in a partial manner as shown by [36]. In the partial substitution approach we replace a response variable with the drive counterpart only in certain locations. The choice of locations will depend on which will cause stable synchronization and which are accessible in the actual physical device we are interested in building. An example of replacement is the following system based on the Lorenz system,

xi =a(yi-xi),

yi= Rxi-yi-xizi,

zi=x1y1-bz1

X2 =
2/2 = Rx2 - 2/2 - £ 2 2 2 ,

z2 - Z22/2 - bz2.

(10 6)

Note the underlined driving term 2/1 in the second system. The procedure here is to just replace 2/2 in this term and not in the entire response. This leads to a variational Jacobian for the stability which is now 3 x 3 , but with a zero where 2/1 is in the ±2 equation. This will create a stability which is in general different than CR drive-response. There may be times when this is benificial.

Unidirectional Coupling We can approach drive-response synchronization from a more general viewpoint and that is as a one-way (unidirectional) coupled system. In this case we start with two identical dynamical systems, say dx^/dt = F (1 )(xW) and dx^/dt - F<2)(x(2)), where F^)(x) = F^2^(x). Then we modify F^2) so that it also depends on x^1) and so that F (1 )(x (1) ) = F ( 2 '(x ( 2 ) ,x ( 1 ) whenever x(2> = x(1>. There are obviously very many ways to do this and we concentrate on the most common, diffusive coupling. This is also known as negative feedback or control in other fields. We modify F(2)(x(2)) by adding the term C(x^) — x(2)), where C is a constant matrix. The most common couplings are to have C be a multiple of the unit matrix or to have one non-zero element located on the diagonal so that coupling is only between one pair ofcomponents of x^ and x^2^. An example of the latter is 2.2 except with C\. Good references that exploit this approach are [28,37,38]. Asimilar approach is taken for controlling chaotic systems by Pyragas [39]. Many other paper have shown variations and generalizations of the diffuse coupling appproach [37,40-44]. The stability of unidirectional coupling is much like the stability of diffusively coupled systems. We get the variational equation much like Eq. 10.5, except that the coupling matrix does not have a factor of 2 in front of it: dz±/dt = (DF-C)z±,

(10.7)

This means that the many of the issues for the stability that arrise in mutually coupled systems (see Section V. below) will also arise in many drive-response scenarios. One other thing that should be pointed out is that the case of CR drive-response is a limiting case of the unidirectional, diffusive coupling setup when we only couple one pair of drive-response components. Namely, when we take the coupling

10.4 Synchronous Circuits and Applications

239

constant to be infinite, we effectively slave that one response variable to its drive counterpart and force it to equal the drive variable. This is the same effect as the complete replacement strategy we first showed. For example, for a Lorenz system with unidirectional, diffuse coupling between x components, x\ - ^1(2/1 ~ xi)

X2 = ^2(2/2 - x2) + c(xi - x2) yi

y2 =-x2z2-{-r2x2 z2 - x2y2 - b2z2

- y2

,

(10.8)

the general stability equation will be Eq. 10.7 with C=diag{0,c,0}. If we let c ->• 00, we will end up with Eq. 10.3, since in the limit of large c we will be forcing (slaving) X2 —> x\. Now, it may seem intuitive that in the c —> 00 limit we will also force the systems to synchronize. As we will see below this is not always the case and the intuitive ideas of sharp thresholds for synchronization and simple bifurcations no longer hold. Several counter-intuitive behaviors exist when we deal with synchronizing chaotic systems.

Control Theory Approaches Recently experts in control theory have begun to apply control-theory concepts to the task of synchronizing chaotic systems. We won't go into details here, but good overviews and explanations on the stability of such approaches can be found in [45-48]. A control theory approach to observing a system is a similar problem since often the underlying goal is the synchronization of the observer with the observed system so the observed system's dynamical variables can be determined. So and Ott follow such approaches in [49]. Finally, an experimental application of control theory ideas to a particular system that is much like the Lorenz system is [50].

10.4 Synchronous Circuits and Applications When developing the mathematical theory of synchronization, one needs to ask how relevant synchronization is for the physical world. It is necessary to study synchronization in real, physical experiments to confirm that the theory holds even in the presence of noise or mismatch between systems. Some of the easiest physical systems to experiment with are electronic circuits. One can define a circuit to a fairly high precision and one can measure any system variable. We have used many simple circuits to demonstrate the synchronization of chaotic systems and to explore possible applications for synchronized chaos.

Simple Synchronization Circuit If one drives only a single circuit subsystem to obtain synchronization, as in Fig. 10.4, then the response system may be completely linear. Linear circuits have been well studied and are easy to match. Fig. 10.7 is a schematic for a simple chaotic driving circuit driving a single linear subsystem [51]. This circuit is similar to the

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10 Synchronization in Chaotic Systems, Concepts and Applications

x

x3

l

i

y2 z2

P

p

P

Drive

Response 1

Response 2

Yi z

Z

3

Figure 10.5 Building block view of cascaded synchronous system. Parameter p is varied in the drive causing the difference x\ — X2 to deviate from the null. By varying p simultaneously in the two responses the x differences can be nulled again and the receivers (responses 1 and 2) now know the internal parameter value of p in the drive, something that is not obvious from the transmitted signal xi.

circuit that we first used to demonstrate synchronization [5] and is based on circuits developed by Newcomb [52]. The circuit may be modeled by the equations: ^

= a [-1.35xi + 3.54x2 + 7.8g (z2) + 0.77zi] ^

=/3[2a?i+1.35a? 2 ].

(10.9)

The function g{x2) is a square hysteresis loop which switches from -3.0 to 3.0 at x2 = -2.0 and switches back at x2 = 2.0. The time factors are a = 103 and /3 — 102. Eq. 10.9 has two x\ terms because the 2nd x\ term is an adjustable damping factor. This factor is used to compensate for the fact that the actual hysteresis function is not a square loop as in the g function. The circuit acts as an unstable oscillator coupled to a hysteretic switching circuit. The amplitudes of x\ and xi will increase until X2 becomes large enough to cause the hysteretic circuit to switch. After the switching, the increasing os-

10.4 Synchronous Circuits and Applications

241

dilation of x\ and x2 begins again from a new center. The response circuit in Fig. 10.7 consists of the x2 subsystem along with the hysteretic circuit. The X\ signal from the drive circuit is used as a driving signal. The signals x'2 and x's are seen to synchronize with x2 and xs. In the synchronization, some glitches are seen because the hysteretic circuits in the drive and response do not match exactly. Sudden switching elements, such as those used in this circuit, are not easy to match. The matching of nonlinear elements is an important consideration in designing synchronizing circuits.

phase adjuster

Drive

Response 1

Response 2

Figure 10.6 Nonautonomous synchronization using 3D systems driven by a sinusoidal signal. The drive and responses have their own phases for the sinusoid (
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10 Synchronization in Chaotic Systems, Concepts and Applications

x2'

response

Figure 10.7 Chaotic drive and response circuits for a simple chaotic system described by Eq. 10.9

Cascading Circuit When all the drive subsystems are reproduced in the response circuit, we are able to cascade the subsystems and reproduce all of the drive signals, as in Eq. 10.3. It is important in a cascaded response circuit to reproduce all nonlinearities with sufficient accuracy, usually within a few percent, to observe synchronization. Many nonlinear elements available for circuits depend on material properties, which vary considerably between different batches of the material. We have designed circuits around piecewise linear functions generated by diodes and op amps. These nonlinear elements (originally used in analog computers [53] are easy to reproduce. Fig. 10.10 shows schematics for drive and response circuits similar to the Rossler system but using piercewise linear nonlinearities [54]. The drive circuit may be described

10.4 Synchronous Circuits and Applications

243

by ^f = -a (x - 7j/ + 0.022/)

% = -a(-g(x)-z) 0

(10.10)

r <

where the time factor a is 104 s" 1 , T is 0.05, (3 is 0.5, A is 1.0 , 7 is 0.133 and // is 15. In the response system, the y signal drives the (x,z) subsystem, after which the y subsystem is driven by x' and y to produce y" . The extra factor of 0.02 y in the second of Eq. 10.10 becomes 0.02 y" in the response circuit in order to stabilize the op amp integrator. Cuomo-Oppenheim Communications Scheme Cascading synchronization was first applied to a simple communications scheme by Cuomo and Oppenheim [55,56]. They built a circuit version of the Lorenz equations Eq. 10.3 using analog multiplier chips. They transmitted the x signal from their drive circuit and added a small speech signal. The speech signal was hidden under the broad band Lorenz signal in a process known as signal masking. At their cascaded receiver, difference x — x1 was taken and found to be approximately equal to the masked speech signal (as long as the speech signal was small). Other groups later demonstrated other simple communications schemes [57-60]. It has been shown that the simple chaotic communications schemes are not secure [61,62]. Other encoding schemes may be harder to break, although one must consider that decryption usually works by finding patterns, and chaotic systems, because there are deterministic, are pattern generators. Nonautonomous and Filtered Synchronization Circuit The chaotic synchronization mentioned above may be extended to nonautonomous chaotic systems, such as periodically forced systems. Nonautonomous synchronization was described in section 2 above. We have also demonstrated that it is possible to filter the chaotic drive signal through a comb filter that removes the forcing frequency and its harmonics [63]. The periodic parts of the signal may be restored using the concepts of synchronous substitution described below in section VI. Filtering is useful when the periodic components in the power spectrum of the transmitted signal are large compared to the rest of the spectrum, as is often true for periodically forced circuits. In principle filtering out parts of the transmitted signal should also work for autonomous circuits, but, as will be seen in section VI, the response system must be stable when synchronous substitution is used to restore the drive signal. Restoring the signal in an autonomous system tends to destabilize one or more periodic orbits, depending on the filter profile.

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Volume-Preserving Circuit Most of the chaotic systems we describe here are based on flows. It may also be useful to work with chaotic circuits based on maps. Using map circuits allows us to simulate volume-preserving systems. Since there is no attractor for a volumepreserving map, the map motion may cover all of the phase space, generating very broad band signals. It seems counter intuitive that a non-dissipative system may be made to synchronize, but in a multidimensional volume-preserving map, there must be at least one contracting direction so that volumes in phase space are conserved. We may use this one direction to generate a stable subsystem. We have used this technique to build a set of synchronous circuits based on the standard map [64]. In hyper chaotic systems, there are more than one postive Lypunov exponent and for a map this may mean that the number of expanding directions exceeds the number of contracting directions, so that there are no simple stable subsystems for a one-drive setup. We may, however, use the principle of synchronous substitution (described in section VI below) to generate various synchronous subsystems. We have built a circuit to simulate the following map [65]:

V»+i = (|) yn + zn = Xn + Vn

\ mod 2 - 4 ,

(10.11)

where "mod(2)-4" means take the result modulus 12 and subtract 4. This map is quite similar to the cat map [66]. The Lyapunov exponents for this map (determined from the eigenvalues of the Jacobian) are 0.683, 0.300, and -0.986. We may create a stable subsystem of this map using the method of synchronous substitution [67]. We produce a new variable wn = zn + Txn from the drive system variables, and reconstruct a driving signal zn at the response system: wn = zn + Txn 2/n+l = (|)2/n + 2n, where the modulus function is assumed. In the circuit, we used T= -4/3, although there is a range of values that will work. We were able to synchronize the circuits adequately in spite of the difficulty of matching the modulus functions. The transmitted signal from this circuit has essentially a flat power spectrum and a delta-function autocorrelation, making the signal a good replacement for a conventional pseudo-noise signal. Our circuit is in essence a self-synchronizing pseudo-noise generator.

10.5 Stability and Bifurcations of Synchronized, Mutually Coupled Chaotic Systems

245

10.5 Stability and Bifurcations of Synchronized, Mutually Coupled Chaotic Systems 10.5.1 Stability for Coupled, Chaotic Systems Let's examine the situation in which we have N diffusively coupled, n-dimensional chaotic systems: = F(x«)

(10.13)

where z=l,2,...,iV and the coupling is circular (N 4- 1=1). We want to examine the stability of the transverse manifold when all the "nodes" of the system are in synchrony. This means that x^1) = x^2^ = ... = x ^ which defines an ndimensional hyperplane, the synchronization manifold. We show in Ref. [10] that the way to analyze the tranverse direction stability is to transform to a basis in Fourier spatial modes. We write A& = (l/N)J2ix^e~2nlk^NWhen N is even (which we assume for convenience), we have N modes which we label with k = 0, 1, ...N/2. For k=0 we have the synchronizouse mode equation, since this is just the average of identical systems: (10.14)

= F(A),

which governs the motion on the synchronization manifold. For the other modes we have equations which govern the motion in the transverse directions. We are interested in the stability of these modes (near their zero value) when their amplitudes are small. This requires us to contruct the variational equation with the full Jacobian analogous to Eq. 10.5. In the original x^) coordinates the Jacobian is, DF-2C C 0

C DF-2C C

0 C 0 DF-2C C 0

\

(10.15)

C DF-2C/

but in the mode coordinates the Jacobian is block diagonal, which simplifies finding the stability conditions, fDF 0

0 0 £>F-4Csin2[7r/iV] 0 0 £>F-4Csin2[7rA;/iV]

\ (10.16)

where each value of k ^ 0 or ^ N/2 occurs twice, once for the "sine" and once for the "cosine" modes. To put it another way, we want the transverse modes represented by sine and cosine spatial disturbances to die out, leaving only the

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10 Synchronization in Chaotic Systems, Concepts and Applications

Response

Drive

Figure 10.8 Piecewise linear Rossler circuits arranged for cascaded synchronization. Rl=100 kfi, R2=200 kft, R3=R13 =2Mft, R4=75 kft, R5=10 kfi, R6=10 kft, R7=100 kO, R8=10 kfi, R9=68 kft, R10=150 kQ, Rll=100 kft, R12=100 kQ Cl=C2=C3=0.001 ^F, and the diode is a type MV2101.

k = 0 mode on the synchronization manifold. At first sight what we want for stability is for all the blocks with k / 0 to have negative Lypunov exponents. We will see that things are not so simple, but let us proceed with this naive view. Fig. 10.9 shows the naive view of how the maximum Lypunov exponent for a particular mode block of a transverse mode might depend on coupling c. There are four features in the naive view that we will focus on. One is that as the coupling increases from 0 we go from the Lypunov exponents of the free oscillator to decreasing exponents until for some threshold coupling csync the mode becomes stable. Two, above this threshold we have stable synchronous chaos. Three, we suspect that as we increase the coupling the exponents will continue to decrease. Four, we can now couple together as many chaotic oscillators as we like using a coupling c > c8ync and always have a stable synchronous state. Below we will use a particular coupled, chaotic system to show that there are counter-examples to all four of these "features". We first note a scaling relation for Lypunov exponents of modes with different fc's. Given any Jacobian block for a mode k\ we can always write it in terms of the block for another mode &2, viz: DF - 4Csin2

= DF - 4C

sin2

(10.17)

.Hence, where we see that the effect is to shift the coupling by the factor gl given any mode's stability plot (as in Fig. 10.9) we can obtain the plot for any other

10.5 Stability and Bifurcations of Synchronized, Mutually Coupled Chaotic Systems

247

Stability for the kth mode (naive view of maximum Lypunov exponent)

max

sync Figure 10.9 Naive view of the stability of a transverse mode in an array of synchronous chaotic systems as a function of coupling c.

mode by rescaling the coupling. In particular, we need only calculate the maximum Lypunov exponent for mode 1 Ajnaa. and then the exponents for all other modes k > 1 are generated by "squeezing" the \max plot to smaller coupling values. This scaling relation, first shown in [10], shows that as the mode's Lypunov exponents decrease with increasing c values the longest wavelength mode, k\ will be the last to become stable. Hence, we first get the expected result that the longest wavelength (with the largest coherence length) is the least stable for small coupling.

10.5.2 Coupling Thresholds for Synchronized Chaos and Bursting To test our ideas we examine the following system of four Rossler-like oscillators diffusively coupled in a circle which has a counterpart in a set of four circuits we built for experimental tests [10], = a[g{x)

_

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10 Synchronization in Chaotic Systems, Concepts and Applications

where g(x) is a piece-wise linear function which "turns on" when x crosses a threshold and causes the spiralling out behavior to "fold" back toward the origin,

For the values a = 104 s " \ T = 0.05, 0 = 0.5, A = 1.0, 7 = 0.133, and /x = 15.0 we have a chaotic attractor very similar to the Rossler attractor (see Fig. 10.11). We couple four of these circuits through the i/-component by adding the following term to each system's y equation: c(y^+1^ + y^~1^ — 2y(^), where the indices are all mod 4. This means the coupling matrix C has just one nonzero element, C22 = c. A calculation of the mode Lypunov exponents indeed shows that the longest wavelength mode becomes stable last at csync=0.063. However, when we examine the behavior of the so-called synchronized circuits above the threshold we see unexpected behaviors. If we take x to be the instantaneous average of the 4 circuits' x components then a plot of the difference of circuit x^ from the average d = x^ — x versus time should be « 0 for synchronized systems. Such a plot is shown for the Rossler-like circuits in Fig.10.14. We see that the difference d is not zero and shows large bursts. These bursts are similar in nature to on-off intermittency [68-70]. What causes them? Even though we are above the Lypunov exponent threshold, c8ync we must realize that this exponent is only an ergodic average over the attractor. Hence, if the system has any invariant sets which have stability exponents greater than the Lypunov exponents of the modes, even at couplings above csync these invariant sets may still be unstable. When any system wanders near them the tendency will be for individual systems to diverge by the growth of that mode which is unstable on the invariant set. This causes the bursts in Fig.10.14. We have shown that the burst can be directly associated with UPO's in the Rossler-like circuit [71]. These burst do subside at greater coupling strengths, but eveb then some deviations can still be seen which may be associated with unstable portions of the attractor which are not invariant sets (e.g. part of a UPO). The criteria for guaranteed synchronization is still under investigation [72,73], but the lesson here is that the naive views (one and two, above) that we have a sharp threshold for synchronization and that above that threshold synchronization is guaranteed, are incorrect. The threshold is actually a rather "fuzzy" one. It might be best drawn as an (infinite) number of thresholds [74,75]. This is shown in Fig. 10.13, where we are plotting a more realistic picture of the stability diagram near the mode 1 threshold. We see that, at the minimum we need to have the coupling be above the highest threshold for invariant sets (UPO's and unstable fixed points). A better synchronization criteria, above the invariant sets one, has been suggested by Gauthier et al. [73]. Their suggestion, for two diffusively coupled systems (x^1) is to use the criteria d\Ax\/dt. A similar suggestion regarding "monodromy" in perturbation decrease was put forward by Kapitaniak [76]. There would be generalizations of this to mode analysis for N coupled systems, but these have not been worked out. Research is still on- going in this area [72].

10.5 Stability and Bifurcations of Synchronized, Mutually Coupled Chaotic Systems

249

Figure 10.10 Schematic of the Rossler-like circuit system with nearest neighbor diffusive coupling.

10.5.3 Desynchronization Thresholds at Increased Coupling Let us look at the full stability diagram for modes 1 and 2 for the Rossler-like circuit system when we couple with the x coordinates diffusively, rather than the y's. This is shown in Fig. 10.14. Note how the mode-2 diagram is just a rescaled mode-1 diagram by a factor of 1/2 in the coupling range. We can now show another, counterintuitive feature that we missed in our naive view. Fig. 10.14 shows that the modes go unstable as we increase the coupling. The synchronized motion is Lypunov stable only over a finite range of coupling. Increasing the coupling does not necessarily guarantee synchronization. In fact we couple the system by the z variables we will never get synchronization, even when c = oo. The latter case of infinite coupling is just the CR drive-response using z. We already know that in that regime both the z and x drivings do not cause synchronizations in the Rossler system. We now see why. Coupling through only one component does not guarantee a synchronous stata and we have found a counterexample for number three in our naive views, that increasing the coupling will guarantee a synchronous state. 10.14. Now, let's look more closely at how the synchronous state goes unstable. In

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10 Synchronization in Chaotic Systems, Concepts and Applications

Figure 10.11 Rossler-like circuit attractor.

finding the csync threshold we noted that mode 1 was the most unstable and was the last to be stabilized as we increased c. Near Cdesync we see that the situation is reversed: mode 2 goes unstable first and mode 1 is the most stable. This is also confirmed in the experiment [77] where the 4 systems go out of synchronization by having, for example, system-l=system-3 and system-2=system-4 while system-1 and system-2 diverge. This is exactly a spatial mode-2 growing perturbation. It continius to rather large differences between the systems with mode-1 perturbations remaining at zero, i.e. we retain the system-l=system-3 and system-2=system-4 equalities. Since for larger systems (N > 4) the higher mode stability plots will be squeezed further toward the ordinate axis, we may generalize and state that if there exists a Cdesync upon increasing coupling, then the highest order mode will always go unstable first. We call this a short- wavelength bifurcation [77]. It means that the smallest spatial wavelength will be the first to grow above Cdesync This is counter to the usual cases where the longest or intermediate wavelengths go unstable first. What we have in the short-wavelength bifurcation is an extreme form of the Turing bifurcation [78] for chaotic, coupled systems. 10.14. Note that this type of bifurcation can happen in any coupled system where each oscillator or node has "internal dynamics" which are not coupled directly to other nodes. In our experiment, using x-coupling, y and z are internal dynamical variables. In biological modeling where

10.5 Stability and Bifurcations of Synchronized, Mutually Coupled Chaotic Systems

-Near —Burst Sync.

251

time

Figure 10.12 Bursting in the difference d between the number 1 circuit x and the average value x.

cells are coupled through voltages or certain chemical exchanges, but there are internal chemical dynamics, too, the same situation can occur. All that is required is that the uncoupled variables form an unstable subsystem and coupling can be pushed above Cdesync If this were the case for a continiuous system (which would be modeled by a PDE), then the short-wavelength bifurcation would produce a growing perturbation which had an infinitesimal wavelength. So far do not wer know of any such findings, but they would surely be of interest and worth looking for.

10.5.4 Size Limits on Certain Chaotic Synchronized Arrays When we consider the cases in which (TV > 4) we come to the following surprising conclusion which counters naive example four. Whenever there is desynchronization with increasing coupling there is always an upper limit on the number of systems we can add to the array and still find a range of coupling in which synchronization will take place. To see this examine Fig. 10.15 which comes from an N = 8 Rossler-like circuit system. We see that the scaling laws relating the stability diagrams for the modes eventually squeeze down the highest mode's stability until just as the 1st mode is becoming stable, the highest mode is going unstable. Or in other words c8ync and Cdesync cross on the c axis. Above N = 8 we never

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10 Synchronization in Chaotic Systems, Concepts and Applications

Infinite number of Synchronization Thresholds UPO's which can cause bursts

max

o

0

c

sync

C

coupling strength Figure 10.13 A more realistic view of stability vs. coupling near the mode 1 synchronization threshold.

have a situation in which all modes are simultaneously stable. In Ref. [77] we refer to this as a size effect.

10.5.5 Riddled Basins of Synchronization There is still one more type of strange behavior in coupled chaotic systems and this comes from two features. One is the existence of unstable invariant sets (UPO's) in a synchronous chaotic attractor and the other is the simultaneous existence of two at tractors, a chaotic synchronized one and another, unsynchronized one. In our experiment these criteria held just below Cdesync where we had a synchronous chaotic attractor containing unstable UPO's and we had a periodic attractor (see 10.16). In this case, insteaed of attractor bursting or bubbling, we see what have come to be called riddled basins. When the system burst apart near an UPO, they are pushed off the synchronization manifold. In this case they have another attractor they can go to, the periodic one. The main feature of this behavior is that the basin of attractor for the periodic attractor is intermingled with the synchronization basin. In fact, the periodic attractor's basin riddles the synchronized attractor's basin.

10.5 Stability and Bifurcations of Synchronized, Mutually Coupled Chaotic Systems

253

x-coupling

0

sync

desync

coupling strength Figure 10.14 The full stability diagrams for mode's 1 and 2 in the Rossler circuit.

This was first studied theoretically by Alexander et al. [79] and followed by several papers describing the theory of riddled basins [70,74,80-83]. Later direct experimental evidence for riddled basins was found by Heagy et al. [84]. Since then Lai [85] has shown that parameter space can be riddled and other's have studied the riddling Phenomena in other systems [86,87]. In our experiment with four coupled, chaotic systems we used a setup that allowed us to examine what might be called a cross-section of the riddled basin. We varied initial conditions of the four oscillators so as to produce a 2D basin map which was consistent with the short-wavelength instability that showed up in the bursts taking the overall system to the other attractor off the synchronization manifold. All z variables were set to the same value for all initial conditions. All four x componets were set to the same value which was varied from -3.42 to 6.58. A new variable u representing the mode-2 perturbation was varied from 0.0 to 7.0 for each initial condition and the y variables were set to values that matched the mode-2 waveform: yW = yW = u and y^ = y^ = —u. The variables x and u made up the 2D initial condition "grid" which was originally suggested by Ott [88]. Varying x changed all the system's x components and kept the systems on the synchronization manifold. Varying u away from zero lifted the systems from the synchronization manifold. When one of the initial conditions led to a final state of synchronization, it was colored yellow. When the final state was the periodic, nonsynchronized attractor

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10 Synchronization in Chaotic Systems, Concepts and Applications

Size Effect in Coupled Arrays

1,8 0 max

0

^ desync sync

coupling strength Figure 10.15 Stability diagrams for mode's 1 and 8 in the Rossler circuit array with 16 oscillators.

it was colored blue. Fig. 10.16 shows the result of this basin coloring for both experiment and numerical simulation [84]. The basin of the synchronized state is indeed riddled with points from the basin of the periodic state. The riddling in these systems is extreme in that even infinitesimally close to the synchronization manifold there are points in the basin of the periodic attractor. To put it another way, any open set containing part of the synchronization manifold will always contain points from the periodic attractor basin and those points will be of non-zero measure. Ott et al. [81] have shown that near the synchronization manifold the density p of the other attractor's basin points will scale as p « ua . In our numerical model we found a = 2.06 and in the experiment we found a = 2.03. The existence of riddled basins means that the final state is uncertain, even more uncertain than where there exist "normal" fractal basin boundries [89-92].

10.6 Transformations, Synchronization, and Generalized Synchronization

X

255

X

Figure 10.16 View of two simultaneous attractors in the study of riddled basins.

10.6 Transformations, Synchronization, and Generalized Synchronization The possibilty of using coordinate transformations to attain a synchronizable system, say using the CR scheme was realized in our first synchronization papers [5,8]. In that work we had a circuit (and its model) that had no stable subsystem, so we used a (linear) transformation to change the coordinates to new dynamical variables in which the new ODE had a stable response system. Although a linear transformation may not always work we can see that an arbitrary nonlinear transformations can also be used to change the system into one with a stable subsystem. To date, very little has been done to examine when this might be done and what transformations would be best. Some exceptions to this are control-theory approaches to synchronization [45-49], although most of these use an added control signal in the vector field as an attempt to gain synchronization. This is not the most general way, nor is it always likely to be the most efficient or, perhaps, attainable in real systems. More studies need to be done to work out methodologies for accomplishing this goal. In the next section we study, briefly, some other approaches to synchronization, including some attempts to generalize the synchronization scenario. Some of them are similar to transformations mentioned above.

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10 Synchronization in Chaotic Systems, Concepts and Applications

10.6.1 Synchronizing with Functions of the Dynamical Variables Using Functions of Drive Variables and Information An interesting approach to generating new synchronizing vector fields was taken by Kocarev [93,94]. This is an approach that uses an invertible function of the drive dynamical variables and the information signal to drive the response, rather than just using one of the variables itself as in the CR approach. Then on the response end we invert the function using the fact that we are close to synchronization to provide estimates of the drive variables to extract the information. Schematically, this will look as follows. On the drive end we have a dynamical system x = F(x, s), where s is the transmitted signal and is a function of x and the information i(t), s = h(x,i). On the receiver end we have an identical dynamical system set up to extract the information: x = F(x,s) and iR = /i~ 1 (y,s). When the systems are in sync iR = i.

Using Synchronous Substitution to Recover Drive Variables that are Transmitted by a General Function to a Synchronized Response Another approach we developed in Ref. [67] which is simiar to the above section comes out when we note the following opportunity. Let x be the n-dimensional drive and y be an identical system as the response, i.e. in the synchronized state yi = Xi. As in the Kocarev et at. approach above we can apply a transformation to the drive variables and create a new drive variable w = T(x\,X2, ...,x n ), where we write out all the x coordinates for a reason. At the response end we would like to recover the signals Xi, but we only have access to one scalar signal w. For example, we might want to drive one of the response components with x\.. We note that if we are near the synchronous state and if DXl ^ 0 we can "recover" x\ by using the synchronized variables 2/2> 2/3,..., yn in place of #2, #3) •••#n and solving %\ = TXl(w,y2,y3, ...,yn), where TXl is the inverse function of T with respect to x\. What we are doing here is using the synchronization of the two systems to get approximations (yi) to the variables (x^ that we don't have access to. dx

ldt = 10(y - *),

dy

/dt = ~xz + 60s - y,

dz

ldt = xy-

w = T(x,y,z) = y + x

2.667*,(10.20) (10.21)

The response system (with primed variables) is: dx

ldt = io(y - *),

d

y/dt = ~xz + 60* - y,

y = Ty{w,x,z) =w-x

dz

/dt = *v-

2.667z,(10.22) (10.23)

Note that the x1 and z1 equations are driven. The y' equation appears superfluous since the x' and z' equations do not depend on it, but it can be used to

10.6 Transformations, Synchronization, and Generalized Synchronization

257

test against a good synchronization by checking \y — y'\. Other combinations of variables and drives are possible. The use of transformations opens up a larger range of possible synchronization systems. It may also be useful in extending the types of circuit systems, where one may be limited by the existence of good, reproducible nonlinear devices. The question that remains is that of stability. Just transforming variables does not guarantee that we will get synchronization. Using the above formulation we can write the general form of the variational problem for the response stability. If the vector field of the response with the transformation is F(y,xi), then the variational equations become d6r

/dt = [DxF^^

+ DzFfrx^DxTr^Sr

(10.24)

where <5r = y — x. The first term in brackets is the usual Jacobian that results in a variational problem. The second term depends on the transformation and the synchronous substitution. The latter can cause changes in stability and allow more interesting and varied synchronization schemes to be developed. Note that the second term will have a column of zeroes in the x\ position. Obviously this approach is not limited to using the variable x\ as the drive and can be applied in situations where more than one signal, say w\ and w2, are transmitted.

10.6.2 Hyper chaos Synchronization Most of the drive-response synchronous, chaotic systems studied so far have had only one positive Lypunov exponent. More recent work has shown that systems with more than one positive Lypunov exponent (called hyperchaotic systems) can be synchronized using one drive signal. We have already seen one example above in the volume-preserving system. Here we display three other approaches. A simple way to construct a hyperchaotic system is to use two, regular chaotic systems. They need not be coupled, just the amalgam of both is hyperchaotic. Tsmiring and Suschick [95] recently made such a system and considered how one might synchronize a duplicate response. Their approach has elements similar to the use of synchronous substitution we mentioned above. They transmit a signal which is the sum of the two drive systems. This sum is coupled to a sum of the same variables from the response. When the system's are in sync the coupling vanishes and the motion takes place or an invariant hyperplane and hence is identical synchronization. An example of this situation using one-dimensional systems is the following [95]: a?i(n + 1) = x2(n+l) = / 2 (a: 2 (n)), w = fi(xi(n)) + / 2 (i2(n)) - fi(yi(n)) - / 2 fain)) = transmitted signal Vl{n + 1) = /i(yi(n)) + e(/i(m(n)) + / 2 (x 2 (n)) - fi(yi(n)) - / 2 (y 2 (n))), + e(/i(xi(n)) + / 2 (z 2 (n)) - /i(yi(n)) - / 2 (|/ 2 (n))), y2(n + 1) = f2fain)) (10.25)

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10 Synchronization in Chaotic Systems, Concepts and Applications

Linear stability analysis, as we introduced above, shows that the synchronization manifold is stable [95]. Tsimring and Sushchick investigated several onedimensional maps (tent, shift, logistic) and found that there were large ranges of coupling (e) where the synchronization manifold was stable. For certain cases they even got analytic formulae for the Lyapunov multipliers. However, they did find that noise in the communications channel, represented by noise added to the transmitted signal w, die degrade the synchronization severely causing bursting. The same features showed up in their study of a set of drive-response ODE's (based on a model of an electronic synchronizing circuit). The reason for the loss of synchronization and bursting are the same as in our study of the coupled oscillators above, namely, local instabilities which throw the systems out of sync, even about Lyapunov synchronization thresholds, since slight noise tends to keep them apart and ready to deverge when the systems visit the unstable portions of the attractors. Whether this can be "fixed" in practical devices so that multiplexing can be used is not clear. Our above study of synchronization thresholds for coupled systems suggests that for certain systems and coupling schemes we can avoid bursting, but more study of this phenomena for hyperchaotic/multiplexed systems has to be done. The issue of synchronizing hyperchaotic systems was addressed by Peng et al. [96]. They started with two identical hyperchaotic systems x = F(x) and y = F(y). Their approach was (similar to above) to use a drive signal that was a linear combination of drive variables. Following Peng et al. we write this "transmitted" signal as w = K T x and we add a coupling term to the y equations of motion: y = F(y) -h ~B(w — u), where v = K T y. This is a common approach for stabilization of systems in control theory [51,97]. Peng et al. show that for many cases one can choose K and B that the y system synchronizes with the x system. This solves a long-standing question about the relation between the number of drive signals that need to be sent to synchronize a response and the number of positiv Lypunov exponents, namely there is no relation, in principle. Many systems with a large number of positiv exponents can still be synchronized with one drive signal. Practical limitations will surely exist, however. The latter still need to be explored. Finally, we mention that synchronization of hyperchaotic systems has been achieved in experiments. Tamasevicius et al. [53] have shown that such synchronization can be accomplished in a circuit. They built circuits that consisted of either mutually coupled or unidirectionally coupled 4D oscillators. They show that for either coupling both positive conditional Lypunov exponents of the "uncoupled" subsystems become negative as the coupling is increased. They go on to further show that they must be above a critical value of coupling which is found by observing the absence of a blowout bifurcation [70,81,98]. Such a demonstration in a circuit is important since this proves at once that hyperchaos synchronization has some robustness in the presence of noise and parameter mismatch.

10.6 Transformations, Synchronization, and Generalized Synchronization

259

10.6.3 Generalized Synchronization In their original article on synchronization Afraimovich et al. investigated the possibliity of some type of synchronization when the parameters of the two coupled systems do not match. Such a situation will certainly occur in real, physical systems and is an important question. Their study showed that for certain systems, including the 2D forced system they studied, one could show that there was a more general relation between the two coupled systems. This relationship was expressed as a one-to-one, smooth mapping between the phase space points in each subsystem. To put this more mathematically, if the full system is described by a 4D vector (£1,2/1, £2,2/2) then there exists smooth, invertible function $ from (£1,2/1) to (£2,2/2).

Thus, knowing the state of one system enables one, in principle, to know the state of the other system and vice versa. This situation is similar to identical synchronization and has been called generalized synchronization. Except in special cases, like that of Afraimovich et a/., rarely will one be able to produce formulae exhibiting the mapping . Proving generalized synchronization from time series would be a useful capability and sometimes can be done. This is beyond the scope of this chapter, but the interest reader should examine Refs. [38,99,100] in which statistics are developed to determine when there are continuous and/or smooth maps between two subsystems. Recently, several attempts have been made to generalize the concept of general synchronization itself. These begin with the papers by Rul'kov et al [38,40] and on to a paper by Kocarev and Parlitz [101]. The central idea in these papers is that for the drive-response setup, if the response is stable (all Lypunov exponents are negative), then there exists a manifold in the joint drive-response phase space such that there is a function from the drive (X) to the response (Y), (j> : X -» Y. In plane language, this means we can predict the response state from that of the drive (there is one point on the response for each point on the drive's attractor) and the point of the mapping $ lie on a smooth surface (such is the definition of a manifold). This is an intriguing idea and it is an atttempt to answer the question we posed in the beginning of this chapter, namely, does stability determine geometry? These papers would answer yes, in the drive-response case the geometry is a manifold that is "above" the drive subspace in the whole phase space. The idea seems to have some verifiation in the studies we have done so far on identical synchronization and in the more particular case of Afraimovich-Verichev-Rabinovich generalized synchronization. However, there are counter-examples that show that the conclusion cannot be true. First, we can show that there are stable drive-response systems in which the attractor for the whole system is not a smooth manifold. Consider the following system, x = F(x) z = -TJZ + XI, where x is a chaotic system and 77 > 0. The z system can be viewed as a filter (LTI or low-pass type) and is obviously a stable response to the drive x. It is now known that certain filters of this type lead to an attractor in which there is a map (often

260

10 Synchronization in Chaotic Systems, Concepts and Applications

NUMERICAL SIMULATION

I g

3

COUPLED CIRCUIT EXPERIMENT

§

distance along synchronization plane Figure 10.17 "Cross-section of riddled basins near the desynchronization threshold for chaotic, Rossler-like circuits and numerical model.

called a graph) <j> of the drive to the response, but the mapping is not smooth. It is continuous and so the relation between the drive and response is similar to that of the real line and the Weierstrass function above it. This explains why certain filters acting on a time series can increase the dimension of reconstructed attractor [102,103]. We showed that certain statistics could detect this relationship [103]. Several other papers have proven the nondifferentiability property rigorously and have investigated several types of stable filters of chaotic systems [104-108]. We note that the filter is just a special case of a stable response. The criteria for smoothness in any drive-response scenario is that the least negative conditional Lypunov exponents of the response must be less than the most negative Lypunov exponents of the drive [108,109]. One can get a smooth manifold if the response is uniformly contracting, that is the stability exponents are locally always negative [108,110]. Note that if the drive is a non-invertible dynamical system, then things

References

261

are "worse". The drive-response relation may not even be continuous and may be many valued, in the latter case there is not even a function (f> from the drive to the response. There is an even simpler counter-example that no one seems to mention that shows that stability does not guarantee that exists and this is the case of period-2 behavior (or any multiple period behavior). If the drive is a limit cycle and the response is a period doubled system (or higher multiple-period system), then for each point on the drive attractor there are two (or more) points on the response attractor. One cannot have a function under such conditions and there is no way to predict the state of the response from that of the drive. Note that there is a function from the drive in this case. Actually, any drive-response system that has the overall attractor on an invariant manifold that is not diffeomorphic to a hyperplane will have the same, multi-valued relationship and there will be no function <\>. Hence, the hope that a stable response results in a nice, smooth, predictable relation between the drive and response cannot always be realized and the answer to our question of whether stability determines geometry is "no," at least in the sense that it does not determine one type of geometry. Many are possible. The term general synchronization in this case may be misleading in that it implies a simpler driveresponse relation than may exist. However, the stable drive-response scenario if obviously a rich one with many possible dynamics and geometries. It deserves more study.

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269

11 Synchronization of Chaotic Systems U. Par lit z1 and L. Kocarev2 1. Drittes Physikalisches Institut, Universitat Gottingen, Biirgerstrafie 42-44, D-37073 Gottingen, Germany 2. Department of Electrical Engineering, St. Cyril and Methodius University, Skopje, PO Box 574, Republic of Macedonia

11.1 Introduction The first observation of a synchronization phenomenon in physics is attributed to C. Huygens (1673) during his experiments for developing improved pendulum clocks (Huygens, 1673). Two clocks hanging on the same beam of his room were found to oscillate with exactly the same frequency and opposite phase due to the (weak) coupling in terms of the almost imperceptible oscillations of the beam generated by the clocks. Huygens also observed that both clocks were able to synchronize only in those cases where their individual frequencies almost coincided - a prerequisite that turned out to be typical for synchronization of periodic oscillations. Since Huygens' early observation synchronization phenomena were discovered and practically used by many physicists and engineers. Rayleigh investigated synchronous oscillations of vibrating organ tubes and electrically or mechanically connected tuning forks in acoustics (Rayleigh, 1945) and Van der Pol and Van der Mark constructed radio tube oscillators where they observed entrainment when driving such oscillators sinusoidally (Van der Pol and Van der Mark, 1926).* These phenomena were investigated theoretically by Van der Pol (Van der Pol, 1927). Synchronization plays also an important role in celestial mechanics where it explains the locking of revolution periods of planets and satellites. Furthermore, synchronization was not only observed in physics but also in (neuro-) biology where rhythms and cycles may entrain (Glass and Mackey, 1988) or synchronizing clusters of firing neurons are considered for being crucial for information processing in the brain (Schechter, 1996; Schiff et al., 1996). In this and other cases synchronization can play a functional role because it establishes some special relation between coupled systems. For periodic oscillations, for example, the frequencies of interacting systems may become the same or lock with a rational ratio due to synchronization. More complex relations can be expected and have been found for coupled chaotic systems. At first glance, synchronization of chaotic systems seems to be rather surprising because one may naively expect that the sensitive dependence on initial conditions would lead to x

This was probably also one of the first experiments where chaotic dynamics was observed, although the authors didn't investigate this aspect.

272

11 Synchronization of Chaotic Systems

an immediate breakdown of any synchronization of coupled chaotic systems. This, however, is not the case. Before possible meanings of the term "chaos synchronization" will be discussed, an example is given for illustration. This example consists of two bi-directionally coupled Lorenz systems (Lorenz, 1963; Ott, 1993; Schuster, 1995): xi x2 X3

= = =

a(x2 - a ? i ) -c(x2 rxi - x2 - X1X3 X1X2- bx3

-2/2) and

2/i 2/2 y3

= = =

^(2/2 - 2/i) + c(x2 - 2/2) r2/i - 2/2 - 2/12/3 2/12/2 - by3.

(11.1.1)

with a — 10, r = 28, and b = 2.666. The parameter c is a coupling constant. Numerically we find that for c > 1.96 both Lorenz systems (11.1.1) synchronize (lim ||x(£) — y(t)\\ = 0) and generate the wellknown Lorenz attractor. For c = a/2 t—¥OO

this can even be proved analytically. In this case the difference e\ = 2/1 — #i converges to zero, because e\ — —ae\. For x\ = 2/1 the synchronization errors e2 — 2/2 — #2 and e3 = 2/3 - £3 of the remaining variables also vanish, because the Lyapunov function L = e\ + e\ decreases monotously with \L — — e2 — be% < 0. Both coupled dynamical systems of this example were assumed to be exactly the same or identical. Therefore, it is possible to observe identical synchronization (IS) where the states of both Lorenz systems converge to the same trajectory and x(t)— y(t) —> 0 for t -¥ 00. Of course, in general two coupled systems are not exactly the same or even may be of completely different origin (e.g., an electrical circuit coupled to a mechanical system). What does "synchronization" mean in such a more general case? Different periodic systems are usually called synchronized if either their phases are locked, \n(j>i — m(f)2\ < const,

n , m G IN

(11.1.2)

or the weaker condition of frequency entrainment uji : OJ2 = n : ra, n,m G IN

(11.1.3)

holds where u\ = (d(t)i/dt)t and UJ2 — (d(j)2/dt)t are the natural frequencies of both oscillators. For chaotic systems the notions of "frequency" or "phase" are in general not well defined and can thus not be used for characterizing synchronization.2 But there is another feature of synchronized periodic oscillations that can also be found with coupled chaotic systems: the dimension of the attractor in the combined state space of the coupled systems reduces as soon as the coupling exceeds some threshold values. For two periodic oscillators synchronization leads to a transition from a quasiperiodic (torus) attractor (incommensurate frequencies, dimension 2) to a common periodic orbit (dimension 1). In systems consisting of N coupled periodic oscillators with N independent frequencies a similar transition from an iV-dimensional torus attractor to lower dimensional tori (partial entrainment) and finally to a common periodic orbit may occur if some effective coupling is increased. 2

For some class of chaotic systems a phase variable can be introduced and used to quantify phase synchronization, see Sec. 11.6.4.

11.1 Introduction

273

In this sense synchronization of periodic systems means that the dimension of the resulting attractor of the coupled system is smaller than the sum of the dimensions of all individual attractors of the elements. This notion of synchronization can of course also be applied to coupled chaotic systems. Landa and Rosenblum suggest to call two coupled chaotic systems to be synchronized if the dimension of the attractor in the combined state space equals the dimensions of its projections into the subspaces corresponding to the coupled systems (Landa and Rosenblum, 1993). A reduction of the dimension means that the degrees of freedom are reduced for the coupled system due to the interaction of its components. In particular, for extended systems consisting of many coupled elements (e.g., nonlinear oscillators) the investigation of synchronization phenomena may thus result in a better understanding of the typical occurrence of low-dimensional cooperative dynamics. Dimension reduction due to coupling is an important indicator for (chaos) synchronization but turned out to be a too coarse-grained tool to be able to resolve important details of the relation between the coupled systems. Due to the coupling there may exist, for example, a function mapping states of one of the coupled systems to those of the other. This feature is called generalized synchronization (see Sec. 11.6) and its occurence cannot be proved using dimension estimates of the attractors involved. Also the occurence of certain (transversal) instabilities leading to intermittent breakdown of synchronization (see Sec. 11.3) is difficult to characterize with any dimension analysis. It is one goal of this article to give an overview of typical synchronization phenomena of chaotic systems and concepts for describing and understanding them. Both systems of the previous example are bi-directionally coupled. This is, however, not necessary, because chaos synchronization occurs already for unidirectionally coupled systems. This case is in some sense easier to investigate and may be viewed as an important building block for understanding bi-directionally coupled systems. Furthermore, uni-directionally coupled systems are important for potential applications of chaos synchronization in communication systems (transmitter-receiver or encoder-decoder pairs) or time series analysis where the information flow is also in a single direction only. Therefore, only uni-directionally coupled (chaotic) systems are investigated in the following. In some cases it is evident how the presented concepts and results can be extended to bi-directionally coupled systems and in others not. Although the case of uni-directional coupling is already very rich of interesting phenomena we expect that bi-directional coupling leads to even more complex dynamics. In the presentation of chaos synchronization of uni-directionally coupled systems we start in Sec. 11.2 with pairs of identical systems. Although in physical reality two systems will in general never happen to be exactly the same, this case is very useful to explain basic features of chaos synchronization and to show how synchronizing pairs can be derived from a given chaotic model. Furthermore, identical systems play an important role with some of the potential applications of chaos synchronization that will be discussed in Sec. 11.7. From a more general

274

11 Synchronization of Chaotic Systems

point of view synchronization phenomena between different dynamical systems are discussed in Sec. 11.6 in terms of generalized synchronization and phase synchronization. Generalized synchronization can lead to the existence of a function that maps (asymptotically for t -> oo) states of the drive system to states of the response system. In this case the chaotic dynamics of the response system can be predicted from the drive system. A weaker notion of synchronization is phase synchronization where only a (partial) synchronization of some phase variable occurs while amplitudes may remain uncorrelated. In Sec. 11.5 we demonstrate that chaos synchronization can also be achieved for spatially extended systems like coupled nonlinear oscillators or partial differential equations. Here, a typical goal is synchronization with a minimum of information flow from the drive to the response system and we make use of the concept of sporadic driving that is introduced in Sec. 11.4. In Sec. 11.7 we briefly discuss potential applications of chaos synchronization in communication systems and time series analysis. In chaos-based encoding schemes the synchronization is used to recover a signal that is necessary to decode the message and in the field of data analysis model parameters are estimated by minimizing the synchronization error. The problem of controlling (chaotic) dynamical systems can also be investigated in the framework of synchronization. In this case one of the coupled systems is typically implemented on an analog or digital computer (called controller) that is connected to an experimental or technical device. The main goal of control is to make this device to follow some prescribed dynamics given by the controller. In this sense the controlled process synchronizes with the controller and this may happen for bi-directional as well as uni-directional coupling (open-loop control). If the control algorithm is based on the knowledge of the (full) state of the dynamics a (nonlinear) state observer is necessary to recover the information about the current state of the process from some (scalar) time series. Such a state observer may be considered as a dynamical system that is implemented on a computer and that synchronizes with some external process due to a uni-directional coupling. Note that in contrast to synchronization phenomena between (experimental) physical systems here we a have a "free choice" how to implement the dynamical system which is implemented on the controller with full access to all state variables and parameters. Therefore, rather powerful synchronization methods can be implemented that allow to recover the state of the external process provided a sufficiently exact model exists that may be based on first principles or empirical approximations.

11.2 Synchronization of identical systems In this chapter the synchronization features of pairs of identical systems are investigated. Although in physical systems and experiments coupled systems are in general never exactly the same, this idealized case is of importance for developing suitable concepts for describing chaos synchronization. These concepts and results can directly be applied to systems which are (almost) identical including configu-

11.2 Synchronization of identical systems

275

rations that are of practical interest as will be discussed in more detail in Sec. 11.7. They also play an important role in the case of generalized synchronization of nonidentical systems that will be treated in Sec. 11.6. Pioneering work on chaos synchronization was done by Fujisaka and Yamada (Fujisaka and Yamada, 1983), Pikovsky (Pikovsky, 1984), Afraimovich, Verichev and Rabinovich (Afraimovich et al., 1986) and Pecora and Carroll (Pecora and Carroll, 1990) who presented the first examples of synchronization of uni-directionally coupled chaotic systems. This work had a strong impact and stimulated very intense research activities on chaos synchronization and related questions (He and Vaidya, 1992; Murali and Lakshmanan, 1994; Parlitz et al., 1996). Identical synchronization of chaotic systems was also demonstrated experimentally in particular using electronic circuits (Anishchenko et al., 1992; Rulkov et al., 1992; Kocarev et al., 1992) and laser systems (Roy et al., 1994; Sugawara et al, 1994; Tsukamoto et al., 1996). Synchronization properties of chaotic phase-locked loops have been investigated in Refs. (Endo, 1991; De Sousa Vieira et al., 1991, 1992, 1994a and 1994b).

11.2.1 Constructing pairs of synchronizing systems For some practical applications and for studying chaos synchronization it is useful to derive pairs of synchronizing systems from a given chaotic model. The most popular method for constructing synchronizing (sub-) systems was introduced by Pecora and Carroll (Pecora and Carroll, 1990; Carroll and Pecora, 1993). With that approach a given dynamical system li = g(u) is decomposed into two subsystems v

= gv(v,w) w = gw(v,w). with v = (t/i, ...,Uk) and w = (tifc+i, ...,UJV) such that any second system w' = g w (v,w') that is given by the same vector field gw5 the same driving v, but different variables w' synchronizes (||w;—w|| -> 0) with the original w-subsystem. The coupling is uni-directional and the v-system and the w-system are referred to as the drive system and the response system, respectively. Note that only a finite number of possible decompositions exists that is bounded by the number of different subsystems N(N — l)/2. In general, only a few of the possible response subsystems possess negative conditional Lyapunov exponents and may be used to implement synchronizing systems using the method of Pecora and Carroll. This method and almost all other coupling schemes for achieving synchronization can formally be described as a decomposition of a given (chaotic) system into

276

11 Synchronization of Chaotic Systems

an active and a passive part, where different copies of the passive part synchronize when driven by the same active component (Kocarev and Parlitz, 1995; Parlitz et al., 1996). Consider an arbitrary iV-dimensional (chaotic) dynamical system z = F(z).

(11.2.1)

The goal is to rewrite this autonomous system as a non-autonomous system that possesses certain synchronization properties. Formally, we may write x = f(x,s)

(11.2.2)

where x is the new state vector corresponding3 to z and s is some vector valued function of time given by s = h(x)

or

s = h(x,s).

(11.2.3)

The pair of functions f and h constitutes a decomposition of the original vector field F (see also the example that follows). The crucial point of this decomposition is that for suitable choices of the function h any system y = f(y,s)

(11.2.4)

that is given by the same nonautonomous vector field f, the same driving s, but different variables y, synchronizes with the original system (11.2.2), i.e., ||x — y|| -» 0 for t -» oo. More precisely, synchronization of the pair of (identical) systems (11.2.2) and (11.2.4) occurs if the dynamical system describing the evolution of the difference e = y — x, e = f (y, s) - f (x, s) = f (x + e, s) - f (x, s) possesses a stable fixed point at the origin e = 0. In some cases this can be proved using stability analysis of the linearized system for small e, e = £>/ x (x,s).e,

(11.2.5)

or using (global) Lyapunov functions. In general, however, the stability has to be checked numerically by computing so-called transversal or conditional Lyapunov exponents (CLEs) using the linearized equation (11.2.5).4 Synchronization occurs if all conditional Lyapunov exponents of the nonautonomous system (11.2.2) are negative.5 In this case system (11.2.2) is a passive system and we call the decomposition an active-passive decomposition (APD) of the original dynamical system (11.2.1). 3 If S is given by a (static) function S = /i(x) then X = Z, but for coupling signals s that are generated by an ODE S = h(s,X) the dimension of X may be smaller than that of Z. 4 The notion transversal is due to the fact that e = y — x describes the motion transversal to the synchronization manifold x = y. Using (11.2.5) one obtains the Lyapunov exponents of the response system under the condition that it is driven by the signal s (Pecora and Carroll, 1990). For uni-directionally coupled systems the CLEs are a subset of the Lyapunov spectrum of the coupled system (see Appendix A). 5 In the presence of noise this condition is not sufficient as will be discussed in Sec. 11.3.

i.# Synchronization of identical systems

277

The stability of the passive parts does not exclude chaotic solutions. To illustrate the APD we consider a decomposition of the Lorenz model that may be written as: Xi

=

- 1 ()Xi + S(t) — 28x i - x 2 - xi = X1X2 x x x'2 -— 2.666X3 =

x2 xX3

(11.2.6)

with s(t) = 10x2. The corresponding response system is given by: 2/i = y2 = 2/3 =

— IO2/1 + s{t) 28yi - y2 - 2/12/3 2/12/2 - 2.6661/3.

(11.2.7)

To estimate the temporal evolution of the difference vector e = y — x of the states of the two systems we note that the difference e\ = y\ — xi of the first components converges to zero, because e\ = — 10ei. Therefore, the remaining two-dimensional system describing the evolution of the differences e2 = y2 — x2 and e% — ys — X3 can, in the limit t -» 00, be written as: e2

=

—e2 — x\{i)e%

£3

=

x\(i)e2 — 2.666e3.

Using the Lyapunov function L = e2 + e| it can be shown that L — —2(e2 + 2.666e|) < 0. This means that the synchronization is globally stable provided that other perturbations like noise added to the driving signal or parameter mismatch between drive and response can be excluded (or are at least of very small magnitude) (Brown et al., 1994). The CLEs of this decomposition are given by Ai = -1.805, A2 = —1.861 and A3 = —10 with respect to the natural logarithm. As it was outlined in Ref. (Parlitz et al., 1996) APD-based synchronization methods are also very closely related to the open-loop control method proposed by Hiibler and Liischer (Hiibler and Liischer, 1989; Jackson and Hiibler, 1990) and Pyragas' chaos control approach (Pyragas, 1993; Kittel et al., 1994). In general, synchronization and controlling chaos are very closely related although the goals may sometimes be different (Kapitaniak, 1994). In particular for the synchronization of identical systems, many methods that have been developed in control theory for nonchaotic dynamics may also be applied to chaotic systems. Instead of decomposing a given chaotic system one may also synthesize it starting from a stable linear system x = A • x where some appropriate nonlinear function s = h(x) is added such that the complete system x = A•x+s is chaotic (Wu and Chua, 1993). It is easy to verify that in this case the error dynamics is given by the stable system e = A e , and synchronization occurs for all

278

11 Synchronization of Chaotic Systems

initial conditions and arbitrary signals s. In this way synchronized chaotic systems may be designed with specific features for applications. Another approach to construct synchronizing systems is based on cascades of low-dimensional systems. In this way one may generate hyper chaotic systems that can be synchronized using a single scalar signal (Kocarev and Parlitz, 1995; Parlitz et al., 1996; Giiemez and Matias, 1995 and 1996). Synchronizing hyperchaotic systems have also been presented in Refs. (Peng et al., 1996; Tamasevicius and Cenys, 1997) and occur typically when spatially extended dynamics are investigated (see Sec. 11.2). The function h that defines the coupling signal may depend not only on the current state x(t) of the drive system but also on its pre-history. In this way linear filters may be incorporated in the coupling scheme, such that the coupling signal possesses some specific spectral properties (e.g. bandlimited for a bandlimited transmission channel). In view of potential applications where amplitude quantization of the driving signal occurs (e.g. due to an A/D-converter) the case of a step function h has also been investigated (Stojanovski et al., 1996a and 1997b). It turned out that synchronization is always possible, but in general the chaotic dynamics turned into periodic oscillations when the number of discretization levels becomes (too) small. In the previous discussion of synchronization methods, the function s was assumed to be vector valued in general. For the examples and in the following, however, we will consider only cases with scalar signals s that are most interesting for practical applications of synchronization. Usually when studying synchronization phenomena all parameters of the systems involved are kept fixed. In the case of parameter mismatch, however, the synchronization error typically increases quite rapidly with the parameter differences6. In such a case one might ask: "Can I find an additional dynamical system for the parameters of the response system such that they adapt automatically to the parameter values of the drive in order to achieve (perfect) synchronization?" The answer is "yes", and this case is called autosynchronization or adaptive synchronization (Mossayebi et al., 1991; John and Amritkar, 1994; Caponetto et al., 1995; Dedieu and Ogorzalek, 1995; Parlitz, 1996; Parlitz et al., 1996; Chua et ah, 1996, Cazelles et al., 1996).

11.3 Transversal instabilities and noise In Sec. 11.2 it was stated that synchronization occurs, if all conditional Lyapunov exponents (CLEs) of the response system are negative. This condition turned out not to be sufficient for many cases in the following sense (Ashwin et al., 1994; Gauthier and Bienfang, 1996; Venkataramani et al.,1996; Yang, 1996; Lai et al., 1996; 6 The error dynamics (11.2.5) is in this case given by the linearized ODE e = D/ X (x,s,p) • e + Dfp(x, s, p) • Ap where p is the parameter vector of the drive and Ap denotes the parameter difference between drive and response. Since in general the additional term doesn't vanish, perfect synchronization e —> 0 cannot occur with parameter mismatch.

11.3 Transversal instabilities and noise

279

Heagy et al., 1996). Negative CLEs with respect to the chaotic dynamics of the drive mean that the synchronization manifold M = {(x, y) : x = y} is attracting on average. There may, however, exist locations on M where it is repelling, and when the orbit comes close to such a region, it is pushed away from the synchronization manifold and synchronization breaks down. As long as the orbit of the coupled system lies exactly on the synchronization manifold, such a local transversal instability at some point (xo,yo) has no effect. Practically, however, typical orbits are never on but only close to M, because they converge asymptotically from some initial condition to M and/or noise kicks them off the synchronization manifold (Brown et al., 1994a). In these cases any trajectory moves away from M as long as it stays in the region of transversal instability. The effective growth of the synchronization error e = x — y depends on the size of this region, the strength of the transversal instability, and the period of time the orbit spends in the unstable region. As soon as the trajectory leaves the region of instability it is attracted again by the synchronization manifold and thus an averaged negative CLE exists that implies e -> 0 in the noiseless case. With (arbitrary small) amounts of noise (added to the coupling signal) the difference ||e|| cannot (on average) become smaller than some constant S which is given by the noise level. In this way the action of the contracting part of the synchronization manifold is limited, and short or long passages through the repelling regions will lead to small or large amplifications of the minimum distance S given by the noise. Since sufficiently long trajectory segments in the repelling region occur only unfrequently the breakdown of synchronization due to transversal instabilities is an intermittent phenomenon. This can be seen in the example given in Fig. 11.1 that was computed using a uni-directionally coupled pair of chaotic one-dimensional Gaussian maps: xn+l

=

y^nj

(11.3.1)

where f(x) = exp(-a 2 [x - bf) with a = 3.5, b = 0.5, c = 0.4. The largest CLE of this system is negative for c > 0.3. In order to stimulate the intermittent bursts of the synchronization error en = yn — xn, uniformly distributed noise rn e [—10~5,10~5] is added to the driving signal. The origin of the regions of transversal instability are for example unstable fixed points or unstable periodic orbits (UPOs) of the driving system7 that fail to entrain the corresponding fixed points or periodic orbits (POs) of the response system. When driven with one of these UPOs the counterparts of the response system possess positive CLEs and no identical synchronization of the UPOs occurs. Such unstable response fixed points or orbits may occur, for instance, due to period doubling bifurcations. To illustrate this mechanism we consider the fixed point (XF,XF) = (0.6781,0.6781) of the coupled system (11.3.1) that is located on the synchronization manifold M. The stability features of this fixed point are given by the 7

Any typical chaotic attractor contains an infinite number of unstable periodic orbits (Ott, 1993; Schuster, 1995) that may also be used for driving and synchronizing a response system (Gupte and Amritkar, 1993).

280

11 Synchronization of Chaotic Systems

0.4 -

0.2 -

2000

4000

n

6000

8000

104

Figure 11.1 Intermittent bursts of the synchronization error e due to transversal instabilities.

eigenvalues \i\ = f'{xp) and /x2 = (1 - c)f'(xF) of the Jacobian matrix of (11.3.1) at (XFIXF)> The first eigenvalue /ii describes the instability within the synchronization manifold and does not depend on the coupling. The second eigenvalue, however, reflects the transversal (in)stability and depends on c. For c > 0.662 the stability criterion |//2| < 1 holds, and at the critical value cp& « 0.662 a period doubling bifurcation occurs. Thus for c < CPD the unstable fixed point (period-1 UPO) of the drive fails to entrain a fixed point of the response but leads to a period-2 cycle (at least for values of c that are sufficiently close to CPD- If c is decreased furthermore, a complete period doubling cascade occurs.) For c = 0.4, for example, the response of the fixed point xp is given by the period-2 orbit {...,0.778,0.505,...} with CLE A = -0.32. Such a subharmonic entrainment of periodic orbits turns out to be also of importance for generalized synchronization as will be discussed in Sec. 11.6. Note that subharmonic entrainment means that some stable periodic response occurs in contrast to a possible chaotic response to the periodic UPO-driving. Another way to illustrate the transversal instability is presented in Fig. 11.2 where the gray shaded regions consist of points (x, y) that are moved away from the synchronization manifold by the map (11.3.1). The thick dot on the diagonal gives the position of the fixed point (XF,XF) and the other two thin dots represent a period-2 UPO of the driving map. For a coupling with c = 0.4 (that was also used for Fig. 11.1) two large regions of transversal instability occur (Fig. 11.2a) that contain the period-1 and the period-2 UPO. If the coupling constant is increased to c = 0.666 the unstable region shrinks to two small spots as can be seen in Figure 11.2b. For this value of the coupling all UPOs entrain periodic orbits on the synchronization manifold with negative CLEs. The remaining small gray shaded areas indicate regions where positive local Lyapunov exponents occur. For c > 0.667 the gray shaded regions vanish and the synchronization manifold is everywhere transversally stable.

11.4 Sporadic driving

281

0.2 -

0.2

0.4

0.6

0.8

1

Figure 11.2 Regions of transversal instability (gray shaded) and unstable periodic orbits (dots) for (a) c = 0.4 and (b) c = 0.666.

Finally, we would like to note that typically the phenomenon of riddled basins is associated with transversal instabilities if another attractor exists off the synchronization manifold. In this case attractor basins occur which are of positive measure but contain no open sets, i.e. in any neighbourhood of a point of the basin of the first attractor are points of the basin of the second attractor (see Refs. (Alexander et al., 1992; Ott et al., 1993; Parmenter and Yu, 1994; Ashwin et al., 1994; Heagy et al., 1994; Ott and Sommerer, 1994; Lai et al., 1996; Ding and Yang, 1996; Venkataramani et al., 1996; Lai and Grebogi, 1996) for details).

11.4 Sporadic driving To achieve synchronization of two continuous systems it is not necessary to couple them continuously. Even if the coupling is switched on at discrete times tn = nT only, synchronization may occur if the coupling and the time interval T are suitably chosen (Amritkar and Gupte, 1993; Stojanovski et al., 1996a, 1997a and 1997b; Chen, 1996; Parlitz et al., 1997). This kind of sporadic driving leads to synchronization, for example in those cases where for the given coupling signal the corresponding continuous driving would lead to synchronization. In this case synchronization occurs for all T < Ts where Ts is a threshold value that depends on the dynamical system and the particular coupling. In general, sporadic driving may be defined in the following way. Let x = g(x)

(11.4.1)

y = g(y)

(11.4.2)

and

be two continuous dynamical systems that are given by the same vector field g but different state vectors x and y, respectively. Like in the previous sections system

282

11 Synchronization of Chaotic Systems

(11.4.1) will be called drive and system (11.4.2) response. Let the signal be a scalar function of the state of the drive that is, however, available only at discrete times tn = nT where T is some sampling time. In order to describe how the resulting discretely sampled time series {sn} with sn = s(tn) = h(x(tn)) is used to drive the response system (11.4.2) we consider now both continuous systems as discrete systems that are given by the flow 0 T : xn+1

=

0 T (x n )

(11.4.3)

V-+ 1

=

4>T{yn).

(11.4.4)

The flow (j)T is obtained by integrating the ODEs (11.4.1) and (11.4.2) over the period of time T where x n = x(tn) and y n = y(tn) are the states of the drive and the response system at time tn, respectively. During this period of time both systems are not coupled and run freely. To achieve (chaos) synchronization of drive and response we apply now the concept of an active-passive decomposition (APD) to the discrete systems (11.4.3) and (11.4.4) and rewrite the map x n + 1 = 0 T (x n ) as xn+1 =f(xn,sn) where sn = h(xn).

(11.4.5)

An example for such a formal decomposition is:

f(x n , sn) = 0 T (x n + [sn - h(xn)] c)

(11.4.6)

where c is a vector containing some coupling constants. If the APD is successful the new (formally) nonautonomous system (11.4.5) is passive, i.e., it possesses only negative conditional Lyapunov exponents when driven by {sn}. This, however, implies that any copy of (11.4.5) y n+l = f ( y n ? 5 n )

(H.4.7)

will synchronize (||x n - y n || -+ 0) for n -» oo. To illustrate this type of coupling we will use in the following as drive and response two Lorenz systems that are given by: Xi

=

10(^2 — X\)

— x2 — x\Xz -2.666x3.

(11.4.8)

The coupling signal sn is x 2 (t n ) and the APD is given as

x n + 1 = f(x n ,5 n ) = 0 T f S" J \

X

3

(11.4.9)

)

where x n = x(£n) and (j)T denotes the flow generated by the Lorenz system. Both Lorenz systems run independently during a period of time T = 0.4. Then the variable y2 of the response system is replaced by x2 of the drive, i.e., the discrete coupling takes place. After this coupling the systems oscillate independently again

11.4 Sporadic driving

244

245

249

283

250

Figure 11.3 Synchronization of two chaotic Lorenz systems (11.4.8) due to sporadic driving. Shown is the response variable 2/2 as a function of time t. The vertical dashed lines denote the times tn = nT = n • 0.4 when the coupling is active.

and so on. Figure 11.3 shows the variable 2/2 of the response system. The times tn where the coupling takes place are denoted by the vertical dashed lines and the diagram shows the oscillation after some synchronization transient. The time evolution of 2/2 (t) coincides exactly with the corresponding evolution of the drive variable X2(t) (not shown here). The synchronization due to the sporadic driving does not only lead to a convergence \x2(tn) — 2/2{tn)I —» 0 at the coupling times tn for n -» 00, but also to a perfect interpolation of the time evolution of all state variables between the coupling times. With this coupling the largest conditional Lyapunov exponent is negative for coupling times T G [0,0.45]. For other systems more than one T-interval with negative CLEs have been observed (Stojanovski et al., 1997a) and in general it can be shown that for sporadic driving with a /.-dimensional driving signal (here: k = 1) the k smallest CLEs of the response system equal —00 (Stojanovski et al., 1997a). Numerical simulations have shown that it becomes more difficult (or even impossible) to find a suitable discrete APD of the flow if T is chosen too large. On the other hand, sporadic driving may lead to synchronization in cases where the corresponding continuous coupling fails (Amritkar and Gupte, 1993; Stojanovski et al., 1996a and 1997a). Another important issue is the sensitivity of synchronization with respect to noise added to the driving signal. Very similar to the case of continuous coupling synchronization due to sporadic driving may lead to either high quality synchronization or intermittency phenomena depending on the dynamical systems and details of the coupling (Stojanovski et al., 1997a). Furthermore, synchronization due to sporadic coupling can be realized using bandlimited channels for the coupling signal despite the fact that chaotic spectra typically possess no finite support (Stojanovski et al., 1997a). The fact that in this way chaotic signals can be transmitted through bandlimited channels is not in contradiction with the well-known sampling theorem since the chaotic signals are generated by deterministic differential equations which in addition are known at the receiver. Only the initial

284

11 Synchronization of Chaotic Systems

conditions of the driving system are not known at the receiving end of the channel, i.e., at the interpolating side. The sampling theorem, on the other hand, assumes no knowledge about the information source. Another feature of sporadic driving which is important for practical implementations is the fact that synchronization can also be achieved when the coupling is switched on for short but finite periods of time (Stojanovski et al., 1997a). Therefore, the basic mechanism seems to be robust enough to be observed also in real physical systems where instantaneous variations of some variable are impossible. In the following sections two potential applications of sporadic driving will be discussed: synchronization of spatially extended systems (Sec. 11.5) and parameter identification from time series (Sec. 11.7).

11.5 Spatially extended systems Spatially extended systems like coupled oscillators or partial differential equations are usually governed by (very) high dimensional chaotic attractors. Nevertheless, it is possible to achieve synchronization of such hyperchaotic systems if they are coupled in a suitable way8 (see Refs. (Lai and Grebogi, 1994; Gang and Zhilin, 1994; Kocarev et al., 1997)). From a practical point of view, however, some constraints apply because often a coupling at all spatial locations and at all times is not feasible. Instead of such a continuous coupling in space and time it is desirable to use discrete coupling schemes where the coupling is active only at certain locations and/or at discrete times. Discrete coupling not only simplifies experimental realizations of synchronized spatio-temporal dynamics but also leads to a reduction of the information flow from the driving system to the response system that may be used to characterize the underlying chaotic dynamics or to store the full chaotic evolution in terms of the control signals. To illustrate the synchronization of spatially extended systems we shall consider now two coupled Kuramoto-Sivashinsky (KS) equations (Hyman and Nicolaenko, 1986)) ut + 4uxxxx + 7 (uxx + ^(ux)2 j + -^ = 0

(11.5.1)

with

Here x G [0,2?r] with periodic boundary conditions and Eq. (11.5.2) is used to subtract the meanvalue from the dynamics. Figure 11.4a shows the spatio-temporal chaos that occurs for 7 = 200. In order to couple Eq. (11.5.1) to another KSequation we assume that we can measure 10 signals s* from k = 1,.., 10 sensors that 8

Here we are considering uni-directionally coupled pairs of spatially extended systems. Internal synchronization phenomena within such systems have also been observed but are beyong the scope of this article.

11.5 Spatially extended systems

285

(•)

(b)

Figure 11.4 Synchronization of uni-directionally coupled Kuramoto- Sivashinsky equations, (a) Gray scaled values of the drive solution u{x^ t) vs. time * and space x.(b) The same as in (a) for the solution v(x,t) of the response equation.(c) Difference |w(#,£) — v(x,t)\.(d) Spatially averaged synchronization error vs. time.

are equidistantly distributed in the ^-interval [0,2?r]. These sensors are assumed to possess a finite resolution such that they measure a local average of the dynamics. The sensor signal at position xk = k • 2TT/10 and time tn = nT is given by =

/

Jxxk-Ax

u(x,tn)dx

(11.5.3)

where the width Ax = 4?r/25 specifies the spatial resolution of the sensor and T is the period of time between the coupling events. The same set of sensor signals {sk(tn)} is assumed to exist for the response KS-equation that is identical to the drive and generates a solution v(x, t). The discrete coupling consists in an updating of the ^-values in the k = 1,..., 10 sensor intervals [xk — Ax,xk + Ax] at times tn = nT with some coupling constant c: Vn = 1,2,3,...VA;= l,...,10 Vx e [xk - Ax,xk + Ax] : V(x,tn) = tf(Mn) + C [Sku(tn) - Sk(tn)] where v stands for the value of v immediately after the moment when the coupling was active. Figure 11.4b shows the temporal evolution of the response KS-equation and in Fig. 11.4c we have plotted the synchronization error \v(x,t) - u(x,t)\. The spatially averaged synchronization error as a function of time is given in Figure 11.4d.

286

11 Synchronization of Chaotic Systems

11.6 Synchronization of nonidentical systems Until now we have considered pairs of identical systems where identical synchronization (lim ||x(£) - y{t)\\ — 0) may occur. If two different systems are coupled t—>oo

identical synchronization is in general not possible (i.e. not a solution of the coupled system), but other types of synchronization may be observed. In this section we present and compare different definitions of generalized synchronization (GS) that have been proposed during the last years.9'10

11.6.1 Generalized synchronization I For chaotic systems Afraimovich et al. (Afraimovich et al., 1986) gave the first definition for what was later called generalized synchronization (GS) by Rulkov et al. (Rulkov et al., 1995). In their definition Afraimovich et al. called two systems synchronized if in the limit M o o a homeomorphic function exists mapping states of one system to states of the other (including some time shift a(t) with lim (t + a(t))/t = 1). Later the assumption of a homeomorphism was relaxed t-yoo

and two uni-directionally systems are said to be in synchrony if their states x and y are asymptotically related by some function H so that ||H(x(£)) - y(t)|| -> 0 for t —> oo. This definition of generalized synchronization (GS) was used in Refs. (Kocarev and Parlitz, 1996; Hunt et al., 1997) and can be verified using time series based methods as suggested by Rulkov et al. (Rulkov et al., 1995). Mathematically it may be formulated as follows: Def.I: Generalized synchronization of the uni-directionally coupled systems drive x response y

= =

f(x) g(y,x)

(x€lR n ) (y G IRm)

( n 6 1}

'

occurs for the attractor Ax C 1R of the drive system if an attracting synchronization set M = {(x,y) € Ax x lRm : y = H(x)} exists that is given by some function H : Ax -» Ay C Htm and that possesses an open basin B D M such that: lim ||y(t) - H(x(t))|| = 0

V(x(0),y(0)) € B.

t->oo

An analogous definition can be given for discrete dynamical systems. Whether the function H is continuous or even smooth depends on the features of the drive and response system and the attraction properties of the set M (Davies, 1996; 9 Closely related are investigations of IIR-filters, see (Davies and Campbell, 1996; Stark, 1997) and the references cited therein. 10 In this section we consider only ideal systems without noise. In the presence of noise similar effects like those discussed in Sec. 11.3 have to be taken into account.

11.6 Synchronization of nonidentical systems

287

Hunt et al., 1997; Stark, 1997). This definition may be motivated by the requirement that statements about synchronization should be independent of the coordinate system used. As an example we consider two uni-directionally coupled systems: xi x2

= =

(T(X2~X1)

rxi -x2 - x i x 3 - bx3

and

yy = (r(x2-f~1(yi))/(f~1)l(yi) l y2 = f~ (yi)(ra - y3) - y2 2/3 = Z" 1 (2/1)2/2 - fy/3

where / l and (/ *)' denote the inverse and its derivative, respectively, of an invertible function / : IR —» IR. n To prove that for this pair of systems GS occurs with H(x) = (/(#i),ax 2 ,axz) we consider first the difference e\ = f~l{y\) — x\. Using Eq. (11.6.1) it is easy to show that e\ = —oe\ and thus e\ -» 0 for t ->• oo if a > 0. Therefore, asymptotically f~1(yi) = X\ or y\ = f(xi). For the remaining two-dimensional system given by e2 = y2 — ax2 and e3 = y$ — ax$ one can show with a Lyapunov function L = (e22 + e32)/2 that L = -e2 — 6e2 < 0 for b > 0 and thus y - H(x) -» 0 for all initial conditions. In this case the synchronization set M is a globally attracting submanifold with y = H(x) and its basin of attraction B is the whole product space IRn x H m of both state spaces. This example is based on example (11.2.6)-(11.2.7) and was constructed starting from a pair of two identical Lorenz systems that synchronize. Then, the response system was subject to a change of the coordinate system given by H. Since any diffeomorphic transformation doesn't change stability properties the new synchronization manifold (here: y = H(x)) remains stable. Thus identical synchronization implies GS in any diffeomorphic equivalent coordinate system. On the other hand, if GS is observed between two dynamical systems with a diffeomorphic function H this function can be used to perform a change of the response coordinate system such that in the new coordinate system the response system synchronizes identically with the drive system. To check for which values of the coupling parameter a GS occurred one may apply nearest neighbors statistics (Rulkov et al., 1995; Pecoraet al., 1995) to detect the existence of a continuous function relating states of the drive to states of the response. This approach for identifying generalized synchronization can be applied to uni- and bi-directionally coupled systems if the original (physical) state spaces of drive and response are accessible. If only (scalar) time series from the drive and the response system can be sampled, then delay embedding (Takens, 1980; Sauer, 1991) may be used to investigate neighborhood relations in the corresponding reconstructed state spaces (Rulkov et al., 1995). In this case, however, only generalized synchronization of wm-directionally coupled systems can be detected by predicting the (reconstructed) state of the response system using a time series from the drive system! In the opposite direction a prediction of the evolution of the drive system based on data from the response system is always possible (i.e. 11

As an examplethe reader may substitute f{x\) = exp(zi) with f~1{yi)

= ln(r/i) and

288

11 Synchronization of Chaotic Systems

with and without generalized synchronization), because (almost) any time series measured at the response system may also be viewed as a time series from the combined systems drive and response and may thus be used to reconstruct and predict the dynamics of drive and response.12 Generalized synchronization in the sense of Def. I can also be found or established using the following result (Kocarev and Parlitz, 1996): Proposition: GS occurs for an attractor A of the coupled systems (11.6.1) if an open basin of synchronization B C H n x Rm exists with M C B such that for all initial values (xo,yo) G B the driven system y = g(y,x) is asymptotically stable in the sense that V(xo,yio),(xo,y2o) e B : lim ||y(t,xo,yio)-y(*,xo,y2o)|| = 0 . t—yoo

Some remarks concerning this proposition are in order: (i) The construction of H in the proof given in Ref. (Kocarev and Parlitz, 1996) is based on the uniqueness of the inverse of the flow generated by (11.6.1). Therefore, the analogous proposition for discrete systems holds only if the drive is given by an invertible map. (ii) The assumptions of the proposition are not fulfilled for subharmonic entrainment of periodic orbits, because in this case different basins of attraction occur. An entrainment with ratio TD • TR = 1 : p (p > 1), for example, results in p basins for the initial values of the response system. For the proposition, however, it is assumed that a single basin B exists. This exclusion of subharmonically entrained periodic orbits is necessary, because for these solutions H is not a function. If, for example, a periodic orbit of the drive entrains a stable periodic orbit of the response with twice the period (i.e. To : TR = 1 : 2) then any point on the attractor of the drive is mapped to two points on the response orbit and H is in this case a relation but not a function. This multivaluedness always occurs for subharmonic periodic entrainment with Tr> < TR. Recall that subharmonic entrainment was also the origin of transversal instabilities as discussed in Sec. 11.3. (iii) Asymptotical stability (as defined in the proposition) can be proved analytically using Lyapunov functions. Numerically this condition can be checked by computing the (largest) Lyapunov exponent of the response system. In this case, however, it has to be made sure additionally that all (unstable) periodic orbits of the drive entrain a stable periodic orbit of the response system with the same period, because (U)POs that lead to subharmonic entrainment or even chaotic response orbits cannot be excluded using a stability 12

Note that without GS a time series of the drive system contains no information about the response system.

11.6 Synchronization of nonidentical systems

289

analysis based on (globally averaged) Lyapunov exponents. Practically, this requirement can be checked using nearest neighbors statistics. (iv) In Ref. (Pyragas, 1996) it was argued that H is a smooth function if the conditional Lyapunov exponents of the response system are smaller than the smallest Lyapunov exponent of the drive, because in this case the Lyapunov dimension of the attractor of the coupled system would not depend on the driven response system. This condition seems, however, not to be sufficient, because (i) for some unstable periodic orbits it may locally be not fulfilled resulting in a local loss of smoothness (Hunt et al., 1997) and (ii) there may also exist subharmonically entrained orbits where H is not a function (Parlitz et al., 1997). In order to illustrate the proposition given above we show with this example that GS occurs for a uni-directionally coupled system consisting of a Rossler system driving a Lorenz system. The equations of the drive and the response system are: X\

=

x2 X3

=

2 + :ri(x 2 - 4 ) -x3 x 2 4-• 0.45x3

-2/2)

^i

=

—O"(j/1

2/2 2/3

= =

ru(^) -- 2/2 - u(t)y3 u(t)y2 ~by3

(11.6.2)

where u(t) is an arbitrary scalar function of X\, x2, x3, and a, b > 0. In order to show that GS occurs we consider the difference e = y — y', where the primed variables describe a second trajectory of the response system starting from different initial conditions. Using the Lyapunov function L = (ef /a + e2 + ef)/2 one obtains

L — —e\ + eie2 — e2 — be\ = — (ei—e2/2)2 — Se2/4 — bel < 0, i.e. the response system is asymptotically stable for arbitrary drive signals u and arbitrary initial conditions. Therefore, GS always occurs although drive and response are completely different systems.

11.6.2 Generalized synchronization II There are many cases where drive and response are not related by a function but a weaker notion of synchronization applies that may be defined as follows13: Def. II: Generalized synchronization of uni-directionally coupled systems drive response

x y

= =

f(x) g(y,x)

(x€Bn) (y G E m )

( n 6 3)

^ n

m

occurs if there exists an open synchronization basin B C IR x IR such that V(xo,yio),(xo,yio) € B : lim ||y(t;xo,yio) -y(*;xo,y2o)|| = 0. t—><x>

This definition says that GS occurs if the response system is asymptotically stable with respect to the driving signal and at first glance the definition may look very 13

This definition is motivated by the auxiliary system method of Abarbanel et al. (Abarbanel et al., 1996).

290

11 Synchronization of Chaotic Systems

similar to the proposition given in the previous section. But there is a crucial difference, because we do not assume here that the complete attractor is contained in the basin B. Therefore, this definition for synchronization includes also the case of subharmonic entrainment of periodic oscillations where several basins coexist (for entrainment with T& : TR = I : p, p basins B{ (i = 1, ...,p) occur). Practically, the occurrence of this type of synchronization can be checked by computing the CLEs of the response system. Again, a rigorous investigation should include not only the (averaged) CLEs of the chaotic dynamics but also the Lyapunov exponents of (all) response orbits that are generated by UPOs embedded in the chaotic drive attractor. If there exists an UPO of the drive that leads to a chaotic response, for example, then two response trajectories will diverge when the drive comes close to this UPO. Another practical method for checking the existence of GS (in the sense of Def. II) that is a direct implementation of the criterion of asymptotic stability was suggested by Abarbanel et al. (Abarbanel et al, 1996). It is based on the use of a second auxiliary response system. The investigated pair of coupled systems is said to have the property of GS if starting from different initial conditions both response systems converge to the same trajectory. The advantage of this approach is the fact that it can directly be applied to experimental systems provided a second copy of the response system is available like for electronic circuits (Rulkov, 1996; Rulkov and Sushchick, 1996) or laser systems. If both response systems differ slightly one even obtains additional informations about the robustness of the synchronization.

11.6.3 Non-identical synchronization of identical systems Usually (generalized) synchronization is studied if two different systems / ^ g are coupled but it may also occur with identical systems f = g that show no identical synchronization (x ^ y). As an example, we consider two uni-directionally coupled identical Lorenz systems ±i x2

= =

10(x2-xi) 28xi - x2 - xixz - 2.666x3

and

i/i = 2/2 = 2/3 =

10(2/2 -2/i) + c(xi 28t/i - y2 - 2/12/3 2/12/2 - 2.6662/3.

-yi)

For c > 7.7 this coupled system shows the wellknown identical synchronization. There exists, however, also a parameter interval for the coupling c where no identical but generalized synchronization in the sense of Def. II occurs. Figure 11.5 shows an example for this case. Figs. 11.5a and Fig. 11.5b show the variables x\ and 2/1 of the drive and the response system, respectively, and in Fig. 11.5d the synchronization error |#i(£) — 2/1 (01 1S plotted. It is obvious that there is no identical synchronization. Figure 11.5c shows the dynamics of a second (auxiliary) response system that started with different initial conditions. After some transient both response systems (Figs. 11.5b and 11.5c) synchronized mutually as can also be seen in Fig. 11.5e. Usually when using conditional Lyapunov exponents for verifying

11.6 Synchronization of nonidentical systems

291

(•) 20 -,

Figure 11.5 Generalized synchronization of two identicalLorenz systems, (a) Drive variable x\. (b) First response variable y\. (c) Second response variable z\. (d) Difference between drive and response variable \x\ — 2/i |- (e) Difference between both response variables \y\ — z\\.

synchronization of identical systems the Jacobian matrix of the vector field is computed using the state vector of the drive system. These Lyapunov exponents will be called IS-LEs in the following. However, for investigating the mutual synchronization of a pair of response systems the Jacobian matrix has to be determined using a response state. The resulting set of Lyapunov exponents are related to GS and are therefore called GS-LEs. If identical synchronization occurs both sets of LEs coincide but not for generalized synchronization. This difference is illustrated in Fig. 11.6a where the largest IS-LE is plotted as a dotted curve and the maximum GS-LE is given by the solid line. For coupling values c < —6.7 the GS-LE becomes negative and GS occurs while the IS-LE remains positive. The onset of GS can also be seen when computing the averaged drive-response and response-response synchronization errors E, respectively that are given in Fig. 11.6b.

292

11 Synchronization of Chaotic Systems

Figure 11.6 Uni-directionally coupled Lorenz systems, (a) GS-LE (solid) and IS-LE (dotted) conditional Lyapunov exponents detecting response and identical synchronization, respectively, (b) Averaged synchronization errors of drive and response (dotted) and between both response systems (solid).

11.6.4 Phase synchronization Another generalization of the notion of identical synchronization is the phenomenon of phase synchronization (PS) (Stone, 1992; Rosenblum et al., 1996; Parlitz et al., 1996; Pikovsky et al, 1996, 1997; Osipov et al., 1997; Fabiny et al., 1993). It is best observed when a well defined phase variable can be identified in both coupled systems. This can be done heuristically for strange attractors that spiral around some particular point (or "hole") in a two-dimensional projection of the attractor. In such a case, a phase angle <j){t) can be defined that de- or increases monotonically. Phase synchronization of two coupled systems occurs if the difference \(j>i{t) — 02(£)| between the corresponding phases is bounded by some constant.14 This phenomenon may be used in technical or experimental applications where a coherent superposition of several output channels is desired (Fabiny et al., 1993; Roy and Thornburg, 1994). In more abstract terms PS occurs when a zero Lyapunov exponent of the response system becomes negative. This leads to a reduction of the degree of freedom of the response system in the direction of the flow. For systems where a phase variable can be defined the direction of theflowcoincides in general with the coordinate that is described by the phase variable. A zero LE that becomes negative reflects in this sense a restriction that is imposed on the motion of the phase variable. If the zero LE that decreases is the largest LE of the response system then phase synchronization occurs together with GS. If there exist, however, in addition to the formerly zero LE, other LEs which are and remain positive, PS occurs but no GS. Another phenomenon that is closely related to PS is lag synchronization that was observed only recently by Rosenblum et al. (Rosenblum et al., 1997) and leads to synchronization with some time delay between drive and response. For more details see the contribution of Pikovsky et al. in this volume. 14 A more general definition includes rational relations \n\ —mfol < const for arbitrary integers n and m. Compare also the definitions (11.1.2) and (11.1.3) for periodic oscillations.

11.7 Applications and Conclusion

293

11.7 Applications and Conclusion During the last six years chaos synchronization has become one of the most intensely studied topics in nonlinear dynamics. This development was mainly stimulated by the seminal paper of Pecora and Carroll (Pecora and Carroll, 1990), although some very interesting results were already found in the 80s (see Sec. 11.2). One reason for the success of the approach suggested by Pecora and Carroll was probably the fact that they mentioned already in their first paper in 1990 the possibility of using uni-directionally coupled systems in communication systems based on chaos. Once a unidirectionally coupled pair of synchronizing systems (drive and response) has been found it can be used in different ways for encoding and masking messages. The basic idea is to transmit an information signal with a broadband chaotic carrier signal and to use synchronization to recover the information at the receiver. Different implementations of this general concept have been suggested (Kennedy, 1997): (a) Chaotic masking The information is added to a chaotic carrier and the synchronization of the response system in the receiver is used to recover the message (Kocarev et al., 1992; Cuomo and Oppenheim, 1993; Cuomo et al., 1993; Murali and Lakshmanan, 1993; Lozi and Chua, 1993). An improved scheme based on dissipative pseudorandom dynamics was suggested in (Gershenfeld and Grinstein, 1995). (b) Chaos modulation The information signal is contained in the transmitted signal and (in contrast to (a)) drives the transmitter system and the receiver system in exactly the same way (Halle et al., 1993; Volkovskii and Rulkov, 1993; Wu and Chua, 1993; Wu and Chua 1994; Kocarev and Parlitz, 1995; Parlitz et al., 1996). This method is related to the inverse systems approach (Feldmann et al., 1996) and in principle it allows to recover the information exactly. (c) Chaos shift keying Binary information signals are encoded by switching between different drive systems. At the receiver the message can be recovered by monitoring the synchronization of the corresponding response systems of the receiver (Parlitz et al., 1992; Dedieu et al., 1993). (d) Parameter modulation The information signal is used to modulate a parameter of the drive system and the receiver uses auto-synchronization to recover the messages by reproducing this modulation (Carroll and Pecora, 1993; Parlitz et al., 1996; Parlitz and Kocarev, 1996). Practically this scheme can be implemented by adding a feedback loop to the response system that controls the dynamics of the modulated parameter. The variations of the information signal have to be slow compared to the convergence properties of the parameter controlling loop.

294

11 Synchronization of Chaotic Systems

(e) Encoding using generalized synchronization Transmitter and receiver possess a common keysequence that they use to drive (identical) response systems. Due to generalized synchronization the output of the response systems is a (complicated) function of the key sequence that is used to encode and decode the message (Xiao et aL, 1996; Parlitz et al., 1997). An important feature of the masking and modulation methods (a) and (b) is the fact that the frequency spectrum of the chaotic carrier can (and should) be the same as that of the information signal. Both signals can therefore not be separated using linear data analysis tools like linear filters. Using numerical and experimental examples it was demonstrated that synchronization can in principle be used for decoding suitably encoded messages, where "chaos" has the important task to scramble the data such that they cannot be easily deciphered by third parties. On the other hand, for low-dimensional chaos methods have been suggested to break such an encryption (Short, 1994; Perez and Cerdeira, 1995; Stojanovski et al., 1996b). Whether chaos synchronization based encryption methods will be developed in a way that they can compete with the very powerful algorithms already known in (standard) cryptography will turn out in the future. Important prerequisites for achieving this ambitious goal have been demonstrated to be fulfilled, like synchronizing very highdimensional chaotic systems using only a scalar (discretely sampled) signal. Another application of synchronization consists in model verification (Brown et al., 1994b) and parameter estimations from time series (Parlitz et al., 1996). Assume that a (chaotic) experimental time series has been measured and that the structure of a model is known, but not the parameters and those state variables that have not been measured. The goal is to find these unknown parameters and perhaps also the time evolution of the variables that have not been measured. This problem can be solved by minimizing the synchronization error (Parlitz et al., 1996) or using auto-synchronization (see Sec. 11.2). In the same way response systems can be established that monitor slow changes of technical devices provided that a sufficiently exact model of the process of interest is available. In the previous sections different approaches were presented for synchronizing a pair of identical dynamical systems. Although these methods are already quite general they are not succesful in all cases. Given a coupling signal and two systems to be synchronized the Pecora-Carroll approach may, for example, fail, because no appropriate stable subsystem can be found. On the other hand, other coupling schemes may be succesful for the same configuration. One may therefore ask the general question: "For which dynamical systems and for which coupling signals is chaos synchronization possible?" At first glance, it may look hopeless to answer this question. However, one can easily show that in general any pair of uni-directionally coupled dynamical systems can be synchronized using (almost) any (smooth) coupling signal (Stojanovski, 1997c). This is an immediate consequence of the state space reconstruction theorems in nonlinear time series analysis (Takens, 1980; Sauer, 1991). There, delay or derivative coordinates are used to

11.7 Applications and Conclusion

295

reconstruct the states of a dynamical process (here: drive system) from scalar time series. More precisely, the reconstructed states are diffeomorphic images of the original states and the theorems provide rather general conditions for the existence of the underlying diffeomorphism. Applying the inverse of this diffeomorphism to the reconstructed states one may thus recover the original states of the drive. The knowledge of these states can then be exploited in diiferent ways to synchronize the response system with the drive, the simplest method being a replacement of the states of the response by the states of the drive at discrete times (Stojanovski, 1997c). The main technical difficulty of such an approach is the computation of the inverse reconstruction map. If a sufficiently large number of pairs {reconstructed state, original state} is available one may, for instance, fit an approximating function based on polynomials, radial basis functions or neural networks in order to describe the map: reconstructed state -» original state. Using the knowledge about the dynamical equations of the drive system it is also possible to formulate for each state a fixed point problem in the reconstructed state space that may be solved by a (quasi) Newton algorithm. With this method as well as with the approximation method the states of the drive can be computed directly, i.e. without any (exponential) transient like in the case of (identical) synchronization. The problem of estimating states of a system from measured data has also a long history in control theory where the recovered states are used as input of a controller that tries to drive the system towards some goal dynamics. In this context algorithms that yield the states of the system are called observer and were introduced in 1966 by Luenberger (Luenberger, 1966). The first work on this topic for linear systems dates back to 1960 when Kalman laid the foundations for this field (Ogata, 1990). Later different methods for (special classes) of nonlinear systems have been proposed (Thau, 1973; Kou et al., 1975; Ciccarella et al., 1993; So et al., 1994; Morgiil and Solak, 1996) and in the following we give an example for such an approach to synchronization. However, not only observer algorithms from control theory can be used in the context of synchronization. Controlling strategies themselves may be used to drive a response system into a synchronized state (Lai and Grebogi, 1993; Newell et al., 1994 and 1995). The relation between synchronization and (standard) control theory was pointed out by Konnur (Konnur, 1996) (see also (Levine, 1996; Nijmeijer and van der Schaft, 1990; Isidori, 1989)). In this article we focussed on synchronization methods and mechanisms of unidirectionally coupled systems. Of course, many physical, biological or technical systems consist of bi-directionally interacting elements or components. We hope that the future investigation of different couplings in combination with sophisticated types of synchronization15 will provide a deeper insight into the variety of cooperative phenomena observed in nature.

15

That are yet to be discovered ?!

296

11 Synchronization of Chaotic Systems

Acknowledgments The authors thank L. Junge, T. Stojanovski, N. Rulkov, M. Sushchik, L. Pecora, H.D.I. Abarbanel, A. Pikovsky, M. Rosenblum, J. Kurths, R. Roy, S. Strogatz, R. Mettin and W. Lauterborn for stimulating discussions on chaos synchronization. This work was supported by the Deutsche Forschungsgemeinschaft (Pa 643/1-1), the German-Macedonian grant MAK-004-96, and the Macedonian Ministry of Science (08-1870).

Appendix In the following we show that for a uni-directionally coupled system x

=

f(x)

(11.7.1)

y

=

g(y,x)

(n.7.2)

the conditional Lyapunov exponents of the response system (11.7.2) are a subset of the complete Lyapunov spectrum of the combined system z = F(z)

(11.7.3)

with z = (y,x), F = (g,f) and y G IRm, x G IRn. The Lyapunov exponents of (11.7.3) can be computed via a Qi?-decomposition of the linearized flow matrix Y which is a solution of the matrix variational equation Y = DF(z(t))-Y

(11.7.4)

for the initial condition Y = I = diag(l,..., 1) (unit matrix). The block structure of the Jacobian matrix DF of F

< U ™>

Df.) leads to the same structure for the matrix Y:


-6)

The fact that the inverse R~x of the upper triangular matrix R of the QRdecomposition is again an upper triangular matrix can now be used to show that the orthonormal matrix Q possesses also a block structure

D -

Y • R'1 - ( YYAA

Q

Y R

~

YB YB\ YD

0

~ V 0 YD ) { 0 YA

•

RB1 YD

1

(( RRAA ) ' \ 0

\

+ YB Rn - Rn

1

>

R^ 1 \\ ^B

ni 7

R

(1L7

) D l )) R~

QA

0

QB QD

11.7 Applications and Conclusion

297

Furthermore, using the identities Q • Qtr = / = Qtr • Q one can show that QB = 0, and QA and QD are orthonormal. This gives the matrix equations YA

=

QARA

(11.7.9)

YD

=

QDRD.

(11.7.10)

Using these equations and the block structure of £>F and F is easy to show that YA and YD are given by the following matrix variational equations YA

= Dgy-YA

(11.7.11)

YD

= DfxYD.

(11.7.12)

These are, however, exactly the ODEs that are used for computing the Lyapunov exponents of the drive (11.7.11) and the conditional Lyapunov exponents of the response system (11.7.12). Since the Q-R-decomposition is unique the diagonal elements of RA and RD give the exponents of the drive systems \P = Km -tHRA-ii) and the conditional Lyapunov exponents of the response system

Xf = lim which all together constitute the Lyapunov spectrum of the coupled system (11.7.3).

References 1. Abarbanel, H.D.I., N.F. Rulkov, and M.M. Sushchik (1996) Phys.Rev.E 53(5), 4528. 2. Afraimovich, V.S., N.N. Verichev, and M.I. Rabinovich (1986) Radiophys. Quantum Electron 29, 795. 3. Alexander, J.C., J.A. Yorke, Z. You, and I. Kan (1992) Int.J.Bif. Chaos 2(4), 795. 4. Amritkar, R.E., and N. Gupte (1993) Phys.Rev.E 47(6), 3889. 5. Anishchenko, V.S., T.E. Vadivasova, D.E. Postnov, and M.A. Safonova (1992) Int. J. Bif. Chaos 2(3), 633. 6. Ashwin, P., J. Buescu, and I. Stewart (1994) Phys. Lett. A 193, 126. 7. Brown, R., N. Rulkov, and N.B. Tufillaro (1994a) Phys.Rev.E 50(6), 4488.

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12 Phase Synchronization of Regular and Chaotic Oscillators A. S. Pikovsky, M. G. Rosenblum, M. A. Zaks, and J. Kurths Department of Physics, Potsdam University, Am Neuen Palais 19, PF 601553, D-14415 Potsdam, Germany, http://www.agnld.uni-potsdam.de

12.1 Introduction Synchronization, a basic nonlinear phenomenon, discovered at the beginning of the modern age of science by Huygens [1], is widely encountered in variousfieldsof science, often observed in living nature [2] and finds a lot of engineering applications [3, 4]. In the classical sense, synchronization means adjustment of frequencies of self-sustained oscillators due to a weak interaction. The phase of oscillations may be locked by periodic external force; another situation is the locking of the phases of two interacting oscillators. One can also speak on "frequency entrainment". Synchronization of periodic systems is pretty well understood [3, 5, 6], effects of noise have been also studied [7]. In the context of interacting chaotic oscillators, several effects are usually referred to as "synchronization". Due to a strong interaction of two (or a large number) of identical chaotic systems, their states can coincide, while the dynamics in time remains chaotic [8, 9]. This effect is called "complete synchronization" of chaotic oscillators. It can be generalized to the case of non-identical systems [9, 10, 11], or that of the interacting subsystems [12, 13, 14]. Another well-studied effect is the "chaos-destroying" synchronization, when a periodic external force acting on a chaotic system destroys chaos and a periodic regime appears [15], or, in the case of an irregular forcing, the driven system follows the behavior of the force [16]. This effect occurs for a relatively strong forcing as well. A characteristic feature of these phenomena is the existence of a threshold coupling value depending on the Lyapunov exponents of individual systems [8, 9, 17, 18]. In this article we concentrate on the recently described effect of phase synchronization of chaotic systems, which generalizes the classical notion of phase locking. Indeed, for periodic oscillators only the relation between phases is important, while no restriction on the amplitudes is imposed. Thus, we define phase synchronization of chaotic system as appearance of a certain relation between the phases of interacting systems or between the phase of a system and that of an external force, while the amplitudes can remain chaotic and are, in general, noncorrelated. The phenomenon of phase synchronization has been theoretically studied in [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. It has been observed in experiments with electronic circuits [30] and lasers [31] and has been detected in physiological

306

12 Phase synchronization of regular and chaotic oscillators

systems [28, 32, 33]. We start with reviewing the classical results on synchronization of periodic selfsustained oscillators in sect. 12.2. We use the description based on a circle map and on a rotation number to characterize phase locking and synchronization. The very notion of phase and amplitude of chaotic systems is discussed in Section 12.3. We demonstrate this taking famouse Rossler and Lorenz models as examples. We show also that the dynamics of the phase in chaotic systems is silimar to that in noisy periodic ones. The next section 12.4 is devoted to effects of phase synchronization by periodic external force. We follow both a statistical approach, based on the properties of the invariant distribution in the phase space, and a topological method, where phase locking of individual periodic orbits embedded in chaos is studied. Different aspects of synchronization phenomena in coupled chaotic systems are described in sect. 12.5. Here we give an interpretation of the synchronization transition in terms of the Lyapunov spectrum of chaotic oscillations. We discuss also large systems, such as lattices and globally coupled populations of chaotic oscillators. These theoretical ideas are applied in sect. 12.8 to the data analysys problem. We discuss a possibility to detect phase synchronization in the observed bivariate data, and describe some recent achievments.

12.2 Synchronization of periodic oscillations In this section we remind basic facts on the synchronization of periodic oscillations (see, e.g.,[34]). Stable periodic oscillations are represented by a stable limit cycle in the phase space, and the dynamics (f)(t) of a phase point on this cycle can be described by

g=u, 0 ,

(12.1)

where UJO = 2TT/T0, and To is the period of the oscillation. It is important that starting from any monotonically growing variable 0 on the limit cycle (so that at one rotation 0 increases by 0), one can introduce the phase satisfying Eq. (12.1). Indeed, an arbitrary 0 obeys 0 — 7(0) with a periodic "instantaneous frequency" 7(0 -f 0) = 7(0). The change of variables (j) = u0 Jo [l(0))~ld0 gives the correct phase, with the frequency UJQ being defined from the condition 2?r = UJO JQ [v(0)]~~ld0. A similar approach leads to correct angle-action variables in Hamiltonian mechanics. We have performed this simple consideration to underline the fact that the notions of the phase and of the phase synchronization are universally applicable to any self-sustained periodic behavior independently on the form of the limit cycle. From (12.1) it is evident that the phase corresponds to the zero Lyapunov exponent, while negative exponents correspond to the amplitude variables. Note that we do not consider the equations for the amplitudes, as they are not universal. When a small external periodic force with frequency v is acting on this periodic oscillator, the amplitude is relatively robust, so that in the first approximation one

12.2 Synchronization of periodic oscillations

307

can neglect variations of the amplitude to obtain for the phase of the oscillator and the phase of the external force ip the equations ^=LJo

+ eG(4,il>),

^

= u,

(12.2)

where G(-, •) is 27r-periodic in both arguments and e measures the strength of the forcing. For a general method of derivation of Eq. (12.2) see [35]. The system (12.2) describes a motion on a 2-dimensional torus that appears from the limit cycle under periodic perturbation (see Fig. 12.1a,b). If we pick up the phase of oscillations <\> stroboscopically at times tn — n^j1, we get a circle map 0n+i = K + eg(4>n)

(12.3)

where the 27r-periodic function g(<j)) is defined via the solutions of the system (12.2). According to the theory of circle maps (cf. [34]), the dynamics can be characterized by the winding (rotation) number p = hm — n-»oo

27T71

which is independent on the initial point 0o and can take rational and irrational values. If it is irrational, then the motion is quasiperiodic and the trajectories are dense on the circle. Otherwise, if p = p/q, there exists a stable orbit with period q such that and the forcing frequency v

LJ =< 4: >- 9V -

( 12 -4)

at For p irrational and rational one has, respectively, a quasiperiodic dense orbit and a resonant stable periodic orbit on the torus (Fig. 12.1a,b). The main synchronization region where u = v corresponds to the winding number 1 (or, equivalently, 0 if we apply mod 2?r operation to the phase; for frequencies this means that we consider the difference u — u), other synchronization regions are usually rather difficult to observe. A typical picture of synchronization regions, called also "Arnold tongues", for the circle map (12.3) is shown in Fig. 12.1c. Several remarks are in order. 1) The concept of phase synchronization can be applied only to autonomous continuous-time systems. Indeed, if the system is discrete (i.e. a mapping), its period is an integer, and this integer cannot be adjusted to some other integer in a continuous way. The same is true for forced continuous-time oscillations (e.g. for the forced Duffing oscillator): here the frequency of oscillations is intrinsically coupled to that of the forcing and cannot be adjusted to some other frequency. We can formulate this also as follows: in discrete or forced systems there is no zero Lyapunov exponent, so there is no corresponding marginally stable variable (the phase) that can be governed by small external perturbations.

308

12 Phase synchronization

of regular and chaotic oscillators

(C)

1/4 1/3 2/5

1/2

2/3

OJ

Figure 12.1 Quasiperiodic (a) and periodic flow (b) on the torus; a stable periodic orbit is shown by the bold line, (c): The typical picture of Arnold tongues (with winding numbers atop) for the circle map.

2) The synchronization condition (12.4) does not mean that the difference between the phase <\> of an oscillator and that of the external force I/J (or between phases of two oscillators) must be a constant, as is sometimes assumed (see, e.g. [36]). Indeed, (12.2) implies, that to enable this, the function G should depend not on separate phases but only on their difference: G(0,^) = G(
\q<j)(t) — pij){t)\ < const .

3) The winding number is a continuous function of system parameters; typically it looks like a devil's staircase. Take the main phase-locking region. Continuity means that near the de-synchronization transition the mean oscillation frequency is close to the external one. As the external frequency v is varied, the de-synchronization transition appears as saddle-node bifurcation, where a stable p/q - periodic orbit collides with the corresponding unstable one, and both disappear. Near this bifurcation point, similarly to the type-I intermittency [37], a trajectory of the system spends a large time in the vicinity of the just disappeared periodic orbits; in the course of time evolution the long epochs when the phases are locked according to (12.5), are interrupted with relatively short time intervals where a phase slip occurs. 4) In the presence of external noise £(£) one can consider instead of (12.2) the Langevin equation = V

(12.6)

12.3 Phase of a chaotic oscillator

309

Equivalently, one can model the effect of noise by adding to the mapping (12.3) the noisy term rj: n + eg((/)n) + rjn

(12.7)

If the noise is small, the frequencies can be nearly locked, i.e. the averaged relation (12.4) is fulfilled. Large noise can cause phase slips, so that the phase performs a random-walk-like motion. In the case of unbounded (e.g. Gaussian) noise the mean phase drift is generally non-zero and, strictly speaking, the synchronization region vanishes. Nevertheless, the largest phase-locking intervals survive as regions of nearly constant mean frequency LJ. For detailed description of a simple model of synchronization in the presence of noise see [7].

12.3 Phase of a chaotic oscillator 12.3.1 Definition of the phase The first problem in extending the basic notions from periodic to chaotic oscillations is to properly define a phase. There seems to be no unambiguous and general definition of phase applicable to an arbitrary chaotic process. Roughly speaking, we want to define phase as a variable which is related to the zero Lyapunov exponent of a continuous-time dynamical system with chaotic behavior. Moreover, we want this phase to correspond to the phase of periodic oscillations satisfying (12.1). To be not too abstract, we illustrate a general approach below on the wellknown Rossler system. A projection of the phase portrait of this autonomous 3-dimensional system of ODEs (see eqs. (12.14) below) is shown in Fig. 12.2. Suppose we can define a Poincare map for our autonomous continuous-time system. Then, for each piece of a trajectory between two cross-sections with the Poincare surface we define the phase just proportional to time, so that the phase increment is 2TT at each rotation: (t) = 2TT * ~ * w + 27m, tn < t < tn+l. (12.8) tn+l — tn Here tn is the time of the n-th crossing of the secant surface. Note that for periodic oscillations corresponding to a fixed point of the Poincare map, this definition gives the correct phase satisfying Eq. (12.1). For periodic orbits having many rotations (i.e. corresponding to periodic points of the map) we get a piecewise-linear function of time, moreover, the phase grows by a multiple of 2?r during the period. The second property is in fact useful, as it represents the organization of periodic orbits inside the chaos in a proper way. The first property demonstrates that the phase of a chaotic system cannot be defined as unambiguously as for periodic oscillations. In particular, the phase crucially depends on the choice of the Poincare surface. Nevertheless, defined in this way, the phase has a physically important property: its perturbations neither grow nor decay in time, so it does correspond to the direction with the zero Lyapunov exponent in the phase space. We note also, that

310

12 Phase synchronization of regular and chaotic oscillators

6.5 6.4

(c)

6.3

(a)

0.0

0

20

X

Figure 12.2 Projection of the phase potrait of the Rossler system (a). The horizontal line shows the Poincare section that is used for computation of the amplitude mapping (b) and dependence of the return time (rotation period) on the amplitude (c).

this definition of the phase directly corresponds to the special flow construction which is used in the ergodic theory to describe autonomous continuous-time systems [38]. For the Rossler system Fig. 12.2(a) a proper choice of the Poincare surface may be the halfplane y = 0, x < 0. For the amplitude mapping xn -» xn+i we get a unimodal map Fig. 12.2(b) (the map is essentially one-dimensional, because the coordinate z for the Rossler attractor is nearly constant on the chosen Poincare surface). In this and in some other cases the phase portrait looks like rotations around a point that can be taken as the origin, so we can also introduce the phase as the angle between the projection of the phase point on the plane and a given direction on the plane (see also [22, 39]): = 3ictan(y/x) .

(12.9)

Note that although the two phases <\> and <\>p do not coincide microscopically, i.e on a time scale less than the average period of oscillation, they have equal average growth rates. In other words, the mean frequency defined as the average of d
12.3 Phase of a chaotic oscillator

311

12.3.2 Dynamics of the phase of chaotic oscillations In contrast to the dynamics of the phase of periodic oscillations, the growth of the phase in the chaotic case cannot generally be expected to be uniform. Instead, the instantaneous frequency depends in general on the amplitude. Let us hold to the phase definition based on the Poincare map, so one can represent the dynamics as (cf. [20]) An+l

=

M{An),

(12.10)

^

=

w(An) = u,o + F(An) .

(12.11)

As the amplitude A we take the set of coordinates for the point on the secant surface; it does not change during the growth of the phase from 0 to 2n and can be considered as a discrete variable; the transformation M defines the Poincare map. The phase evolves according to (12.11), where the "instantaneous" frequency u = 2n/(tn+i—tn) depends in general on the amplitude. Assuming the chaotic behavior of the amplitudes, we can consider the term uj(An) as a sum of the averaged frequency LJO and of some effective noise F(A); in exceptional cases F(A) may vanish. For the Rossler attractor the "period" of the rotations (i.e. the function 27r/oj(An)) is shown in Fig. 12.2(c). This period is not constant, so the function F(A) does not vanish, but the variations of the period are relatively small. Hence, the Eq. (12.11) is similar to the equation describing the evolution of phase of periodic oscillator in the presence of external noise. Thus, the dynamics of the phase is generally diffusive: for large t one expects < ((t) - 0(0) - u0t)2 >oc Dpt, where the diffusion constant Dp determines the phase coherence of the chaotic oscillations. Roughly speaking, the diffusion constant is inversely proportional to the width of the spectral peak calculated for the chaotic observable [40]. Generalizing Eq. (12.11) in the spirit of the theory of periodic oscillations to the case of periodic external force, we can write for the phase

Here we assume that the force is small (of order of e) so that it affects only the phase, and the amplitude obeys therefore the unperturbed mapping M. This equation is similar to Eq. (12.6), with the amplitude-depending part of the instantaneous frequency playing the role of noise. Thus, we expect that in general the synchronization phenomena for periodically forced chaotic system are similar to those in noisy driven periodic oscillations. One should be aware, however, that the "noisy" term F(A) can be hardly explicitly calculated, and for sure cannot be considered as a Gaussian ^-correlated noise as is commonly assumed in the statistical approaches [7, 41].

312

12 Phase synchronization of regular and chaotic oscillators

12.4 Phase synchronization by external force 12.4.1 Synchronization region We describe here the effect of phase synchronization of chaotic oscillations by periodic external force, taking as examples two prototypic models of nonlinear dynamics: the Lorenz x = 10(y-x), y = 2Sx-y-xz, (12.13) z = —8/3 • z + xy + E cos vt. and the Rossler x = — y - z -f Ecosvt , y = x + 0.15y, z = 0.4 + z ( z - 8 . 5 ) .

(12.14)

oscillators. In the absence of forcing, both are 3-dimensional dissipative systems which admit a straightforward construction of the Poincare maps. Moreover, we can simply use the phase definition (12.9), taking the original variables (x,y) for the Rossler system and the variables (y/x2 -f y2 — uo, z — zo) for the Lorenz system (where UQ = 12\/2 and ZQ = 27 are the coordinates of the equilibrium point, the "center of rotation"). The mean rotation frequency can be thus calculated as ft = lim 2TT—

(12.15)

t where Nt is the number of crossings of the Poincare section during observation time t. This method can be straightforwardly applied to the observed time series, in the simplest case one can, e.g., take for Nt the number of maxima (of x(t) for the Rossler system and of z(t) for the Lorenz one). Dependence of the obtained in this way frequency fi on the amplitude and frequency of the external force is shown in Fig. 12.3. Synchronization here corresponds to the plateau fi = v. One can see that the synchronization properties of these two systems differ essentially. For the Rossler system there exists a well-expressed region where the systems are perfectly locked. Moreover, there seems to be no amplitude threshold of synchronization (cf. Fig. 12.1c, where the phase-locking regions start at e = 0). It appears that the phase locking properties of the Rossler system are practically the same as for a periodic oscillator. On the contrary, for the Lorenz system we observe the frequency locking only as a tendency seen at relatively large forcing amplitudes, as this should be expected for oscillators subject to a rather strong noise. In this respect, the difference between Rossler and Lorenz systems can be described in terms of phase diffusion properties (see Sect. 12.3.2). Indeed, the phase diffusion coefficient for autonomous Rossler system is extremely small Dp < 10~4, whereas for the Lorenz system it is several oder of magnitude larger, Dp « 0.2 [24]. This difference in the coherence of the phase of autonomous oscillations implies different response to periodic forcing. In the following sections we discuss the phase synchronization of chaotic oscillations from the statistical and the topological viewpoints.

12.4 Phase synchronization by external force

v-Q

313

v-Q

Figure 12.3 The phase synchronization regions for the Rossler (a) and the Lorenz (b) systems.

12.4.2 Statistical approach We define the phase of an autonomous chaotic system as a variable that corresponds to invariance with respect to time shifts. Therefore, the invariant probability distribution as a function of the phase is nearly uniform. This follows from the ergodicity of the system: the probability is proportional to the time a trajectory is spending in a region of the phase space, and according to the definition (12.8) the phase motion is (piecewise) uniform. With external forcing, the invariant measure depends explicitly on time. In the synchronization region we expect that the phase of oscillations nearly follows the phase of the force, while without synchronization there is no definite relation between them. Let us observe the oscillator stroboscopically, at the moments corresponding to some phase tpo of the external force. In the synchronous state the probability distribution of the oscillator phase will be localized near some preferable value (which of course depends on the choice of ^o). In the non-synchronous state the phase is spread along the attractor. We illustrate this behavior of the probability density in Fig. 12.4. One can say that synchronization means localization of the probability density near some preferable time-periodic state. In other words, this means appearance of the long-range correlation in time and of the significant discrete component in the power spectrum of oscillations. Let us consider now the ensemble interpretation of the probability. Suppose we take a large ensemble of identical copies of the chaotic oscillator which differ only by their initial states, and let them evolve under the same periodic forcing. After the transient, the projections of the phase state of each oscillator onto the plane x, y form the cloud that exactly corresponds to the probability density. Let us now consider the ensemble average of some observable. Without synchronization the cloud is spread over the projection of the attractor (Fig. 12.4b), and the average is small: no significant average field is observed. In the synchronous state the probability is localized (Fig. 12.4a), so the average is close to some middle point of the cloud; this point rotates with the frequency v and one observes large regular

314

12 Phase synchronization of regular and chaotic oscillators

Figure 12.4 Distribution inside (a) and outside (b) the synchronization region for the Rossler system, shown with black dots. The autonomous Rossler at tract or is shown with gray.

oscillations of the average field. Hence, the synchronization can be easily indicated through the appearance of a large (macroscopic) mean field in the ensemble. Physically, this effect is rather clear: unforced chaotic oscillators are not coherent due to internal chaos, thus the summation of their fields yields a small quantity. Being synchronized, the oscillators become coherent with the external force and thereby with each other, so the coherent summation of their fields produces a large mean field. An important consequence of the statistical approach described above is that the phase synchronization can be characterized without explicit computation of the phase and/or the mean frequency: it can be indicated implicitly by the appearance of a macroscopic mean field in the ensemble of oscillators, or by the appearance of the large discrete component in the spectrum. Although there may be other mechanisms leading to the appearance of macroscopic order, the phase synchronization appears to be one of the most common ones.

12.4.3 Interpretation through embedded periodic orbits In order to understand structural metamorphoses of attracting chaotic sets under the action of the synchronizing force, it is convenient to look at the properties of individual periodic orbits embedded into the strange attractors. Unstable periodic orbits are known to build a kind of "skeletons" for chaotic sets [34]; in particular, each of the systems (12.13) and (12.14) in the absence of forcing possesses infinite number of periodic solutions with two-dimensional unstable manifolds. Let us pick up one of these solutions and consider the dynamics on its two-dimensional global stable manifold. Prom this point of view, there is no difference from familiar problem of the synchronization of stable periodic oscillations by external driving force (see Sect. 12.2 above): the winding number p can be introduced, and in

12-4 Phase synchronization by external force

315

the parameter space one should observe synchronization inside the Arnold tongues (locking regions) which correspond to rational values of p (cf. Fig. 12.1). Like in the situation described in Sect. 12.2 above, an invariant torus evolves from the periodic orbit of the autonomous chaotic system. Trajectories wind around this torus; inside the Arnold tongues there are two closed orbits on its surface: the attracting one which will call below "phase-stable", and the repelling one, called "phase-unstable". On the border of the locking region these two orbits coalesce and disappear via the tangent bifurcation. Outside the tongues the motion corresponding to this particular periodic orbit is not synchronized and the trajectories are dense on the torus.

\ .\\ *

\

\

\

\\

A // / •' \ ! i •'

!

V / •'

1.020 frequency v

Figure 12.5 The Arnold tongues for the unstable periodic orbits in the Rossler system with different number of rotations around the origin. In the shadowed region the mean frequency of oscillations virtually coincides with the forcing frequency v.

Since in the entire phase space of the autonomous system the considered periodic solution is unstable, the torus in the weakly driven system is also unstable. On the plane of the parameters E and v the tip of the main Arnold tongue lies in the point E — 0, v = u)i where ui is the individual mean frequency of the considered autonomous orbit.This frequency differs from the formally defined frequency of the periodic solution 2n/Ti where Ti is a period of the orbit: we take here into account also the number of round trips U{ of the orbit and write U{ = limijTi). Naturally, the values of Ui differ for different periodic orbits; however, in many autonomous dissipative systems (like in Eq. (12.14)) chaos manifests itself in the form of nearly isochronous rotations, and the frequencies Ui are very close to each other. Respectively, the Arnold tongues overlap (Fig. 12.5), and one can find the parameter region in which all periodic motions are locked by the external force. If the forcing remains moderate, this is the overlapping region for the leftmost and the rightmost Arnold tongues which correspond to the periodic orbits of the

316

12 Phase synchronization of regular and chaotic oscillators

autonomous system with, respectively, the smallest and the largest values of uj{. Inside this region the chaotic trajectories repeatedly visit the neighborhoods of the tori; moving along the surface of a torus they approach the phase-stable solution and remain there for a certain time before the "transverse" (amplitude) instability bounces them to another torus. Since all periodic motions are locked, the phase remains localized within the bounded domain: one observes phase synchronization. (a)

(c)

(b)

\ s •.:• •:•*. '.. '-S" -J

-12

-8

-4

-12

-8

-4

-12

Figure 12.6 The Poincare maps of the forced Rossler oscillator inside and outside the synchronization region. The markers denote the points belonging to the period-2 cycle, they lie apparently "outside" the attractor. In Fig. 12.6 we show the phase portraits of the forced Rossler oscillator in the synchronized and non-synchronized states. The Poincare maps are presented taken at the secant surface y = 0, the coordinates are the variable x of the Rossler system and the phase of the external force ip (note that this representation is complementary to Fig. 12.4, where the phase of the external force is fixed). On these mappings the phase-stable orbits are represented by finite invariant subsets of points, they form a kind of the "skeleton" for the attractor. Similarly, phaseunstable orbits are a skeleton of the repeller (which is not shown in Fig. 12.6a); the latter plays a role of a barrier which separates the attraction domains of the two equivalent attractors whose phases differ by 2ir. On approaching the boundary of the locked region from inside, the corresponding phase-stable and phase-unstable periodic orbits come closer. When they coalesce, attractor and repeller collide in the points of the "glued" orbit. After the bifurcation, a "channel" appears in the barrier, enabling phase slips during which the phase changes by ±2n. These slips appear in Fig. 12.6b as the rare points with the phases I/J < 3 and I/J > 5.5. Since the Arnold tongues for different periodic orbits do not coincide, the onset

12.4 Phase synchronization by external force

317

of frequency lockings for these orbits occurs at different values of the frequency of external force. As a result, close to the threshold the synchronized segments of the trajectory alternate with the non-synchronized ones, and the whole transition to phase synchronization is smeared. The behavior observed at this transition is a specific kind of intermittency which we call "eyelet" since the seldom leakages from the locked state require the very precise hitting of certain small regions in the phase space. The following description sketches the features of the transition mechanism (see [27] for the accurate derivation). The dynamics of the phase can be reasonably well approximated by the circle mapping. Just outside the tangent bifurcation the characteristic time intervals between the phase slips obey the inverse square root law: r « C\\v — i/c|~1//2 where vc is the bifurcation value of the frequency v. Let the trajectory be reinjected into the vicinity of the respective unstable torus, at a small distance do from it. To exhibit the slip, the trajectory should remain close to the torus for the time interval not smaller than r. Within this time the distance to the torus grows: d r « doeAr where A is the positive Lyapunov exponent of the torus (for the weak forcing it is close to the Lyapunov exponent of the respective unstable periodic orbit in the autonomous system); we require d r to remain small: dT Assuming that the density of invariant probability on the attractor is (locally) uniform we estimate the probability to undergo a phase slip as proportional to the length of the interval do; for the latter holds do < C2 exp(-Ar) « C2 exp(-ACi|i/ - vc\~l/2) . Just outside the border of the synchronization region this interval is an extremely small "eyelet", and phase slips are exceptionally rare. In its turn, the increment S of the rotation number (with respect to the value of p inside the locked region) is proportional to the averaged number of phase slips per mapping iteration, which leaves us with logS « - | i / - vc\~1/2 (Fig. 12.7). The exponentially slow eyelet intermittency is the reason why the region of phase synchronization often appears to be larger than the overlapping part of the Arnold tongues and in certain cases seems to be observed also under small forcing amplitudes, for which there is no full phase synchronization at all. Only after a sufficiently large number of tangent bifurcations the probability of phase slip becomes noticeable, and one observes a deviation of the mean observed frequency from the frequency of the external force. This picture is basically confirmed by numerical comparison of the domain of phase synchronization for the forced Rossler equations with the locked regions of individual periodic orbits (Fig. 12.5,12.7). However, certain peculiarities of the Rossler system do not fit the predictions. Phase synchronization is observed well below the intersection of the outermost tongues, i.e. in the domain where only a part of periodic orbits is synchronized. Thus, it appears that some periodic orbits do not contribute to the phase rotation. In the phase space, these orbits seem to lie outside of the bulk of the attractor (Fig, 12.6a); consequently, their vicinities are visited extremely seldom and possible phase slips are simply not detectable.

318

12 Phase synchronization of regular and chaotic oscillators

Figure 12.7 The average number of phase slips at the border of the synchronization region vs. the deviation of the forcing amplitude e.

Why some periodic orbits under the influence of forcing become "non-observable", remains an open question. Analysis in terms of unstable periodic orbits allows one to understand the fine features of the onset of phase synchronization. We have discussed here the simplest case when the borders of the region of full phase synchronization are given by the phase-locking regions of the periodic orbits. More complex situations can occur if one of these borders is reached on a chaotic everywhere dense trajectory. Then the attractor and the repeller can collide in a dense set of points; similar situation is encountered in a quasiperiodically forced circle map [42, 43].

12.5 Phase synchronization in coupled systems Now we demonstrate the effects of phase synchronization in coupled chaotic oscillators. We start with the simplest case of two interacting systems, and then briefly discuss oscillator lattices, globally coupled systems, and space-time chaos.

12.5.1 Synchronization of two interacting oscillators We consider here two non-identical coupled Rossler systems 2/1,2 £1,2

= =

^1,2^1,2+^2/1,2, / +21,2(^1,2 - c ) ,

(12.16)

where a = 0.165, / = 0.2, c = 10. The parameters u)\^ = wo ± ACJ and e determine the mismatch of natural frequencies and the coupling, respectively. Again, like in the case of periodic forcing, we can define the mean frequencies

12.5 Phase synchronization in coupled systems

319

fii)2 of oscillations of each system, and study the dependence of the frequency mismatch fl2 — fh on the parameters ACJ, e. This dependence is shown in Fig. 12.8 and demonstrates a large region of synchronization between two oscillators. It is in-

e

0.1

0.15

Figure 12.8 Synchronization of two coupled Rossler oscillators; CJQ = 1.

structive to characterize the synchronization transition by means of the Lyapunov exponents (LE). The 6-order dynamical system (12.16) has 6 LEs (see Fig. 12.9). For zero coupling we have a degenerate situation of two independent systems, each of them has one positive, one zero, and one negative exponent. The two zero exponents correspond to the two independent phases. With coupling, the phases become dependent and the degeneracy must be removed: only one LE should remain exactly zero. We observe, however, that for small coupling also the second zero Lyapunov exponent remains extremely small (in fact, numerically indistinguishable from zero). Only at relatively stronger coupling, when the synchronization sets on, the second LE becomes negative: now the phases are dependent and a relation between them is stable. Note that the two positive exponents remain positive which means that the amplitudes remain chaotic and independent: the coupled system remains in the state of hyperchaos. With the increase of coupling one of the positive LE becomes smaller. Physically this means that not only the phases are locked, but the difference between the amplitudes is suppressed by coupling as well. At a certain coupling only one LE remains positive, so one can expect synchronization both in phases and amplitudes. As the systems are not identical (due to the frequency mismatch), their states cannot be identical: x\(t) / X2(t). However, almost perfect correspondence between the time-shifted states of the systems can be observed: xi(t) « X2(t - At). This phenomenon is called "lag synchronization" [26]. With further increase of the coupling € the lag At decreases and the states of two systems become nearly identical, like in case of complete synchronization (see the paper by Kocarev and Parlitz in this volume [14]).

320

12 Phase synchronization of regular and chaotic oscillators

-0.1 0.00

0.20

Figure 12.9 The Lyapunov exponents A (bottom panel, only the 4 largest LEs are depicted) and the frequency difference vs. the coupling e in the coupled Rossler oscillators; UJQ = 0.97, Aa; = 0.02. Transition to the phase (ep) and to the lag synchronization (ei) are marked.

12.5.2 Synchronization in a Population of Globally Coupled Chaotic Oscillators A number of physical, chemical and biological systems can be viewed at as large populations of weakly interacting non-identical oscillators [35]. One of the most popular models here is an ensemble of globally coupled nonlinear oscillators (often called "mean-field coupling"). A nontrivial transition to self-synchronization in a population of periodic oscillators with different natural frequencies coupled through a mean field has been described by Kuramoto [35, 44]. In this system, as the coupling parameter increases, a sharp transition is observed for which the mean field intensity serves as an order parameter. This transition owes to a mutual synchronization of the periodic oscillators, so that their fields become coherent (i.e. their phases are locked), thus producing a macroscopic mean field. In its turn, this field acts on the individual oscillators, locking their phases, so that the synchronous state is self-sustained. Different aspects of this transition have been studied in [45, 46, 47], where also an analogy with the second-order phase transition has been exploited. A similar effect can be observed in a population of non-identical chaotic systems, e.g. the Rossler oscillators =

-LJiVi

- Zi+

eX,

(12.17) coupled via the mean field X = N xY2i xi- Here N is the number of elements in the ensemble, e is the coupling constant, a and uj{ are parameters of the Rossler oscillators. The parameter uj{ governs the natural frequency of an individual system. We take a set of frequencies ui which are Gaussian-distributed around the

12.6 Lattice of chaotic oscillators

321

mean value uo with variance (Au;)2. The Rossler system typically shows windows of periodic behavior as the parameter UJ is changed; therefore we usually choose a mean frequency UJQ in a way that we avoid large periodic windows. In our computer simulations we solve numerically Eqs. (12.17) for rather large ensembles iV = 3000 + 5000. With an increase of the coupling strength e, the appearance of a non-zero macroscopic mean field X is observed [22]. This indicates the phase synchronization of the Rossler oscillators that arises due to their interaction via mean field. This mean field is large, if the attractors of individual systems are phase-coherent (parameter a = 0.15) and the phase is well-defined. On the contrary, in the case of the funnel attractor a = 0.25, when the oscillations look wild, and the imaging point makes large and small loops around the origin, the field is rather small, and there seems to be no way to choose the Poincare section unambiguously. Nevertheless, in both cases synchronization transition is clearly indicated by the onset of the mean field, without computation of the phases themselves.

12,6 Lattice of chaotic oscillators If chaotic oscillators are ordered in space and form a lattice, only the nearest neighbors interact. Such a situation is relevant for chemical systems, where homogeneous oscillations are chaotic, and the diffusive coupling can be modeled with dissipative nearest neighbors interaction [48, 39]. In a lattice, one can expect complex spatiotemporal synchronization structures to be observed. Consider as a model a 1-dimensional lattice of R"ossler oscillators with local dissipative coupling:

Vj = Zj =

VjXj + ayj + e(yj+i - 2yj + yj-i), 0.4+ (XJ -8.5)zj.

(12.18)

Here the index j = 1 , . . . , N counts the oscillators in the lattice and e is the coupling coefficient. To study synchronization in a lattice of non-identical oscillators, we introduce a linear distribution of natural frequencies Uj UJj

= cj! + S(j - 1)

(12.19)

where S is the frequency mismatch between neighboring sites. Depending on the values of 6 we observed two scenarios of transition to synchronization [23]. For small <5, the transition occurs smoothly, i.e. all the elements along the chain gradually adjust their frequencies. If the frequency mismatch is larger, clustering is observed: the oscillators build phase-synchronized groups having different mean frequencies. At the borders between clusters phase slips occur; this can be considered as appearance of defects in the spatio-temporal representation. Both regular and irregular patterns of defects have been reported in ref. [23].

322

12 Phase synchronization of regular and chaotic oscillators

12.7 Synchronization of space-time chaos The idea of phase synchronization can be also applied to space-time chaos. E.g., in the famous complex Ginzburg-Landau equation [49, 50, 51] dta = (1 + iu>0)a - (1 + ia)\a\2a + (1 + i(3)d?a ,

(12.20)

there are regimes where the complex amplitude a rotates with some mean frequency, but these rotations are not regular: the phase deviates irregularly in space and time (this regime is called "phase turbulence"). Let us now add periodic in time spatially homogeneous forcing of amplitude B and frequency ue. Transition into a reference frame rotating with this external forcing (a-¥ A = aexp(—iuet)) reduces Eq. (12.20) to dtA = (1 + iv)A - (1 + ia)\A\2A + (1 + i&)d\A + B ,

(12.21)

where v = UQ — ue is the frequency mismatch between the frequency of the external force and the frequency of small oscillations. An analysis of different regimes in the system (12.21) has been recently performed [52]. As one can expect, a very strong force suppresses turbulence and the spatially homogeneous periodic in time synchronous oscillations are observed, while a small force has no significant influence on the turbulent state. A nontrivial regime is observed for intermediate forcing: in some parameter range the irregular fluctuations of the phase are not completely suppressed but are bounded: the whole system oscillates "in phase" with the external force and is highly coherent, although some small chaotic variations persist. One can easily see an analogy to the phase synchronization of chaotic oscillators, where chaos remains while the phase becomes entrained.

12.8 Detecting synchronization in data The analysis of relation between the phases of two systems, naturally arising in the context of synchronization, can be used to approach a general problem in time series analysis. Indeed, bivariate data are often encountered in the study of real systems, and the usual aim of the analysis of such data is to find out whether two signals are dependent or not. As experimental data are very often non-stationary, the traditional techniques, such as cross-spectrum and cross-correlation analysis [53], or non-linear characteristics like generalized mutual information [54] or maximal correlation [55] have their limitations. Prom the other side, sometimes it is reasonable to assume that the observed signals originate from two weakly interacting systems. The presence of this interaction can be found by means of the analysis of instantaneous phases of these signals. These phases can be unambiguously obtained with the help of the analytic signal concept based on the Hilbert transform (for an introduction see [53, 24]). It goes as follows: for an arbitrary scalar signal s(t) one can construct a complex function of time (analytic signal) C(t) = s(t) + is(t) = A(t)ei(l>HW where s(t) is the Hilbert transform of s{t), s(t) = T T ^ P . V . r

^ d r

>

(12.22)

12.9 Conclusions

323

and A(t) and (j>H{t) are the instantaneous amplitude and phase (P.V. means that the integral is taken in the sense of the Cauchy principal value). As recently shown in [19, 24], the phase defined by this method from an appropriately chosen oscillatory observable practically coincides with the phase of an oscillator computed according to one of the definitions given in Sec. 12.3. Therefore, the analysis of the relationship between these Hilbert phases appears to be an appropriate tool to detect synchronous epochs from experimental data and to check for a weak interaction between systems under study. It is very important that the Hilbert transform does not require stationarity of the data, so we can trace synchronization transitions even from nonstationary data. We recall again the above mentioned similarity of phase dynamics in noisy and chaotic oscillators (see Sect. 12.3.2). A very important consequence of this fact is that, using the synchronization approach to data analysis, we can avoid the hardly solvable dilemma "noise vs chaos": irrespectively of the origin of the observed signals, the approach and techniques of the analysis are unique. Quantification of synchronization from noisy data is considered in [56]. Application of these ideas allowed us to find phase locking in the data characterizing mechanisms of posture control in humans while quiet standing [32, 28]. Namely, the small deviations of the body center of gravity in anterior-posterior and lateral directions were analyzed. In healthy subjects, the regulation of posture in these two directions can be considered as independent processes, and the occurrence of some interrelation possibly indicates a pathology. It is noteworthy that in several records conventional methods of time series analysis, i.e. the cross-spectrum analysis and the generalized mutual information failed to detect any significant dependence between the signals, whereas calculation of the instantaneous phases clearly showed phase locking. Complex synchronous patterns have been found recently in the analysis of interaction of human cardiovascular and respiratory systems [33]. This finding possibly indicates the existence of a previously unknown type of neural coupling between these systems. Analysis of synchronization between brain and muscle activity of a Parkinsonian patient [56] is relevant for a fundamental problem of neuroscience: can one consider the synchronization between different areas of the motor cortex as a necessary condition for establishing of the coordinated muscle activity? It was shown [56] that the temporal evolution of the coordinated pathologic tremor activity directly reflects the evolution of the strength of synchronization within a neural network involving cortical motor areas. Additionally, the brain areas with the tremorrelated activity were localized from noninvasive measurements.

12.9 Conclusions The main idea of this paper is to demonstrate that synchronization phenomena in periodic, noisy and chaotic oscillators can be understood within a unified framework. This is achieved by extending the notion of phase to the case of continuous-

324

12 Phase synchronization of regular and chaotic oscillators

time chaotic systems. Because the phase is introduced as a variable corresponding to the zero Lyapunov exponent, this notion should be applicable to any autonomous chaotic oscillator. Although we are not able to propose a unique and rigorous approach to determine the phase, we have shown that it can be introduced in a reasonable and consistent way for basic models of chaotic dynamics. Moreover, we have shown that even in the case when the phases are not well-defined, i.e. they cannot be unambiguously computed explicitly, the presence of phase synchronization can be demonstrated indirectly by observations of the mean field and the spectrum, i.e. independently of any particular definition of the phase. In a rather general framework, any type of synchronization can be considered as appearance of some additional order inside the dynamics. For chaotic systems, e.g., the complete synchronization means that the dynamics in the phase space is restricted to a symmetrical submanifold. Thus, from the point of view of topological properties of chaos, the synchronization transition usually means the simplification of the structure of the strange attractor. In discussing the topological properties of phase synchronization, we have shown that the transition to phase synchronization corresponds to splitting of the complex invariant chaotic set into distinctive attractor and repeller. Analogously to the complete synchronization, which appears through the pitchfork bifurcation of the strange attractor, one can say that the phase synchronization appears through tangent bifurcation of strange sets. Because of the similarity in the phase dynamics, one may expect that many, if not all, synchronization features known for periodic oscillators can be observed for chaotic systems as well. Indeed, here we have described effects of phase and frequency entrainment by periodic external driving, both for simple and spacedistributed chaotic systems. Further, we have described synchronization due to interaction of two chaotic oscillators, as well as self-synchronization in globally coupled large ensembles. As an application of the developed framework we have discussed a problem in data analysis, namely detection of weak interaction between systems from bivariate data. The three described examples of the analysis of physiological data demonstrate a possibility to detect and characterize synchronization even from nonstationary and noisy data. Finally, we would like to stress that contrary to other types of chaotic synchronization, the phase synchronization phenomena can happen already for very weak coupling, which offers an easy way of chaos regulation.

Acknowledgements We thank G. Osipov, H. Chate, 0. Rudzick, U. Parlitz, P. Tass, C. Schafer for useful discussions. A.P. and M.Z. acknowledge support of the Max-Planck-Society.

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[17] L. Bezaeva, L. Kaptsov, and P. S. Landa. Synchronization threshold as the criterium of stochasticity in the generator with inertial nonlinearity. Zhurnal Tekhnicheskoi Fiziki, 32:467-650, 1987. (in Russian). [18] G. I. Dykman, P. S. Landa, and Yu. I Neymark. Synchronizing the chaotic oscillations by external force. Chaos, Solitons & Fractals, l(4):339-353, 1992. [19] M. Rosenblum, A. Pikovsky, and J. Kurths. Phase synchronization of chaotic oscillators. Phys. Rev. Lett, 76:1804, 1996. [20] A. S. Pikovsky. Phase synchronization of chaotic oscillations by a periodic external field. Sov. J. Commun. Technol. Electron., 30:85, 1985. [21] E. F. Stone. Frequency entrainment of a phase coherent attractor. Phys. Lett. A, 163:367-374, 1992. [22] A. Pikovsky, M. Rosenblum, and J. Kurths. Synchronization in a population of globally coupled chaotic oscillators. Europhys. Lett, 34(3):165-170, 1996. [23] G. Osipov, A. Pikovsky, M. Rosenblum, and J. Kurths. Phase synchronization effects in a lattice of nonidentical Rossler oscillators. Phys. Rev. E, 55(3):23532361, 1997. [24] A. Pikovsky, M. Rosenblum, G. Osipov, and J. Kurths. Phase synchronization of chaotic oscillators by external driving. Physica D, 104:219-238, 1997. [25] A. Pikovsky, M. Zaks, M. Rosenblum, G. Osipov, and J. Kurths. Phase synchronization of chaotic oscillations in terms of periodic orbits. CHAOS, 7(4):680-687, 1997. [26] M. Rosenblum, A. Pikovsky, and J. Kurths. From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett., 78:4193-4196, 1997. [27] A. Pikovsky, G. Osipov, M. Rosenblum, M. Zaks, and J. Kurths. Attractorrepeller collision and eyelet intermittency at the transition to phase synchronization. Phys. Rev. Lett, 79:47-50, 1997. [28] M. Rosenblum, A. Pikovsky, and J. Kurths. Effect of phase synchronization in driven chaotic oscillators. IEEE Trans. CAS-I, 44(10):874-881, 1997. [29] E. Rosa Jr., E. Ott, and M. H. Hess. Transition to phase synchronization of chaos. Phys. Rev. Lett, 80(8): 1642-1645, 1998. [30] U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev. Experimental observation of phase synchronization. Phys. Rev. E., 54(2):2115-2118, 1996. [31] D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg. Experimental evidence of frequency entrainment between coupled chaotic oscillations. Phys. Rev. E, 57(3):3649-3651, 1998.

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[32] M. G. Rosenblum, G. I. Firsov, R.A. Kuuz, and B. Pompe. Human postural control: Force plate experiments and modelling. In H. Kantz, J. Kurths, and G. Mayer-Kress, editors, Nonlinear Analysis of Physiological Data, pages 283-306. Springer, Berlin, 1998. [33] C. Schafer, M. G. Rosenblum, J. Kurths, and H.-H. Abel. Heartbeat synchronized with ventilation. Nature, 392(6673):239-240, March 1998. [34] E. Ott. Chaos in Dynamical Systems. Cambridge Univ. Press, Cambridge, 1992. [35] Y. Kuramoto. Chemical Oscillations, Waves and Turbulence. Springer, Berlin, 1984. [36] D. Y. Tang and N. R. Heckenberg. Synchronization of mutually coupled chaotic systems. Phys. Rev. E, 55(6):6618-6623, 1997. [37] P. Berge, Y. Pomeau, and C. Vidal. Order within chaos. Wiley, New York, 1986. [38] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai. Ergodic Theory. Springer, New York, 1982. [39] A. Goryachev and R. Kapral. Spiral waves in chaotic systems. Phys. Rev. Lett, 76(10):1619-1622, 1996. [40] J. D. Farmer. Spectral broadening of period-doubling bifurcation sequences. Phys. Rev. Lett, 47(3):179-182, 1981. [41] H. Z. Risken. The Fokker-Planck Equation. Springer, Berlin, 1989. [42] A. Bondeson, E. Ott, and T. M. Antonsen. Quasiperiodically forced damped pendula and schrodinger equation with quasiperiodic potentials: implication of their equivalence. Phys. Rev. Lett, 55(20):2103-2106, 1985. [43] U. Feudel, J. Kurths, and A. Pikovsky. Strange nonchaotic attractor in a quasiperiodically forced circle map. Physica D, 88(3-4): 176-186, 1995. [44] Y. Kuramoto. Self-entrainment of a population of coupled nonlinear oscillators. In H. Araki, editor, International Symposium on Mathematical Problems in Theoretical Physics, page 420, New York, 1975. Springer Lecture Notes Phys., v. 39. [45] H. Sakaguchi, S. Shinomoto, and Y. Kuramoto. Local and global selfentrainments in oscillator lattices. Prog. Theor. Phys., 77(5):1005-1010,1987. [46] H. Daido. Discrete-time population dynamics of interacting self-oscillators. Prog. Theor. Phys., 75(6):1460-1463, 1986.

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[47] H. Daido. Intrinsic fluctuations and a phase transition in a class of large population of interacting oscillators. J. Stat. Phys., 60(5/6):753-800, 1990. [48] L. Brunnet, H. Chate, and P. Manneville. Long-range order with local chaos in lattices of diffusively coupled ODEs. Physica D, 78:141-154, 1994. [49] M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium. Rev. Mod. Phys., 65(3):851, 1993. [50] H. Chate. Spatiotemporal intermittency regimes of the one-dimensional complex ginzburg-landau equation. Nonlinearity, 7:185-204, 1994. [51] B. I. Shraiman, A. Pumir, W. van Saarlos, P.C. Hohenberg, H. Chate, and M. Holen. Spatiotemporal chaos in the one-dimensional Ginzburg-Landau equation. Physica D, 57:241-248, 1992. [52] H. Chate, A. Pikovsky, and O. Rudzick. Forcing oscillatory media: Phase kinks vs. synchronization. Physica D (in press), 1998. [53] P. Panter. Modulation, Noise, and Spectral Analysis. McGraw-Hill, New York, 1965. [54] B. Pompe. Measuring statistical dependencies in a time series. J. Stat. Phys., 73:587-610, 1993. [55] H. Voss and J. Kurths. Reconstruction of nonlinear time delay models from data by the use of optimal transformations. Phys. Lett. A, 234:336-344, 1997. [56] P. Tass, M.G. Rosenblum, J. Weule, J. Kurths, A.S. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Preund. Detection of n : m phase locking from noisy data - Application to magnetoencephalography. Phys. Rev. Lett., 1998. submitted.

13 Tools for Detecting and Analyzing Generalized Synchronization of Chaos in Experiment A. Kittel1, J. Parisi1, and K. Pyragas2 1. University of Oldenburg, Oldenburg, Germany 2. Semiconductor Physics, Institute and Vilnius Pedagogical University, Vilnius, Lithuania

13.1 Introduction The cooperative behavior of coupled dynamical systems is becoming to an important field of nonlinear dynamics. Synchronization effects in systems with periodic behavior are widely used in engineering, for example, for improvement of the line width of a high-power generator with the help of a low-power generator having a narrower spectral line. In recent years, the synchronization of coupled chaotic systems has become an area of active research. The motivation for these investigations derived from possible applications of this phenomenon to secure communications [1], the long-term prediction of chaotic systems [2], controlling chaos [3], the model verification of nonlinear dynamics [4], or the estimation of model parameters [5]. Also, understanding the synchronization process is important for efficient control of the spatiotemporal chaos that occurs in various complex systems such as laser arrays [6] or cardiac systems [7]. A generic feature of nonlinear systems exhibiting chaotic motion is extreme sensitivity to initial conditions. This feature, known as the "butterfly effect", would seem to defy synchronization among dynamical variables in coupled chaotic systems. Nonetheless, coupled systems with certain properties of symmetry may exhibit synchronized chaotic motions. Most frequently a situation is studied where the complete system consists of coupled identical subsystems. Many different examples of this type have been introduced [2, 8, 9]. In these cases, the synchronization is easy to detect. It appears as an actual equality of the corresponding variables of the coupled systems as they evolve in time. Geometrically, this implies a collapse of the overall evolution onto the identity hyperplane in the full phase space. As suggested in Ref. [10], we refer to this type of synchronization as an identical synchronization (IS). A more complicated situation arises when coupled non-identical chaotic systems are investigated. For essentially different chaotic systems, the phase space does not contain any trivial invariant manifolds from which one can expect a collapse of the overall evolution. The central questions in this case are (i) how to generalize a mathematical definition of chaotic synchronization for such systems and (ii) how

330

13 Tools for Detecting and Analyzing

to detect it in a real experimental situation. Recently, two approaches have been suggested in order to answer these questions. One of them [11] uses the concept of an analytical signal and introduces an instantaneous phase and amplitude for the chaotic process. The synchronization appears as locking of the phases of the coupled systems, while the amplitudes remain uncorrelated. This type of synchronization is identified as a phase synchronization. Another approach [12] is based on the concept of the functional relationship between the variables of the coupled subsystems. It becomes particularly attractive in connection with a recent publication of Rulkov et al. [13]. They restricted their consideration to the case of forced synchronization. This means that the full system consists of an autonomous driving subsystem that is one-way linked to a response subsystem. Generalized synchronization (GS) is taken to occur if, ignoring transients, the response Y(t) is uniquely determined by the current drive state X(t). That is, Y(t) = $(X(t)), where $ is a mapping that takes the trajectories X(t) of the attractor in the driving space to the trajectories Y(i) in the response space. For non-identical driving and response systems, the map differs from identity, which complicates the detection of GS. To recognize GS in a real experimental situation, Rulkov et al. [13] suggested a practical algorithm based on the assumption that $ is a smooth (differentiate) map. The algorithm was tested on artificially constructed examples with an a priori known map $. Subsequent progress of GS theory was achieved in recent publications [10, 14, 15, 16, 17, 18]. Depending on the properties of the map $, two different types of GS were discovered [16], namely, strong synchronization (SS) and weak synchronization (WS), which are characterized by a smooth and a nonsmooth (fractal) map , respectively. The main goal of this chapter is to review the recent ideas of GS theory. Particularly, we focus on different numerical and experimental tools for detecting GS and analyzing its properties. In Sec. 13.2, we briefly describe the main ideas defining the concept of GS. In Sec. 13.3, we introduce some numerical characteristics to estimate the properties of the synchronization manifold. With the help of various examples, we show that GS may appear in two different states, referred to as WS and SS. In Sec. 13.4, we show that at the threshold of WS the system exhibits a new type of on-off intermittency. Unlike the conventional on-off intermittency, where the system dynamics is determined by the escape of trajectories from an unstable smooth hyperplane, this intermittency is characterized by the escape of trajectories from an unstable fractal manifold. Sec. 13.5 is devoted to the detection of GS from time series analysis. A special algorithm for estimating conditional Lyapunov exponents from two scalar data sets, one taken from the driving system and the other taken from the response system, is described. In Sec. 13.6, we illustrate these ideas on a real experimental situation. Two examples of one-way coupled electronic circuits displaying GS are considered. Section 13.7 contains the conclusions of our review.

13.2 Generalized Synchronization of Chaos

331

13.2 Generalized Synchronization of Chaos Let us consider one-way coupled chaotic systems of the following general form (master-slave configurations or systems with a skew product structure): (13.1) (13.2) Here X = {^i,^, • • • ->xd} is a d-dimensional state vector of the driving system and Y = {2/1,2/2? ••• >2/r} is an r-dimensional state vector of the response system. F and G define the vector fields of the driving and response systems. One can show [10, 14] that there exists some mapping $ (not necessarily smooth [16]) between X and Y if, under the action of driving perturbations, the response system "forgets" its initial conditions, i.e., when the response system becomes a stable system [19]. This suggests an auxiliary system approach [14] as a tool for detecting GS in an experiment. According to this approach, it is supposed that we are able to construct an auxiliary response system Y' that is identical to Y and to link it to the driving system X in the same way that Y is linked to X1: Y' = G(Y',X).

(13.3)

Due to the identity of the original [Eq. (13.2)] and the auxiliary [Eq. (13.3)] response systems, they may exhibit IS. GS between X and Y occurs if there is IS between Y and Y'. To show [16] that IS between Y and Y' results in the relationship Y = $(X), let us denote the solution of Eqs. (13.1-13.2) by X(t) = Vx(X0,t) and Y(t) = $!y(Xo, Yo, t), where X = XQ and Y = YQ are the initial conditions at t — 0. If the driving dynamics is invertible, Xo — tyx(X(t), —t), the response solution can

be rewritten as Y(t) - ^y(^x(X(t),-t),Y0,t) = Vy(X(t),Y0,t). IS between Y and Y1 implies lim \\Y - Y'\\ = lim \\Vy(X(t),Y0,t) - 9y(X(t),Y£,t)\\ - 0 for t—too

i->oo

arbitrary initial conditions YQ and YQ taken in some region of Y space. From this it follows that *&y is asymptotically independent of YQ. AS t -» 00, ^!y is also independent of the explicit time t. Indeed, let YQ = &y(X(i),YQ,i) be the state of the system Y' at an intermediate time i < t. Then the state of the system Y' at time t can be expressed as Y'{t) = tyy(X(t),YQ,t - i) and the synchronization

condition becomes lim \\9y(X(t),Y0,t)

- Vy(X(t),Y£,t - i)\\ = 0 for any i < t.

t-KX>

It follows that as t —> 00, &y is independent of both lo and the explicit time t. Thus, in the limit t —>> 00, we obtain a relationship between X and Y in the form Y = lim Vy(X(t),Y0,t) = $(X(t)). t—too

GS guarantees that the asymptotic dynamics of the response system is independent of its initial conditions and is completely determined by the driving system. Geometrically, this implies a collapse of the overall evolution onto a stable synchronization manifold M — {(X,Y) : $(X) = Y} in the full phase space of the x

The related problem of synchronizing identical systems that are driven by random noise has been considered in Ref. [20]

332

13 Tools for Detecting and Analyzing

two systems X © Y. It is easy to show [14] that the linear stability of the identity manifold Y' — Y in the extended phase space X ® 7 © 7 ' is equivalent to the linear stability of the manifold M = {(X,Y) : $(X) = Y} in the original X 0 Y phase space. The linear equations that govern the evolution of the quantities SY = Y - $(X) and SYf = Y - Y' are equivalent: , X(t))<JY,

(13.4)

(JY = DYG(Y,X(t))8Y'.

(13.5)

1

Here DyG denotes the Jacobian matrix of the response system with respect to the Y variable, where Y = Y(t) = $(X(t)) is denned by Eqs. (13.1-13.2). Therefore, if the manifold of synchronized motions in X 0 Y © Y' is linearly stable for 8Y1 = Y — Y1, than it is linearly stable for <5Y = Y — $(X) and vice versa. Note that the linearized equations for SY1 =Y -Y1 are identical to the equation that defines the conditional Lyapunov exponents (CLEs) for the response system Af > A^ > . . . > A* [8]. Both manifolds Y' = Y and Y = $(X) are stable when all CLEs are negative. Thus, the condition of GS is Af < 0. We have thus demonstrated that to study the transition to GS, the analysis of the stability of the synchronization manifold in the space X 0 Y, which in general may have a very complex shape Y = $(X), can be replaced by the analysis of the stability of the simple identity manifold Y = Y' in Y 0 Y' space.

13.3 Weak and Strong Synchronization 13.3.1 Properties of the Synchronization Manifold Note that IS between Y and Y1 does not guarantee the smoothness of $ [16]. The synchronization manifold M = {(X, Y) : $(X) = Y} can have a fractal structure. Ding et al. [21] have illustrated that nonsmooth (fractal) maps do not preserve the dimension of strange attractors. As a simple example of this type, let us consider the Weierstrass function y = Fw(x) = ]C^Li cos(n^x)/n a . It specifies a continuous (C°) but non-differentiable map of points on the rc-axis (with the dimension equal to 1) to points on the Weierstrass curve x —> [x, y = Fw(x)] with a fractal dimension between 1 and 2 for typical values of a and /? satisfying 1 < a < (3. Recently, Sauer and Yorke [22] gave a criterion for dimension preservation. They provide a theorem which shows that continuously differentiate (C1) maps preserve the dimension of strange attractors. Thus, depending on the properties of the synchronization manifold, GS can be subdivided into two types. For the continuous C° but nonsmooth map $, the global dimension of the strange attractor dG in the whole phase space X 0 Y is larger than the dimension of the driving attractor dD in the X subspace, dG > dD'. We refer to this type of synchronization as WS. For smooth $ with degree of smoothness Cl or higher, we expect that the response system does not have effect on the global dimension, i.e., dG — dP. This type of synchronization we call SS. Obviously, IS is a particular case of SS.

13.3 Weak and Strong Synchronization

333

The threshold of SS can be estimated from the Kaplan-Yorke conjecture [23], in the same way that Badii et al. determined the condition at which a linear low-pass filter does not influence the dimension of filtered chaotic signals [24], Note that for systems with a skew product structure described by Eqs. (13.1-13.2), the CLEs represent a part of the whole Lyapunov spectrum Ai, A 2 ,..., \r+d of this system. The remainder of this spectrum consists of Lyapunov exponents Af > A^ > ... > Xf of the driving system (13.1). In other words, to obtain the whole spectrum of Lyapunov exponents of system (13.1-13.2) in the usual (descending) order, Ai > A2 > ... > Ar+d, the combined spectrum of the driving Lyapunov exponents and the CLEs Af, Af,..., A^, Af, Af,..., Af have to be resorted in order of their numerical size. If the whole spectrum of the Lyapunov exponents is known, then one can extract information about the properties of the synchronization manifold. Using the Kaplan-Yorke conjecture [23], the dimensions dG and dP can be estimated as follows:

^

1

lG

j>,

(13.6)

ID

1=1

where IQ and ID are the largest integers for which the corresponding sums over / are nonnegative. The lower index A indicates that these dimensions are calculated from the Lyapunov exponents. The global Lyapunov dimension is independent of the response system (d^ — d®) at the condition [16] Af < A^ + 1 . If this condition is fulfilled and relations (13.6) and (13.7) are valid, we have SS. The smoothness of $ can be also estimated by a more direct criterion, namely, by determining the mean local "thickness" a of the synchronization manifold [16]. Let us consider a set of points [Xi]^=l = [X(ti)}?=1 and \Y$LX = [Y{U)}f=1 in the spaces of the coupled systems coming from finite segments of trajectories sampled at the moments U — iAt. Pick an arbitrary point Xk and find its Nn > dr neighbors Xj whose distance from Xk is less than e; \\Xj - Xk\\ < e, j = 1,..., Nn. Suppose that, for small e, the points Xj are related to their images Yj by a linear map Yj - Yk — A^{Xj - Xk), where Ak is a d x r matrix, whose elements can be determined by a least-squares fit. Then the square of the local thickness of the synchronization manifold at the point Xk can be estimated as al = J2j=iK ~Yk~ Ak(Xj — Xk)]2- The mean thickness a is obtained by averaging the local values, Now we illustrate some properties of GS with specific examples. As usual in such problems, we start with discrete time systems. At first we consider coupled identical subsystems which can exhibit IS and show that even in this case WS appears for coupling strength below the threshold of IS. The last example illustrates GS in essentially different coupled time-continuous systems.

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13 Tools for Detecting and Analyzing

13.3.2 Numerical Examples Coupled Logistic Maps Let us consider a simple example of two one-way coupled identical one-dimensional logistic maps: x(i + 1) = f(x(i)),

(13.8) )}

(13.9)

with f(x) = 4ax(l - z) and a = 1. Here Eqs.(13.8) and (13.9) describe the driving and response systems, respectively, k is the coupling strength. At 0 < k < 1, the coupling term in Eq.(13.9) preserves the global stability of the response system, since 0 < (1 — k)f(y(i)) + kf(x(i)) < 1 at any x(i) and y(i) lying in the interval [0,1]. To observe GS, we consider an auxiliary response system y'(i + 1) = /(tf'(t)) + * { / ( * « ) - / ( » ' « ) }

(13-10)

identical with the original response system (13.9), but having a different initial condition than that of system (13.9). We emphasize that this system does not influence the dynamics of the original response and driving subsystems described by Eqs. (13.8-13.9). It serves only to detect the properties of the system (13.813.9). At any coupling strength k, Eqs. (13.8-13.9) have an invariant manifold y = x and, hence, admit IS. The case of identical systems is interesting, since it provides a simple criterion for SS. SS for such systems is equivalent to IS. Indeed, the identity diagonal y — x is an invariant manifold of the system (13.8-13.9). If it is a stable manifold, the variables of the response and driving systems are related by the identity map y(i) = x(i), which obviously is smooth. Thus, SS can be simply detected as IS between the driving and response systems. Figure 13.1 shows the phase portraits of the system for the logistic map in x-y and y-y1 coordinates at a = 1 and for various values of parameter k. With the increase offc,synchronization occurs first between y and y' and, later on, between x and y. Thus, GS in the form of WS is observed even for identical systems, and it precedes SS. The thresholds of WS and SS are determined by two different Lyapunov exponents, namely, the CLE

$S|/'((i))|

(13.11)

i=l

defining the stability of the invariant manifold y' = y, and the transverse Lyapunov exponent of the identity manifold y = x 1 /c)+ lim - V l n | / ' ( a ; ( i ) ) | . n—>-oo Tl

(13.12)

. -

The dependence of these exponents on k is shown in Fig. 13.2(a). XR{k) becomes zero at two characteristic values of the coupling strength kw and ks, corresponding

13.3 Weak and Strong Synchronization

335

0 y

0

1

1

(c)

/ /

0 -

0 - /

Figure 13.1 x-y and y-?/ phase portraits of coupled logistic maps for various values of the coupling strength A;: (a) &=0.1, unsynchronized state; (b) k=0A, WS;

(c)fc=0.6,'SS.

to the thresholds of WS and SS, respectively. Above the latter threshold k > ks, these two exponents coincide, XT(k) = XR(k). For the logistic map, Eq. (13.12) transforms to X1(k) = ln(l - k) + XD, where A^ = In 2 is the Lyapunov exponent of the driving system and the threshold of SS is equal to ks = 1 - exp(—XD) = 0.5. In a real experiment, IS between the systems Y and Y1 will be partially disturbed by noise and the mall mismatch between the parameters of these systems. These factors will result in a finite amplitude of the deviation Y1 — Y. Numerical analysis shows that the r.m.s. of this deviation SRR = \f{(Y' — Y)2) depends on the amplitude of the noise an by a power law SRR OC a.%. In the case of Eqs. (13.813.9) and (13.10), 7 « 0.12 for WS and 7 = 1 for SS [see the insert in Fig. 13.2(a)]. The same scaling laws are observed for SRR VS Aa, where Aa is the deviation between the parameters of the systems y and y' [a = 1 for Eqs. (13.8-13.9) and a = 1 — Aa for Eq. (13.10)]. Thus, WS is much more sensitive (7 < 1) to noise and parameter deviations than SS (7 = 1). WS observed with the help of an auxiliary response system y' may show no evidence in x-y coordinates. At kw < k < k8, there exists a relationship y = 3>(x),

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13 Tools for Detecting and Analyzing

0.5

0.0

(b)

0.4

0.0 0.0

Figure 13.2

0.2

0.4

0.6

0.8

1.0

(a) Conditional XR and identity A7 Lyapunov exponents, (b) correlation dimension dc of the attractor in the x-y plane, (c) thickness a and cross correlator Kxy for coupled logistic maps as functions of coupling strength k. \R and A7 are calculated from Eqs. (13.11) and (13.12), respectively. dc and a are determined from N=50000 data points (x(i),y(i),i = 1,..., N). The insert in (a) shows the deviation s vs the amplitude of noise an: (1) unsynchronized state at k = 0.3; (2) WS at k = 0.4; (3) SS at k = 0.6. At every iteration, random numbers uniformly distributed in the interval [-a n /2,a n /2] have been added to the variables of Eqs. (13.8-13.9) and (13.10).

however, the map $ is nonsmooth and has a fractal structure [Fig. 13.1(b), left]. The global correlation dimension [25] df of an attractor lying in the x-y plane does not exhibit any characteristic changes at the threshold kw [Fig. 13.2(b)j. An abrupt dimension decrease is observed only at the threshold ks, where $ is turned to identity. At the threshold of WS, there are no characteristic changes in the cross-correlator Kxy between x and y variables [Fig. 13.2(c)], although here this correlation is rather large, Kxy(kw) « 0.71. WS similarly shows no evidence in the mean local thickness a of the synchronization manifold [Fig. 13.2(c)]. The thickness a decreases abruptly only at k = ks, like the dimension d^. At k > ks, $ becomes a smooth map; the thickness a turns to zero and the global dimension d^ becomes equal to the dimension df of the strange attractor of the driving system, Although this example is based on an noninvertible logistic map, similar effects are observed in coupled invertible Henon maps.

13.3 Weak and Strong Synchronization

337

Coupled Henon Maps The second example represents two identical one-way coupled invertable Henon [26] maps (13.13)

where f[xi,x2] = 1 — ax\ + x2i a — 1.4, b = 0.3, and k is the control parameter defining the coupling strength. At any k, this system (like the to a previous example) has an invariant manifold Y = X and, hence, admits IS which is equivalent to SS. IS appears when the identity manifold Y — X becomes stable. The linear stability of this manifold is described by the variational equations 2(l-*)si(»)

b

l\(8yi(i)

0){Sy2(i)

defining the two transverse Lyapunov exponents A[ and A^. The dependence of the maximal transverse Lyapunov exponent A[ on k is shown in Fig. 13.3. It becomes negative when k exceeds some threshold A; > £3 « 0.34. Before reaching this threshold, the system exhibits GS in the form of WS. This conclusion can be made by analyzing the CLEs of the response system. They are determined from the variational equations _(

-2(1-h)yi(i)

l\(6y[(i)

defining the dynamics of small deviations SY' = Y — Y', where Y' is the variable of the auxiliary response system constructed in accordance with Eq. (13.14). The dependence of the maximal CLE Af on k is also presented in Fig. 13.3. In the general case, the exponent A^ differs from Af when the driving and response systems are not synchronized in the sense of IS. They coincide only in the domain of the control parameter k, where \{(k) < 0. If we suppose that the identity plane Y = X is the only invariant smooth manifold of the system, the conditions of WS can be expressed as Af (&) < 0, \{(k) > 0. The first condition guarantees the existence of a stable synchronization manifold and the second condition shows that the smooth identity manifold Y = X is unstable. Thus, WS is observed in the interval k G [fci,fc2], *i ~ 0.16, k2 « 0.20. Here, the maximal CLE is negative, while the maximal transverse Lyapunov exponent is positive. This means that the systems Y and Y' are synchronized in the sense of IS and there is no IS between X and Y.

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13 Tools for Detecting and Analyzing

0.2 -

0.0 0.4

0.0

Figure 13.3 (a) Maximal conditional Af and identity A{ Lyapunov exponents as functions of the coupling strength k for one-way coupled Henon maps. The interval ki < k < ki corresponds to WS. At k > fe, the systems (13.1313.14) exhibit identical behavior corresponding to SS.

Coupled Rossler and Lorenz Systems As a third example, we present GS in essentially different time-continuous systems: d

(13.17) 0.2

d ~dt

2/3

yi ~V2 -2/12/3 2/12/2 -

(13.18)

. These equations describe the coupling of the Rossler [27] [Eqs. (13.17), driving] and the Lorenz [28] [Eqs. (13.18), response] systems. The multiplier a — 6 is introduced to control the characteristic time scale of the driving system. Here the parameter b is chosen to be b = 28. The perturbation kx2 is applied only to the second equation of the Lorenz system and does not contain any feedback term. In addition to Eqs. (13.17-13.18), we consider an auxiliary response system which is equivalent to the system of Eqs. (13.18) except that the variables yi are replaced with y[. Despite the lack of any symmetry in Eqs. (13.17-13.18) admitting IS, this system can exhibit GS. As in the previous examples, GS can easily be detected as IS between Y and an auxiliary response system Y'. The threshold of GS is determined by Af (&) = 0 and is equal to kw « 6.66 [Fig. 13.4(a)]. In this model, the onset of GS is characterized by a considerable decrease of both the dimension

13.4 On-Off Intermittency

339

[Fig. 13.4(b)] and the thickness of the map [Fig. 13.4(c)]. However, the mean local thickness of the synchronization manifold remains rather large. This means that here we actually have the transition to WS. In the case of a driving system presented by a three-dimensional flow, the condition of SS defining the equality of the global and driving Lyapunov dimensions df = df becomes Af (k) < Af. For the system of Eqs. (13.17), we have Af « 0.41, Af = 0, Af « -37.66 and the driving Lyapunov dimension is equal to d% — 2 + Af /|Af | « 2.01. Because of the large negative value of Af, the condition Af (A:) < Af is not achieved even for very large k « 1000, and we have WS for all k > kw. Although the rigorous criterion of WS df(k) > d® is fulfilled for all k > kw, the global dimension goes down to a value approximately equal to the driving dimension at &~40. Here, the global dimension is df(k) = 2 + Af /|Af (k)\ and, since Af /|Af | < 1 and Af /|Af
13.4 On-Off Intermittency A recent publication of Platt, Spiegel, and Tresser [29] has inspired great interest to a particular behavior of nonlinear systems known as on-off intermittency. This behavior derives its name from the characteristic two-state nature of the intermittent signal. The "off" (laminar) state is nearly constant and can remain so for very long periods of time. The "on" state is a burst, departing quickly from, and returning quickly to, the off state. Such behavior occurs in dynamical systems with certain symmetry properties. The chaotic attractor of such systems lies on a smooth invariant manifold (usually a hyperplane) having a lower dimension than the dimension of the full phase space. This attractor may become a repeller at the blow-out bifurcation [30]. The on-off intermittency is observed just above this bifurcation threshold. Initially, this intermittency was discovered numerically in the system of coupled identical chaotic maps [31]. Later, it was investigated in various mathematical models, such as a set of coupled ordinary differential equations [29], random maps [32], and random map lattices [33], as well as various physical sys-

340

13 Tools for Detecting and Analyzing

2 0 cc

(a) -4

° * 4 o o

(b)

3

2

10

(c)

G 5 0 20

cc10 CO

0.1

10

100

1000

k

Figure 13.4 (a) Maximal CLE Af, (b) global correlation d£ and Lyapunov d^ dimensions, (c) thickness a, and (d) deviation s for coupled Rossler and Lorenz systems as functions of the coupling strength k. dc and a are calculated from N=50000 data points (X(iAt), Y(iAt),i = l,...,iV) with At=0.5. The points in (a) and (b) show the maximal CLE and global Lyapynov dimension, respectively, calculated from time series (see Sec. 13.5). The insert in (c) shows s vs the deviation of the parameter Ab: (1) unsynchronized state at k=5; (2) WS at k = 10; (3) WS at k = 20; (4) SS at k = 50.

13.4 On-Off Intermittency

341

terns, such as particle motion in a symmetrical potential [34], electronic circuits [35], and high power ferromagnetic resonance [36]. Here, we show [18] that the class of nonlinear chaotic systems exhibiting onoff intermittency can be essentially extended. On-off intermittency may appear in systems that do not possess any trivial invariant manifolds. It may occur in any dynamical system consisting of two one-way coupled chaotic subsystems at the threshold of WS. Let us illustrate this with the simple example of two one-way coupled logistic maps [Eqs. (13.8-13.9)] considered in Sec. 13.3. In the region of coupling strength kw < k < ks, we have IS between the original and auxiliary response systems y{i) — y'(i) a n d have no IS between the driving and original response systems y(i) ^ x(i). This corresponds to WS between the driving x and original response y systems. Here, the identity manifold y = x is unstable and the overall dynamics in the x-y plane collapses to another invariant synchronization manifold M — {(x,y) : y = <&(x)} that has a fractal structure. This manifold is shown in Fig. 13.5(a) (left) just above the threshold of WS. The IS between y and y' [Fig. 13.5(a), right] testifies to the stability of this manifold. Just below the threshold of WS (A; < kw), the CLE becomes positive, XR(k) > 0. This means that the fractal synchronization manifold responsible for WS becomes unstable. Close to the threshold, we can expect that the system spends a long time in the vicinity of the manifold and experiences short bursts away from this manifold. Figure 13.5(b) shows the x-y and y-y' phase portraits just below the threshold of WS. The expected intermittent behavior is not seen in x-y coordinates. However, it can be detected with the help of an auxiliary response system (13.10). A typical structure for intermittent behavior is seen in y-y' coordinates. Figure 13.6 illustrates the dynamics of y(i) and the difference y(i) — y'(i) just below the threshold of WS. Intermittency is not seen in the dynamics of the original response system y(i) [Fig. 13.6(a)] but it is evident in the signal y(i) — y'(i) [Fig. 13.6(b)] formed from the difference of the output of the original and auxiliary response systems. Recall that the auxiliary response system y' does not influence the dynamics of the original response y and driving x systems but serves only as an indicator of intermittent behavior in the x-y plane, which is related to the loss of stability of the fractal synchronization manifold, j Now let us compare this intermittent behavior with conventional on-off intermittency. Instead of investigating the escape of trajectories from a fractal synchronization manifold y = $(x) in the x-y plane, we can consider the escape of trajectories from the identity manifold y = y1 in the y-y' plane. In Sec. 13.2, we showed for a general case that the linear Eqs. (13.4) and (13.5) governing the evolution of the quantities SY = Y - $(X) and SY' = Y - Y' are equivalent. Since the main properties of on-off intermittency are determined by these linear equations, we can analyze the system dynamics in y-y1 coordinates rather than in x-y coordinates. We calculated the dependence of the mean laminar length r on the coupling strength k (Fig. 13.7) and the distribution of the laminar lengths P(r) close to the threshold of WS (Fig. 13.8). The states \y(i) - y'(i)\ < 0.1 were interpreted as

342

13 Tools for Detecting and Analyzing

Figure 13.5 x-y and y-y' phase portraits of the coupled logistic maps (a) just above the threshold of WS at k = 0.35 and (b) just below the threshold of WS at k=0.32.

laminar phases and the states \y(i) — y'(i)\ > 0.1 were considered as bursts. The dependence of the inverse mean laminar phase length 1/r on the coupling strength k shows a well-defined linear part corresponding to a power law with the exponent — 1 (r ex (kw — A:)"1), exactly as in conventional on-off intermittency [32]. The distribution of the laminar length P(T) is also the same as in the case of conventional on-off intermittency. For moderate r, it is well approximated by a power law with the exponent —3/2 (P(r) oc r~ 3 / 2 ), and for large r, it has an exponential fall-off. The identical properties of these two different intermittent processes are related to the fact that the problem of trajectory escape from a fractal synchronization manifold y = 3>(x) can be replaced by the problem of trajectory escape from a smooth manifold y = y'. The last problem is typical for conventional on-off intermittency. Let us consider the dynamical equation governing the difference Sy'(i

= 4(1 - *)(1 - 2y(i) + Sy1(i

(13.19)

134 On-Off Intermittency

343

-0.5 -

1000

Figure 13.6 (a) Dynamics of the original response system y(i) and (b) the difference y(i) - y'(i) just below the threshold of WS at k = 0.33.

0.15 0.10 -

0.05 -

0.00 0.26

0.34

Figure 13.7 Dependence of the inverse mean laminar length on the coupling strength close to the threshold of WS. For each k, a time series of 107 data points is used.

344

13 Tools for Detecting and Analyzing

10°

Figure 13.8 Distribution of the laminar phase lengths just below the threshold of WS at k = 0.33. A time series of 107 data points is used. The solid line corresponds to a power law scaling with the exponent —3/2.

The properties of the intermittent process are determined by small | we can rewrite Eq. (13.19) as 5y'(i + l) = z(i)6y'(i), where z(i) = 4(1-fc)(l -2y(i)) is a chaotic process determined by Eqs. (13.8-13.9). This is the standard form of a linear map driven with a chaotic signal that is considered in the theory of conventional on-off intermittency in order to derive the above properties [32]. Note that this intermittency can be considered as chaos-to-hyperchaos intermittency, since below the threshold of WS the second Lyapunov exponent of the system (13.8-13.9) becomes positive (one Lyapunov exponent corresponds to the driving system (13.8); it is always positive, independent of the coupling strength fc).

13.5 Time Series Analysis In Sees. 13.2 and 13.3, we described several tools for detecting GS and analyzing its properties. The most appropriate method for experimental applications is that based on the auxiliary system approach. Unfortunately, this approach is of limited utility. The method fails for systems whose dynamical equations are not available. Even though the dynamical equations are known (e.g., in electronic circuit experiments), the auxiliary response system can be designed only with finite accuracy; it cannot be an exact copy of the original response system. An alternative approach to detecting GS in experiments is to estimate the CLEs A^, i — 1,..., r from an observed time series [17]. Recall that the condition of GS is Af < 0; therefore, to detect GS we need to estimate only the maximal CLE.

13.5 Time Series Analysis

345

Thus, without recourse to an experimental auxiliary response system, we can predict whether an identical copy of the response system connected to the driving system will exhibit behavior identical to the original response system. If we are interested in the properties of the synchronization manifold, we may need to estimate some additional CLEs and perhaps some of the Lyapunov exponents of the driving system Af, i = 1,..., d, in order to compare the global Lyapunov dimension d^ [Eq. (13.6)] with the driving dimension d® [Eq. (13.7)]. As a result of this comparison, we can distinguish between WS and SS. SS (d^ = d®) corresponds to Af < A^ + 1 . Otherwise, we have WS, df > d^. Alternatively, we can evaluate the smoothness of the synchronization manifold by estimating the mean local thickness a of the synchronization manifold. The algorithms for estimating CLEs and a from a time series are similar. Below, we present only the algorithm for estimating CLEs. Note that only a finite number of Lyapunov exponents can be reliably determined from data on the attractor [37]. An appropriate cut-off value for the number of exponents is related to the global Lyapunov dimension and is equal to IQ + 1. The only exponents that are included in Eq. (13.6) are fundamentally important to the character of the attractor and their estimation is available from time series. In the case of WS, at least maximal CLE affects the global dimension, hence, it can be estimated from a time series. The condition of SS Af < A/^+1 corresponds to the case where the global dimension do does not depend on the CLEs. Thus, we cannot expect a reliable estimation of CLEs from time series above the threshold of SS. However, the CLEs can be determined just before this threshold, and it suffices to estimate the characteristic values of the control parameters corresponding to the onset of SS.

13.5.1 Algorithm for Estimating CLEs In experiments, we generally, do not have the luxury of working with the actual vectors of phase space variables. Normally, only the time series of a single variable is available to characterize the behavior of each system. Suppose that an experimental system under investigation can be simulated by Eqs. (13.1-13.2). We imagine that the equations are unknown, but two scalar time series X{ and yi, i = 1,...,7V corresponding to the driving and response subsystems, respectively, are available for observation. We assume that the time interval r between measurements is fixed so that Xi — x(ir) and yi = y(ir). In what follows, r is identified with the delay time of phase space reconstruction in step (a) of our algorithm. In principle, any choice of r is acceptable in the limit of an infinite amount of data. For a small amount of data, the choice of r can be based, for example, on the evaluation of mutual information [38]. Due to the one-way coupling, the X{ series does not contain any information about the response system, while the yi series contains information about both subsystems. Since the CLEs represent a part of the whole Lyapunov spectrum, one can expect that they can be determined by standard algorithms [37, 39, 40] from the yi time series. However, the CLEs may be placed far from the maximal

346

13 Tools for Detecting and Analyzing

exponent in the whole Lyapunov spectrum, while standard algorithms give reliable values for only a few of the largest exponents [37, 39, 40]. Moreover, it is a nontrivial problem to define which exponents belong to the CLEs and which to the driving system, even though the whole spectrum of the Lyapunov exponents is reliably determined. These problems can be solved in the framework of an algorithm that involves information from both scalar time series X{ and y\. Here, we mainly use the ideas of the algorithm proposed by Eckmann et al. [40] based on the construction of local linear maps. The mappings with a higher order Taylor series [37] are beyond our scope. We extend the Eckman-Kamphorst-Ruelle-Ciliberto (EKRC) algorithm to the case of two time series and adopt it for the direct estimation of the CLEs. The reliability of estimating the maximal CLE by our algorithm is comparable to that of estimating the conventional maximal Lyapunov exponent by the EKRC algorithm. To speed up the computation and to bring our consideration closer to a real experimental situation, we represent the time series X{ and yi by integer numbers normed to the same maximal value Mo so that 0 < xi < Mo and 0 < yi < MQ. Typically, we take Mo = 10000, in accordance with a precision of 10~4. Like the EKRC algorithm, our algorithm involves the following three steps: (a) reconstructing the dynamics by the time-delay method [41] and finding the neighbors of the fiducial trajectory, (b) obtaining the tangent maps by a least-squares fit, and (c) deducing the CLEs from the tangent maps. Now let us consider these steps in detail. (a) We choose different embedding dimensions Ex and Ey for the driving and response systems and define (Ex + Ey)-dimensional vectors Ri = {Xi-Ex+i, -.., Xi_2, x^ yi-Ey+i,...,

Vi-2,yi}

(13.20)

for i = %Q = ma,x(Ex,Ey),io + 1,..., N, to construct the dynamics of the fiducial trajectory in the whole X®Y phase space. In view of step (b), we have to determine the neighbors of Ri, i.e., the points Rj of the orbit that are contained in a ball of small radius e$ centered at Ri, \\Rj - Ri\\ < e{.

(13.21)

Here || • || implies the maximal projection of the vector rather than the Euclidean norm. This allows a fast search for the Rj by first sorting the data [40]. Let us denote by J{ the number of neighbors Rj of Ri within a distance e*, as determined by Eq. (13.21). Clearly, Ji depends on e;. In (b) we discuss the choice of these parameters for every i. (b) Having embedded our dynamical system, we want to determine the tangent map that describes how the time evolution sends small vectors around Ri — {Xi, Y]} to small vectors around Yi+m. This problem can be considered in a phase space of reduced dimension [40]. Following Ref. [37], we introduce local dimensions Lx < Ex and Ly < Ey that reflect the number of dimensions necessary to capture the geometry of a small neighborhood of the attractor after it has been successfully embedded (i.e., the time delay representation is diffeomorphic to the original

13.5 Time Series Analysis

347

attractor). The dimensions Lx and Ly are used for constructing local maps and correspond to the number of Lyapunov exponents of the driving system and the CLEs, respectively, produced by the algorithm. The transition from embedding dimensions to local dimensions is performed as in Ref. [40]. We drop the intermediate components in Eq. (13.20) and define the Lx-dimensional X{ and I^-dimensional Y{ vectors as r

{Xi-Ex+l,---,Xi-m,Xi)

(13.22)

{yi-Ey+i,-'-,yi-m,yi)T -

(13.23)

The dimensions Lr Ex and Ly < Ey are determined by the equalities Ex — (Lx — l)ra + 1 and Ey = {Ly - l)m + 1, which we assume to hold for some integer m > 1. The case m — 1 corresponds to Lx = Ex, Ly = Ey. When m > 1, the dimension of the tangent map is reduced with respect to the embedding dimension; this can help to avoid spurious Lyapunov exponents [40]. The tangent map is defined by two matrices Ai and Bi, which are obtained by looking for neighbors Rj of R{ and imposing At(Xj -

(13.24)

Yj+m - Yi+m.

Ai is a rectangular Ly x Lx matrix and Bt is a square Ly x Ly matrix, which, in view of Eqs. (13.22) and (13.23), have the form

( o 0

0 0

0 0

0

0

0

0 \ 0

1

Ai=

0 a

V \

b\

a, L

0

1 Lv

l

Matrix Ai contains Lx unknown elements a k, k — 1,2,... Lx, and matrix B{ contains Ly unknowns b\, k = 1,2,..., Ly. These Lx -h Ly unknowns are obtained by a least-squares fit Ji

mm

Jz

\-m)\\Euo

where || • \\2Euc denotes the square of the Euclidean norm of the vector. This problem reduces to a set of Lx + Ly linear equations in Lx -f Ly unknowns alk, blk, which we solve by the LU decomposition algorithm [42]. Obviously, this algorithm fails if the number of neighbors Rj of the fiducial point R{ is less than the number of unknowns, Ji < Lx + Ly. To avoid this problem, the radius c« has to be chosen to be sufficiently large. For the specific examples discussed below, we have selected 6{ and Ji as follows. We count the number of neighbors Ji of Ri corresponding to increasing values of e* from a preselected sequence of possible values and stop when Ji first exceeds Jmin = 2(LX + Ly). To speed up the calculations, we also stop the

348

13 Tools for Detecting and Analyzing

search for the neighbors when for given e^ the number of neighbors exceeds the maximal value Jmax = 40. Thus, for every z, J{ is in the interval [Jmin, Jmax\(c) Step (b) gives matrices A{ and B{ of the tangent map, which represent the reconstructed Jacobians DXG and DyG of Eq. (13.2) with respect to X and Y variables, respectively. The CLEs are determined by the product of the matrices BioBio+mBio+2m To extract the CLEs from this product, we use the QR decomposition technique [40, 42]. The method recursively defines an orthogonal matrix Qi and an upper triangular matrix Ri, I = 0 , 1 , . . .L — 1, via Bio+imQi = Qz+ii2/+i, where Qo is the unit matrix. The CLEs are given by L-l

A£m = — £>(<3l)nn, 1=0

where K < (N — io)/m is the available number of matrices and (Qi)nn is the diagonal element of the matrix Qi. Note that in the final step we do not require knowledge of the matrix A{. However, the use of this matrix in step (b) is necessary in order to correctly determine the tangent map (13.24) and, hence, the matrix B{ defining the CLEs. Let us now illustrate our algorithm with the two specific examples presented in Sec. 13.3.

13.5.2 Examples Coupled Henon Maps Let us come back to the model of identical one-way coupled Henon maps described by Eqs. (13.13-13.14). To test the algorithm, two scalar time series x\{%) and y\(i) were treated as experimental data. The results presented in Table 13.1 correspond to the fixed value k = 0.1 and different values of the local dimensions Lx and Ly. For comparison, the correct values of the CLEs calculated directly from Eqs. (13.13-13.14) and (13.16) at k = 0.1 are Af « 0.227 and A£ « -1.537. For any Lx > 2 and Ly > 2, the algorithm gives two CLEs close to these correct values. If Ly is chosen correctly [i.e., equal to the dimension of the response system (13.14) Ly — r — 2], we obtain the right number of CLEs whose values weakly depend on Lx provided Lx > 2. For Ly > 2, the algorithm gives spurious CLEs in addition to the valid CLEs. One way of identifying spurious exponents is to analyze the influence of external noise [37]. This is illustrated in Fig. 13.9. Here we have added Gaussian white noise to the data points with the standard deviation an. In Fig. 13.9(a) we used Lx = Ly = 2, while in Fig. 13.9(b) we used Lx = Ly = 3, which gives one spurious CLE. The spurious CLE in Fig. 13.9(b) decreases rapidly as the added noise is increased, going from +0.7 down to -0.9. Figure 13.10 shows a correlation between the dependence of the CLEs on the coupling strength k estimated from time series with that calculated directly from Eqs. (13.13-13.14) and (13.16). Good agreement is observed for k < fc3, especially

13.5 Time Series Analysis

Ly

Lx

2 2 2 3 3 3 4 4 4

2 3 4 2 3 4 2 3 4

A

l

349

A\H

3

0.228 -1.408 0.224 -1.411 0.219 -1.402 0.462 0.203 -1.558 0.459 0.186 -1.547 0.489 0.178 -1.546 0.530 0.206 -0.962 -1.629 0.512 0.189 -0.863 -1.612 0.536 0.191 -0.786 -1.613

Table 13.1 CLEs for coupled Henon maps at k = 0.1 computed from N = 50000 data points evaluated with the sampling time r = 1. We vary the local dimensions Lx and Ly at fixed m = 1 so that they coincide with the embedding dimensions, Ex = Lx, Ey = Ly. The correct values of the CLEs calculated directly from Eqs. (13.13-13.14) and (13.16) are Af « 0.227, A? « -1.537. For I/y > 2, the algorithm gives Ly — 2 spurious CLEs in addition to the two valid CLEs. The values corresponding to the valid CLEs are underlined.

for the maximal CLE. For k > ks, we have SS with the identical time series y\(i) — x\{i) and the algorithm fails. This is in agreement with the general prediction that the CLEs cannot be reliably estimated from time series in the domain of SS. However, the algorithm gives the correct values of the maximal CLE in the immediate vicinity of the threshold k^ks.

Coupled Rossler and Lorenz Systems Let us now consider more the complex system described by Eqs. (13.17-13.18). In testing the algorithm, the variables xi(t) and yi(t) were treated as experimentally available outputs. The maximal CLE and the global Lyapunov dimension obtained from time series analysis are shown by dots in Figs. 13.4(a) and 13.4(b), respectively. The calculations were performed at the following values of the parameters: N = 50000, r = 0.15 for k < 10 and r = 0.03 for k > 10, Lx = Ly = 3, and ra = 1. In Figs. 13.4(a) and 13.4(b), the same characteristics determined directly from Eqs. (13.17-13.18) are shown by solid lines. Good agreement of the corresponding characteristics is observed in a large interval of the coupling strength k. Thus, the results of the time series analysis allow us to correctly predict both the threshold of GS and the smoothness of the synchronization manifold.

350

13 Tools for Detecting and Analyzing

1

c c o Q. X > O

c

g. 5 c o O

'

1

1

1

'

1

• • • • * • • • • • « » • ***»•*««»«• mr» • mwm 0 -

-1 -

"2

1

.

1 1

I

.

1

1

'

1

i

•

i

(b)

.

.

i

•••••••..

-1 -w******

-2 1 -10

•

1

i

-2

Figure 13.9 The effect of external noise on the determination of the CLEs for coupled Henon maps at the same values of parameters as in Table 13.1. crn is the standard deviation of the noise added to the data, (a) The local dimensions are Lx = Ly = 2. (b) Here Lx — Ly — 3. The spurious exponent wanders from about +0.7 to nearly -0.9 as the noise level is increased. The exponents do not cross each other, but switch roles as they become close. The correct values of the CLEs are shown by dashed lines.

13.6 Experimental Examples The aim of this section is to illustrate the above tools of GS analysis for a real experimental situation. We consider [43] two examples of one-way coupled electronic circuits displaying GS. In one of them, the dynamics of both the response and the driving systems represent a double-scroll chaos oscillator. In another example, the double-scroll oscillator is driven by an electronic analog of the Mackey-Glass system. In our studies, we use both an auxiliary system approach and the method of time series analysis.

13.6.1 One-Way Coupled Double-Scroll Oscillators Let us consider synchronization effects in the identical one-way coupled electronic chaos oscillators proposed by Shinriki et al. [44]. The block diagram of the experimental setup is shown in Fig. 13.11, where the driving, the original response, and the auxiliary response systems are denoted by D, R, and R', respectively. All these systems represent Shinriki oscillators designed according to the electronic circuit shown in Fig. 13.12 [3]. The system of coupled circuits is described by the

13.6 Experimental Examples

351

-1.6 0.0

Figure 13.10 Dependence of the CLEs on the coupling strength for coupled Henon maps at m = 1, Lx = Ly = 2. The points correspond to the values of the CLEs estimated from time series and the dashed lines show the correct values of the CLEs calculated directly from Eqs. (13.13-13.14) and (13.16).

r=1/k

r'=1/k'

Figure 13.11 Block diagram of the experimental set-up.

352

13 Tools for Detecting and Analyzing

following set of differential equations:

(13.25)

C2VR = f{VR-VR)-IR,

LI* =

C

j/R'

2^2 T jR!

£f\rR'

=

(13.26)

-IRR3+VR,

r/R'\

jR'

/ i n o'7\

J\Vi

~ V2 ) ~ *3 > JR! T> _I_ irR'

(J.O.27)

Ll3 = - i 3 it3 + l/2 , where Eqs. (13.25), (13.26), and (13.27) describe the driving, the original response, and the auxiliary response systems, respectively. We use the definitions of Ref. [3]. Vi, V2, and /3, with the corresponding upper indices D (driving), R (original response) or R' (auxiliary response), denote the state variables of the individual subsystems. The function f(V) is given by {—•> J\^J

1 V

I V \< Vd\ 1 ^ ^ / T / \ f A A T/3

1 D A T/4 i /°r A T/5l

I T/ l \ T/

^lo.^Oj

where Vd = 2.5F, A = 2.2500, B = -1.9460, C = 0.8188, AV =\ V \ -Vd, and sgn(V) — ±1 for V > 0 and V < 0, respectively. The voltage V is given in volts, and the current f(V) in milliamps. In all three circuits, we have fixed the values of the corresponding parameters to be C\ — 10 n F , C2 = 100 nF, L — 270 mH, R = 6.91 fefi, i?i = 29fen,R2 = 12fen,i?3 = 100 n, with tolerances of the order of 1 %. The last terms in the first Eq. of (13.26) and (13.27) describe the coupling. The coupling strength is denned by the parameter fe = fe; = 1/r, where r = r' is the resistance of the coupling resistors, which we use as a main control parameter. Numerical integrations of the underlying equations where carried out by a fourth-order Runge-Kutta method [42]. Before integrating, the equations were transformed to dimensionless form by normalizing all voltages to VQ = Vd = 2.5V, all currents to Jo = VO/RQ = 1.52 mA for Ro = yjLjCi — 1.64 fen, and the time to to — VLC = 16.4 ms. The step size was fixed at A£ = 0.01. In practice, a small amount of white noise of the order 10~7 was added to the right hand side of Eqs. (13.26, 13.27) during the numerical integrations. This was done in order to prevent the spurious synchronization that stems from finite precision arithmetics and to exclude synchronous states that are unstable with respect to small noise.

13.6 Experimental Examples

353

Figure 13.12 Electronic circuit of the Shinriki oscillator. The only active device in the circuit, the so called negative impedance converter (NIC) is marked with a dashed line.

As pointed out in Sec. 13.3, SS for coupled identical systems is equivalent to IS. Here, the hyperplane {VXD = Vf, V2D = V2R, If = I*} is an invariant manifold of the system of Eqs. (13.25) and (13.26). If it is a stable manifold, the variables of the response and driving systems are related by the identity map. Thus, SS can be simply detected as IS between the driving and response systems. In the limit (5VU8V2,6I3) = (VR - VXD, V2R - V2DJR - If) -> 0, the condition of IS between the driving and the response systems is determined by the variational equations

C2SV2

=

L5h

=

(13.29)

•SV2)-SI3, -R3SI3 + SV2

defining the transverse Lyapunov exponents \*,i = 1,2,3 of the above identity hyperplane. Here, / ; denotes the derivative of the function (13.28). Synchronization occurs if the maximal transverse exponent A[ is negative. Along with IS, the coupled identical systems (13.25) and (13.26) can display WS. Although the identity hyperplane does exist for any value of the coupling strength k, for some k it can become an unstable manifold and another more complex stable synchronization manifold may appear in the system. As pointed out in Sec. 13.2, the condition for GS is IS between the original [Eqs. (13.26)] and the auxiliary [Eqs. (13.27)] response systems. The limit {SVi,6V2,SIs} = {VR - V/*', V2R - V2R>, IR - if } -)• 0 leads to the variational equations

Cl8V1 = (±-

~]

6Vi - f{V? C2SV2 = f{V*

- V2R)(SVt -6V2)-

kSVi,

- V^){8V1 - SV2) - 8I3,

(13.30)

354

13 Tools for Detecting and Analyzing

•T1 0.2 CO

k

i

SS I1 WS

•

-H CC

-0.2

i

i

.

i

1

\

.

T

2

(b) . oc CO

5:

S

RR

S

DR

Q

w

0

i

3

-

.

.

.

.

(c) '

2 i

1E-3

i

i

i

i

IA 0.01

0.1

Figure 13.13 Theoretically obtained dependence of (a) the Lyapunov exponents Af and A{, (b) the deviations SDR, SRR, and (c) the global correlation dimension d® on the coupling strength k for coupled double-scroll oscillators.

defining the three CLEs Af, i = 1, 2,3. The theoretical dependence of the exponents Af and A[ on the coupling strength k is shown in Fig. 13.13(a). As seen from the figure, the conditions of WS Af (A:) < 0 and A{(A:) > 0 are fulfilled for h < k < k2, where kx « 0.011/cfr 1 and k2 ~ 0.029/cn~1. In this region, we have IS between the original and the auxiliary response systems, and we have no IS between the driving and the response systems. As a practical indicator of IS between the original and the auxiliary response systems, we can take the r.m.s. deviation SRR = J ((V/* — V ^ ' ) 2 ) . Similarly, we can take SDR = d{(V® ~ ^i.^)2) as an indicator of IS between the driving and the response systems. The dependence of these characteristics on the coupling strength k is shown in Fig. 13.13(b). At SS, both deviations vanish, SRR —> 0 and SJJR —> 0. At WS, SRR vanishes and SOR remains finite. To estimate the dimension of the synchronization manifold, we calculated the global correlation dimension d^(k) using N = 50000 data points for every k [Fig. 13.13(c)]. At SS, d^(k) has a plateau, the value of which coincides with the correlation dimension of the driving at tractor, d^ « d® (the Lyapunov dimen-

13.6 Experimental Examples

355

sion of the driving attractor is df = 2 + Af /|Af | « 2.0062). At WS, as well as at the nonsynchronized state, d^ is larger than df?.

(a)-

. . . .1

(b) •

X

1E-3

0.01

XXX-X-X-X—X

X

X

0.1 k [kQ 1 ]

Figure 13.14 Experimentally obtained dependence of (a) the deviations SDR, SRR and (b) the correlation dimension d^ on the coupling strength k for coupled double-scroll oscillators. The correlation dimension d^ is calculated from N=50000 data points using two time series V^(iAt) and VxR(iAt),i = 1,...,7V;A* = 0.1ms. The strange attractor has been reconstructed in six-dimensional phase space with the state vectors

{VfliAt], V?[{i + I) At), V?[{i + 2)At)l V?[iddl V?[{%

^

Figure 13.14 shows the experimental dependence of the above characteristics. The experimental results presented in Figs. 13.14(a) and 13.14(b) should be compared with the theoretical results presented in Figs. 13.13(b) and 13.13(c), respectively. Due to the small mismatch between the parameters of the original and the auxiliary response systems, the deviation SRR does not vanish in the region of WS. Nevertheless, WS can be detected by comparison of SRR with SDR- In the region of WS, we have SRR < SDR- The region of SS can be identified as the region, where both of these deviations are small. The experimental results show only one region of SS corresponding to large k. The theoretically obtained region of SS that precedes WS cannot be identified in experiment — only the local minimum in

356

13 Tools for Detecting and Analyzing

> cc >"

0

-4

-3

i

1

-2

/

-

-3

-

-

/ / i

.

i

- (c) -

/ /

-4

1 2

-

/

-2

-5 -4 - 3 - 2 - 1 0

0 i

-1

-4

-1

.

-3 V,

I

.

-2 0

[V]

I

.

i

-1

0

-3

-2

-1

R

v, [V]

Figure 13.15 Experimentally obtained V^-V^ and Vf'-Vf' phase portraits of coupled double-scroll oscillators for various values of the coupling strength k: (a) k = O-OOll/cO"1, nonsynchronized state; (b) k — 0.033/cn"1, WS; (c) 1 , SS.

) near k — 0.01/cfi x gives a weak hint. This finding may be related to the small disagreement of the model equations with the real experimental circuits. The experimental dependence of the correlation dimension shown in Fig. 13.14(b) is in qualitative agreement with the theoretical dependence presented in Fig. 13.13(c). Figure 13.15 shows the projections of the experimental phase portraits in V^-Vfand V^-V^1 coordinates for the cases of a nonsynchronized state [Fig. 13.15(a)], WS [Fig. 13.15(b)], and SS [Fig. 13.15(c)]. For WS, the Vf-Vp' phase portrait collapses to the identity diagonal, whereas the V^-V^ phase portrait does not. For SS, both phase portraits collapse to the identity diagonal.

13.6 Experimental Examples

357

13.6.2 Double-scroll Oscillator Driven with the Mackey-Glass System Let us now, we consider GS in the case where the driving and the response systems differ considerably from one another. We used the same original and auxiliary response systems described in Sec. 13.6.1 and replaced the driving system by an electronic analog of the Mackey-Glass oscillator. An electronic circuit of the MackeyGlass system [45] was designed in the same manner as in Ref. [46]; see Fig. 13.16. The only difference was that here we used the standard, commercially available, integrated "bucket brigade" delay line (National Panasonic, type MN3011) instead of specially designed network of T-type LCL filters used in Ref. [46] to improve the spectral flatness. Furthermore, the nonlinearity was realized with the help of an operational amplifier, so that we were able to tune the characteristics. Our circuit simulates the mathematical Mackey-Glass model that is presented by the nonlinear delay differential equation •

aVM(t-T)

=

b (

/IQQI\

, - VM[t)

(13.31)

with TRC = RMCM ~ 0.27ms, r = 1.37ras, a — 0.44, and b — 8.7. The output voltage VM m Eq. (13.31) is given in volts. We shifted and amplified the output voltage VM as Vf = A(VM - Vo)

(13.32)

before feeding it into the original [Eqs. (13.26)] and auxiliary response [Eqs. (13.27)] systems. The parameters A — 8 and Vo = 0.93F were chosen in such a way that the amplitude and the mean value of the driving signal V® were comparable with the corresponding characteristic values of the signal V/*. The whole system of coupled oscillators is described by Eqs. (13.26), (13.27), (13.31), and (13.32). As in the previous case, we calculated the conditional Lyapunov exponents using the variational equations (13.31). The dependence of the maximal conditional exponent Af on the coupling strength k is shown in Fig. 13.17(a). The condition Af (fc) = 0 defines the threshold of GS, kw « 0.024fcft~1. The region of GS can again be determined by means of the experimentally convenient characteristic SRR — W ((V/* — V^1 ' ) 2 ) , which defines the r.m.s. deviation between the original and the auxiliary response systems. For GS, the deviation vanishes [Fig. 13.17(b)]. To distinguish between WS and SS, we cannot use here the deviation SDR, since the driving and the response systems differ essentially from one another, and IS between them is impossible. SS can be recognized by analyzing the properties of the map <£. Following Sec. 13.3, we calculated the thickness a of the synchronization manifold. The results are presented in Fig. 13.17(b). As can be seen from the figure, the synchronization manifold remains thick behind the threshold of WS and becomes thin (a -»• 0) only at sufficiently large coupling k > ks, where ks « 0.2M)" 1 . These two regions, namely, kw < k < ks and k > ks, can be interpreted as those corresponding to WS and SS, respectively. Our conclusion is confirmed by calculating the global Lyapunov dimension d^, shown in Fig. 13.17(c). In the region of WS, the global dimension df^ is

358

13 Tools for Detecting and Analyzing

Figure 13.16 Electronic circuit of the Mackey-Glass system.

larger than the Lyapunov dimension of the driving attractor d^. For given values of the parameters, the driving Lyapunov dimension d® depends only on the three first largest exponents Af « 0.183ms"1, Af = 0, and Af « -0.472ms" 1 and is equal to df = 2 + Af/|Af | « 2.4. In the region of SS (k > ks), the global dimension df saturates to the dimension of the driving attractor df, d^(k) = d® « 2.4. The condition df(k) = df is fulfilled at Af (fc) < Af. The threshold of SS estimated from the condition Af (ks) — Af gives ks « 0.2/c^"1, which is in good agreement with the characteristic k where the thickness a of the synchronization manifold becomes small [Fig. 13.17(b)]. Figure 13.18 shows the same characteristics as Fig. 13.17, but estimated from experimental data. They are in good qualitative agreement with the numerical results obtained from the model equations. Note one essential difference between the theoretical and experimental dependence of the deviation SRR on k: The theoretical characteristic SRn(k) vanishes at the threshold of WS, k — kw, and is small for all k > kw, while the experimental decreases in two steps with two characteristic thresholds corresponding to the onsets of WS and SS. In the region of WS, the experimental value of SRR remains large and becomes small only at the threshold of SS. This finding is related to the different sensitivities of smooth and nonsmooth synchronization manifolds to the noise and to the small parameter mismatch between the original and the auxiliary response systems (compare with Fig. 13.4(d)). The two characteristic thresholds in the experimental dependence of the deviation SRR on the coupling strength k can be used as a convenient experimental tool for detecting both stages of GS (WS and SS) by means of an auxiliary response system. The experimental results also show that the thresholds of WS and SS can be successfully estimated, even in the case where the design of an auxiliary response system is impossible. This can be done by calculating the CLE from two experimental time series, where one is taken from the driving system and the other from the response system. The maximal exponent Af (k) estimated

13.7 Conclusions

x

(a)

359

'

0.0 -

-0.5

-i i

1 -

J.

a

0.2

S

0.1

RR

V r

i

•

i

• • 1 1

0.3

0.0

• i

(c) • 4 -

K

3 -

1E-3

0.01

0.1

Figure 13.17 Theoretically obtained dependence of (a) the maximal CLE Af, (b) the deviation SRR and the thickness <J, and (c) the Lyapunov dimension d^ on the coupling strength k for a double-scroll oscillator driven with the Mackey-Glass system. To calculate a, the driving and response attractors have been reconstructed in three-dimensional phase space with the state vectors {VxD[iAt], ViD[(t + l)At), V?[{i + 2)At)]} and {Vi*[tAt], V^i + 1)A«), VxR[(i + 2)A*)]} (A* = 0.1ms), respectively.

from these data [Fig. 13.18(a)] correctly reflects the threshold of WS and is in good agreement with the theoretical results [Fig. 13.17(a)]. The same is valid for the two other characteristics a(k) and d^(k) that are based on the same experimental data and which do not require the information from the auxiliary response system. These characteristics can be used to experimentally estimate the threshold of SS.

13.7 Conclusions The generalized synchronization of chaos is a natural generalization of the concept of identical synchronization to the case of nonidentical chaotic systems. This phenomenon is typical for one-way coupled chaotic systems. It appears, when under the action of the driving system, the response system "forgets" its initial conditions and becomes an asymptotically stable system, i.e., when any initial conditions in

360

13 Tools for Detecting and Analyzing

1E-3

Figure 13.18 The same characteristics presented in Fig. 13.17, but obtained from experimental data. As in the case of Fig. 13.14, the strange attractor has been reconstructed in six-dimensional phase space using two time series taken from the driving and the response systems. The number of data points is N=50000.

the response lead to the same asymptotic dynamics. Physically, this means that an ensemble of identical response systems driven with the same chaotic signal should exhibit identical asymptotic behavior. This resembles the well-known physical phenomenon known as the bunching or grouping effect, which is widely used in particle accelerators and other similar systems. Here an ensemble of identical particles started from different initial conditions is grouped to a common trajectory. The difference is that in the bunching effect one usually takes a periodic driving signal; in generalized synchronization, however, we consider chaotic driving. The parallel with the bunching phenomenon makes evident the idea of the auxiliary system approach. To detect the bouncing phenomenon or generalized synchronization, one requires an ensemble of identical response systems. This ensemble should consist of at least of two identical systems, namely, the original and the auxiliary response systems.

References

361

In the phase space interpretation, the asymptotical stability of the response system leads to a stable synchronization manifold M — {(X, Y) : $(X) = Y} that relates the variables of the driving and the response systems. Depending on the coupling strength, the generalized synchronization appears as weak synchronization or strong synchronization. Weak synchronization is characterized by a fractal manifold M with a nonsmooth map $ that increases the global dimension with respect to the dimension of the driving system, dG > dP. Strong synchronization is related to a smooth map $ so that the response system does not influence the global dimension, dG = dP. The threshold of generalized synchronization, in addition to certain properties of the synchronization manifold, can be expressed through the conditional Lyapunov exponents Xf1 and the Lyapunov exponents of the driving system Xf. The mean local thickness a of the synchronization manifold can be used as an alternative characteristic to estimate the smoothness of the manifold. The onset of generalized synchronization is characterized by a new type of on-off intermittency. Unlike the conventional situation, this intermittency can occur in nonsymmetrical systems that do not possess any trivial invariant manifolds. It may appear in any system consisting of two one-way coupled chaotic subsystems. The intermittent behavior appears just below the threshold of weak synchronization and is determined by the loss of stability of the invariant fractal synchronization manifold. It is not noticeable in the phase space of the original response and driving systems; however, it can be detected and analyzed with the help of an auxiliary response system. In experiment, two alternative tools can be used to detect and analyze the generalized synchronization of chaos. One of these is based on an auxiliary response system. Here, the experimental situation should admit the design of a replica of the response system. Another approach is based on time series analysis. Using two scalar time series, one taken from the driving system and the other from the response system, one can estimate the conditional Lyapunov exponents and other parameters that define the existence and the properties of generalized synchronization.

References [1] S. Hayes, C. Grebogy, and E. Ott, Phys. Rev. Lett. 70 (1993) 3031; K.M. Cuomo and A.V. Oppenheim, Phys. Rev. Lett. 71 (1993) 65; T. Kapitaniak, Phys. Rev. E 50 (1994) 1642; G. Perec and H.A. Cerdeira, Phys. Rev. Lett. 74 (1995) 1970; J.H. Peng, E. J Ding, M. Ding, and W. Yang, Phys. Rev. Lett. 76 (1996) 904. [2] K. Pyragas, Phys. Lett. A 181 (1993) 203. [3] A. Kittel, K. Pyragas, and R. Richter, Phys. Rev. E 50 (1994) 262. [4] R. Brown, N.F. Rulkov, and E.R. Tracy, Phys. Rev E 49 (1994) 3784. [5] U. Parlitz, Phys. Rev. Lett. 76 (1996) 1232.

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[26] M. Henon, Math. Phys. 50 (1969) 69. [27] O.E. Rossler, Phys. Lett. A 57 (1976) 397. [28] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [29] N. Platt, E.A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70 (1993) 279. [30] E. Ott and J. Sommerer, Phys. Lett. 188 (1994) 39. [31] A.S. Pikovsky, Z. Phys. B 55 (1984) 149; H. Fujisaka and H. Yamada, Prog. Theor. Phys. 74 (1985) 918; 75 (1986) 1087. [32] J.F. Heagy, N. Platt, and S.M. Hammel, Phys. Rev. E 49 (1994) 1140. [33] H.L. Yang and E.J. Ding, Phys. Rev. E 50 (1994) R3295. [34] J.C. Sommerer and E. Ott, Nature 365 (1993) 138; E. Ott and J.C. Sommerer, Phys. Lett. A 188 (1994) 39. [35] J.F. Heagy, T.L. Carrol, and L.M. Pecora, Phys. Rev. Lett. 73 (1994) 3528; P.W. Hammer, N. Platt, S.M. Hammel, J.F. Heagy, and B.D. Lee, Phys. Rev. Lett. 73 (1994) 1095; T. Kapitaniak, J. Phys. A 28 (1995) L63; Y.H. Yu, K. Kwak and T.K. Lim, Phys. Lett. A 198 (1995) 34; A. Cenys, A. Namajunas, A. Tamasevicms, and T. Schneider, Phys. Lett. A 213 (1996) 259. [36] F. Rodelsperger, A. Cenys, and H. Benner, Phys. Rev. Lett. 75 (1994) 2594. [37] P. Bryant, R. Brown, and H.D.I. Abarbanel, Phys. Rev. Lett. 65 (1990) 1523; R. Brown, P. Bryant, and H.D.I. Abarbanel, Phys. Rev. A 43 (1991) 2787. [38] A.M. Fraser and H.L. Swinney, Phys. Rev. A 33 (1986) 1134. [39] A. Wolf, J.B. Swift, H.L. Swinney, and J.A. Vastano, Physica D 16 (1985) 285. [40] J.-P. Eckmann, S.O. Kamphorst, D. Ruelle, and S. Ciliberto, Phys. Rev. A 34 (1986) 4971. [41] N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, Phys. Rev. Lett. 45 (1980) 712. [42] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, Second Edition (Cambridge University Press, Cambridge, 1994). [43] A. Kittel, J Parisi, and K. Pyragas, Physica D 112 (1998) 459. [44] M. Shinriki, M. Yamamoto, and S. Mori, Proc. IEEE 69 (1981) 394. [45] M.C. Mackey and L. Glass, Science 197 (1977) 287. [46] A. Namajunas, K. Pyragas, and A. Tamasevicius, Phys. Lett. A 201 (1995) 42.

14 Controlling Chaos in a HighDimensional Continuous Spatiotemporal Model E. Tziperman Environmental Sciences,The Weizmann Institute of Science, Rehovot 76100, Israel

14.1 Introduction There has been significant interest in recent years in the control of low-order chaotic dynamical systems using small systematic perturbations that lead to the stabilization of unstable periodic orbits (UPO) [1, 2, 3]. Controlling large- or infinitedimensional systems, however, such as spatiotemporal systems that are governed by partial differential equations is still in its infancy [4, 5, 6, 7, 8, 9, 10, 11]. In this paper, we discuss in some detail the approach proposed in [12] for the control of spatiotemporal systems that are continuous in both space and time. We then describe the application of the proposed method to the control of a complex model, composed of a spatiotemporal system of PDEs, that simulates the El Nino phenomenon in the Equatorial Pacific ocean and atmosphere. Spatiotemporal systems often do not have an accessible adjustable system-wide parameter that can be used for an OGY [1, 11] control. In such systems it often makes more sense and is more practical to apply the control perturbation to one of the dynamical variables, at a specific spatial location. Accordingly, our approach does not require the existence of a global adjustable parameter; rather, it is based on applying OGY-like feedback perturbations to a single dynamical degree of freedom of the system [13], at a single point in space. The spatiotemporal system successfully controlled here seems to be significantly more complicated than the previously controlled discrete systems of coupled chaotic elements [5, 7], or relatively simple or isotropic Id or 2d systems of PDEs [4, 6, 8, 9]. Often, the precise dynamics responsible for the chaotic behavior of systems of such complexity, whether they are models or experimental systems, is not known. Therefore a practical control methodology should not depend on the details of the dynamics or on a manipulation of specific known analytic solutions of the governing system of equations. We therefore use phase space reconstruction for deriving and applying the control law, thus not requiring a knowledge of the detailed dynamics of the system [14]. Both delay-coordinate and non-delay phase space reconstruction are considered here, in an extension to the work presented in [12]. While a knowl-

366

14 Controlling chaos in a highdimensional continuous spatiotemporal model

edge of the specific governing equations of the controlled system is not required by the present methodology, we emphasize that an understanding of the dynamics is, in fact, crucial. As will be seen below, the control of complex spatiotemporal systems may not be achieved using an out-of-the-box algorithm, but requires a good understanding of the relevant dynamics to determine some of the basic features of the control strategy. Previous approaches to the control of continuous systems were often based on a projection of the dynamics onto a discrete map. Our approach, in contrast, controls the continuous spatiotemporal system using a continuous reconstruction of the UPO in phase space. It turns out that this continuous treatment is crucial as we consequently find that not every point on the continuous UPO may be used to apply the control (and therefore that not every projection to a discrete map may be used). We therefore present a novel general criterion for determining the controllability of phase space points along a given UPO. The paper outline is as follows (readers interested only in the chaos-control methodology presented here and not in the specific application to El Nino can read only the sections marked here by an asterisk). We first (section 14.2) briefly review the relevant aspects of El Nino's dynamics (section 14.2.1) and the theories that attribute El Nino's irregularity to low order chaos (section 14.2.2). Next, the El Nino model used here is described in section (14.2.3). Section (14.3*) discusses the choice of control variable (i.e. the variable to which control perturbations are applied) and of control point in physical space (the point in space where the perturbations are applied to the control variable). The actual control algorithm is derived in section (14.4*). The issue of at which point in reconstructed phase space to apply the control corrections, or in other words a criterion for the controllability of phase space points is derived in section (14.5*). The results of applying the proposed control scheme to the El Nino model are presented in section (14.6). Finally, section (14.7*) discusses an extension of the above ideas to phase space reconstruction not based on delay coordinates, and we conclude in section (14.8).

14.2 El Nino's dynamics and chaos While we obviously do not propose here the control of actual El Nino events, we do show that chaos control in a realistic El Nino model can contribute to the understanding of El Nino's dynamics. El Nino events involve a wide-spread warming of the equatorial Pacific Ocean surface water (shown in Fig. 14.1 for the model in which chaos is controlled in this work). The warming events last a few months to a year, occur irregularly in time every 2-6 years, dramatically affect worldwide weather, and have important social and economic implications. El Nino, and a corresponding "Southern-Oscillation" of atmospheric pressure are together termed ENSO [15]. Recent theories [18, 17, 19, 20] attribute El Nino's irregularity to a low order chaos. We thus briefly describe now the relevant aspects of El Nino dynamics (section 14.2.1), survey El Nino chaos theories (section 14.2.2), and describe the structure of the El Nino model used here (section 14.2.3) before

14-2 El Nino's dynamics and chaos

367

120E 140E 160E 180 160W 140W 120W 100W SOW

Longitude Figure 14.1 The deviation of the sea surface temperature from its long-term mean during a peak of a model El Nino (warming) event (lower panel) and during the peak of a La Nina (cooling) model event (upper panel) which typically occurs between El Nino events.

proceeding to the control algorithm.

14.2.1 El Nino's dynamics Let us briefly examine the mechanism of the El Nino cycle, in a simplified version known as the "delayed oscillator" mechanism [23] (The "delayed" in "delayedoscillator" has nothing to do with the delay-coordinate reconstruction of phase space, also used in this paper...). During non-El Nino periods, the wind along the Equatorial Pacific is easterly. The resulting wind-stress acting on the surface ocean water results in a "piling-up" of warm surface water on the western side of the basin where the sea-surface is therefore higher by a few tens of centimeters than in the eastern Pacific. As a result of this piling of warm water, the interface between the warm surface water and cold deep water (known as the thermocline) is also about 100 meters deeper in the western Pacific than in the eastern Pacific ocean. The sea surface temperature (SST) is warmer in the west Pacific ocean, and the east-west SST gradient drives the easterly winds. El Nino events result from an instability of this basic state. The instability and positive feedback that causes it could be triggered by a number of factors, and we describe one possible (yet not unique) simple scenario here. Consider a weakening of the easterly winds in the central equatorial Pacific. The weakening results in

368

14 Controlling chaos in a highdimensional continuous spatiotemporal model

a change to the wind stress curl which causes warm water to shift from higher latitudes towards the equator, creating an excess of warm water (i.e. a thermocline deepening) at the equator and of cold water (i.e. a thermocline shallowing) off the equator [23]. The equatorial thermocline deepening, in turn, increases the distance of the deep cold water from the surface and therefore weakens the mixing of cold deep water with the warm surface water. This results in a warming of the surface water. This thermocline deepening perturbation at the equator propagates eastward as a "Kelvin wave", reaching the eastern boundary after about 1 month. This wave involves eastward-only propagating vertical movements of the thermocline, is restricted to the equatorial domain, and exists due to the Coriolis force [24]. Upon reaching the eastern boundary, the thermocline deepening signal induces a warm SST perturbation there as well. The warming of the eastern Pacific SST reduces the east-west SST gradient, and thus further weakens the easterly winds above the equator that are driven by this gradient, creating a positive feedback (i.e. a coupled ocean-atmosphere instability) that leads to a rapid warming in the eastern equatorial Pacific Ocean, starting an El Nino event. Meanwhile, the initial shallow thermocline perturbations off the equator in the central Pacific, and the corresponding cold SST perturbations, travel westward as Rossby waves. (Rossby waves exist due to the variation of the Coriolis force with latitude, and have a westward phase velocity). The cold Rossby waves are then reflected at the western boundary as cold equatorial eastward-traveling Kelvin waves. Amplified again by the atmospheric feedback, these cold Kelvin waves reach the Eastern Pacific delayed by about 6 months after the original wind perturbation, and terminate the El Nino event. This is an extremely simplistic view of the El Nino mechanism, but it would suffice for our purpose here. For a recent review of current ideas about the El Nino mechanism, see [26].

14.2.2 El Nino's chaos The delayed oscillator mechanism for the ENSO cycle, and its various extensions proposed over the past few years have provided quite a satisfactory explanation for the onset, termination and cyclic nature of ENSO events. However, two basic ENSO characteristics that are still unexplained by these delayed oscillator theories are the irregular occurrence of ENSO events and their apparent partial locking to the seasonal cycle [27]. Figure 14.2 shows the observed sea surface temperature over more than a century, showing the irregularity of ENSO events with regard to both the periodicity of the events and to the amplitude and evolution of each event. Figure 14.3 shows that in spite of this irregularity, the large-scale East Pacific sea surface temperature warming of many of the events tend to peak towards the end of the calendar year. (Data for both plots are from the January 1993 version of the Global Ocean Surface Temperature Atlas (GOSTA) [16]). recent theories [19, 20, 18, 17] have proposed that the irregularity of ENSO may be explained as a low order chaotic behavior driven by the seasonal cycle. According to these ideas, ENSO is viewed as a periodically forced dissipative nonlinear

14-2 El Nino's dynamics and chaos

4i

369

ii i

2 -

1900

1950

-2 1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

2000

Figure 14.2 The observed record of the NIN03 index (sea surface temperature anomaly averaged over 5°S-5°N and 90°W-150°W in the eastern Pacific), showing the irregularity of the events.

oscillator (i.e. the delayed oscillator described above). As such, this system may undergo a transition to chaos according to the universal quasi-periodicity route to chaos [28]. The chaos arises because the natural delayed oscillator of the equatorial Pacific coupled ocean-atmosphere system can enter into a nonlinear resonance with the seasonal cycle at different periods of the oscillator (mostly 2-5 years). In the chaotic regime, each of these nonlinear resonances constitutes an unstable periodic orbits in phase space. The coexistence ("overlapping") of these unstable resonances results in chaotic behavior due to the irregular jumping of the system between the different resonances (unstable periodic orbits). Fig. 14.4 shows several analyses of a time series from the fairly realistic El Nino model whose chaos is controlled in this work. The model is further described in the following section. The time series analyzed in Fig. 14.4 is from a model run using the standard parameter regime where the model is chaotic. The dimension of the attractor was estimated to be about 3.5 [19], so that the behavior is low-order in spite of the many formal degrees of freedom in this continuous spatiotemporal model. The quasi-periodicity route to chaos involves a transition, as the system's nonlinearity is increased, from a periodic behavior with a period that is not commensurate with the forcing period (which in this case is the annual period) to a mode-locked regime characterized by a periodic solution with a period that is commensurate with the forcing period, to a chaotic behavior. In the model used here, this transition may be seen in model experiments as the strength of the seasonal cycle is increased,

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14 Controlling chaos in a highdimensional continuous spatiotemporal model

JMMJSNJMMJSN Figure 14.3 Two-year segments of the observed NINO3 index (sea surface temperature anomaly averaged over 5°S-5°N and 90°W-150°W in the eastern Pacific) during observed ENSO events. The plotted segments begin at January of the years 1888, 1896, 1902, 1911, 1925, 1930, 1951, 1957, 1963, 1965, 1972, 1976 and 1982.

or as the coupling strength between the ocean and the atmosphere is increased. Fig. 14.5 shows a model run in which the coupling strength between the ocean and the atmosphere is reduced (and hence so is the model nonlinearity), and the solution is therefore mode-locked (in a stable nonlinear resonance, or equivalently, in a stable periodic orbit) at a perfectly periodic solution with a four year period (which therefore commensurate with the annual forcing frequency at a ratio of 4:1) rather than chaotic. Fuller description of this transition to chaos may be found in [20]. The significant role played in this scenario by the seasonal cycle provides also a simple explanation for the locking of ENSO events to the seasonal cycle (yet not a detailed mechanism, which is proposed in [29]). The above chaos mechanism due to the periodic seasonal forcing is an attractive explanation for ENSO's irregularity, yet not the only possible one. It is possible, of course, that random excitations in the form of stochastic forcing of the large-scale delayed oscillator dynamics are the cause of the irregularity of ENSO events. The stochastic forcing could be due to short term weather phenomena which may be regarded as a random noise source for the large-scale dynamics because it results from processes that act on very short time scales of a few days and relatively small spatial scales. It is also possible that the large scale dynamics is chaotic even without the seasonal forcing. In fact, the ENSO model in which chaos is controlled in this study is sufficiently nonlinear that it is also (weakly) chaotic in the absence of a seasonal forcing as demonstrated in Fig. 14.6. The chaos mechanism in this El Nino model seems to be chiefly due to the forcing by the seasonal cycle, so that with the periodic forcing present, the model irregularity is enhanced. Below, we shall demonstrate the control of the weakly chaotic model of Fig. 14.6 without the seasonal cycle.

14-2 El Nino's dynamics and chaos

0 4 8

14 20 26

t (years)

.4

.8

1.2

371

1.6

(1/years)

1 2 3 4 5 6 7 8 9

101112

Figure 14.4 Analyzes of the model NINO3 index time series for the standard seasonally forced chaotic model run. (a) A 30 year portion of the model NIN03 time series (thin line) showing the irregularity of El Nino events, and its 12months running average (thick line), (b) Power spectrum characterized by a wide peak around 4 years, (c) A 2d reconstructed delay coordinate phase space Poincare-section obtained by subsampling the NINO3 time time series every year, using a delay r = 1 year, (d) A histogram of the number of ENSO events (vertical axis) per month of the calendar year (horizontal axis), showing the tendency of most events to peak towards the end of the calendar year, as for observed events.

14.2.3 Model description The El Nino model used here (described in detail in [21]) has proven quite successful in predicting El Nino events up to one and a half years in advance [22]. It is based on a set of nonlinear partial differential equations for the ocean and another one for the atmosphere, with specified couplings between them. The equations are written for the deviations from the observed spatially-variable long-term mean state of the Equatorial Pacific. This mean state may be seasonally varying [21], and in the model version used here (Fig. 14.6) is set to the time-independent mean July state. In the model equations t is the time, (x,y,z) are the (east, north, up) coordinates; (u,v,w) the corresponding ocean water velocities; (ua,va) the atmospheric wind velocity in the (east, north) directions; v = (u, v), v a = (ua,va)'i V = (dx,dy); the total depth of warm surface waters in the model, also known as the "thermocline" depth, is h(x,y,t); T(x,y,t) is the sea surface temperature (SST). v, T and w are the spatially variable specified observed long-term July

372

Controlling chaos in a highdimensional continuous spatiotemporal model

4 8

14 20 26

t (years)

.4

.8

1.2 1.6

(1/years)

1 2 3 4 5 6 7 8 9 101112

Figure 14.5 Same as Fig 14.4, for the case of a weaker specified nonlinearity, where the model solution becomes mode-locked at a four-year nonlinear resonance with the seasonal cycle. The Poincare section reduces to four clouds corresponding to the four year period. The evolution of El Nino events is strongly coupled to the annual cycle and event peaks all occur in August-September. The spectrum is composed of sharp peaks at the main frequency of 4 years and its harmonics with the annual forcing frequency.

mean fields; /3 = df jdy is the gradient of the Coriolis parameter; and the terms with r, ra and a represent various dissipation processes; H is a mean thermocline depth and g' denotes gravity acceleration. The model oceanic currents are driven by the atmospheric wind stress, {T^X\T^), which is quadratically related to the wind velocity (ua,va). The atmospheric winds are driven by a heating, Q[T, Vva] which is a nonlinear function of the SST and of the atmospheric wind divergence. The model is composed of four sets of PDEs. The first set is for the vertically averaged ocean momentum and mass conservation equations, ut — f3yv = —g'hx -f T^X' [ua, va] — ru /3yu = -g'hy + r^ [ua, va] - rv ht + HVv = -rh.

{UA)

A second similar set of PDEs is used to simulate the vertical velocity shear in the ocean [21]. A third effectively 2d set of PDEs models the momentum and mass balances of the atmosphere, f3yva

=

-cj)x -

raua

H.3

Choosing a control variable and a control

3.0 , 1.5 0 -1.5 0 4 8

14 20 26

t (years)

0

.4

point in space

373

.8 1.2 1.6

(I/years)

1 2 3 4 5 6 7 8 9

101112

Figure 14.6 Same as Fig 14.4, for the case of a perpetual July forcing (no seasonal cycle), where the model solution is weakly chaotic.

/3yua (j)t+HX7v

=

-<j>y -

=

Q[T,Vva]

rava (14.2)

Finally, the SST is determined by a nonlinear advection - dissipation equation roughly of the form Tt + vVT + vV(T + T) + wTz + (w + w)Tz = -aT

(14.3)

The model's finite-difference discretization is based on many thousands of grid point variables. The model solution is aperiodic (Fig. 14.6) and involves unstable interactions between the ocean and the atmosphere, which are manifested through multiple spatial and temporal scales of all model fields. Yet, the previously demonstrated low-order temporally chaotic behavior of this model [19, 20] makes it a perfect candidate for testing chaos-control ideas in a complex, high dimensional, spatially extended system. The challenge, of course, is to control an UPO that represents the full-domain oscillation of the El Nino cycle without applying the chaos-control corrections at many spatial locations.

14.3

Choosing a control variable and a control point in space

One of the main keys to controlling chaos in a complex spatiotemporal system using the approach suggested here, is the careful identification of the correct spatial

374

14 Controlling chaos in a highdimensional continuous spatiotemporal model

point and degree of freedom (model variable or physical quantity in an experimental system) to which control corrections are applied, based on an understanding of the system's dynamics. The variation of the Coriolis parameter with latitude (fly in (14.1)) results in the equator being a wave guide for trapped ocean wave modes which have the form Hn(y/£)exp(—^y2/£2)expi(kx — ant), with Hn being the Hermite polynomial of order n, and where £ = 3.2 degrees latitude. The mode n — 0 is the eastward propagating equatorial Kelvin Wave discussed in section (14.2.1), and the n > 1 modes are the westward propagating, off equatorial, Rossby modes ([24]). These equatorial ocean Kelvin and Rossby waves play a central role in El Nino's dynamics [15] as seen in section (14.2.1) above. Now, as made clear by the delayed oscillator mechanism of ENSO, the western boundary of the ocean at the equator is a "choke point" which affects the entire tropical Pacific through the reflection of the Rossby waves into Kelvin waves [23]. Remember that it is this reflection of the Rossby waves that bring about the termination of the warming event. We therefore chose to control the entire Equatorial Pacific model ocean and atmosphere by applying small perturbations to the oceanic model fields at the western boundary (x = xw) of the Pacific Ocean. Moreover, we choose the applied control perturbations to have the ^-structure of the Kelvin mode. Because the Kelvin mode amplitude decays like exp(— \y2 jH?) away from the equator, the control correction directly affects the oceanic model fields only in a very small region near the equator, at the western boundary. Furthermore, the perturbation structure implies that our control variable is the Kelvin mode amplitude at the western boundary only, Kw(t). The model solution for the ocean fields, which is obtained numerically over the finite difference model grid, may be expanded in terms of the equatorial Kelvin and Rossby modes (which constitute a complete set of eigen functions for the ocean dynamics (14.1)). In terms of such an expansion, it is clear that we are modifying a single degree of freedom out of thousands which exist in the model. This choice of a choke point may seem somewhat counter intuitive at first sight, because the warming induced by the El Nino events actually occurs in the eastern Pacific (Fig. 14.1), while we are trying to control it by making changes thousands of kilometers away in the western most Pacific. However, this choice clearly makes sense once we bring into account the role of the wave dynamics and of the western boundary of the ocean in the delayed oscillatory mechanism. This choice is thus based on some 10 years of research into El Nino's dynamics, making it clear that one needs a good understanding of the dynamics of such complex spatiotemporal chaotic systems in order to successfully control them.

14-4 A continuous delay-coordinates phase space...

375

14.4 A continuous delay-coordinates phase space approach to controlling chaos in high dimensional, spatiotemporal systems As the next step in applying our method to the control of chaos in this model, we identify its UPOs. The UPOs are determined in an N dimensional delay-coordinate phase space reconstructed from our control variable, which is the Kelvin wave amplitude at the western boundary. Note that the approach presented here is not limited to delay coordinate reconstruction, and the case of non-delay coordinate phase space reconstruction is discussed in section 14.7 below. The phase space coordinates are thus the iV-dimensional vectors X(t) = {Xh i = 1,.., N} = (Kw(t - (TV - l)r), ...,Kw(t - r),Kw{t))J',

(14.4)

where the last coordinate, Xjy, is the present-time Kelvin wave amplitude to which the control corrections will be applied. To identify the UPOs for a given period p, we search for phase space points X{t) that return to the same neighborhood after a period p, so that ||X(*)-X(*-p)||<e

(14.5)

for some small e. Using r = 1 year, and plotting the number of such close pairs as a function of p, the UPOs show up as peaks (Fig. 14.7a). By actually plotting the close pairs of neighbors identified through the criterion (14.5) for a given period as dots in a three dimensional reconstructed phase space (N = 3), we can visualize the UPOs, and two such UPOs are shown in Fig. 14.7c,d. The first UPO corresponds to a relatively weak El Nino event with a period of 4.3 years, while the second corresponds to a strong event followed by a very weak one, repeating with a period of 7.83 years. Next, an 7V x N linear map, M, is least-square-fitted to the model dynamics over a small neighborhood in phase space near a point that is located along the controlled unstable periodic orbit, and that serves as the control point in phase space. The linear map should map the phase space point X(t — p) to the state of the system at a time p later, X(t). The map M is therefore found by minimizing, via a least-square procedure, the following quantity, J(M) = £ | | X ( i ) - M X ( i - p ) | | 2 .

(14.6)

t

Given the map M, we calculate its eigenvectors and thus find its stable and unstable manifolds. The feedback control correction is now calculated so that when the system approaches the control point in phase space, the control correction brings the phase space trajectory X(£) towards the stable manifold of the UPO (see point X 2 (t) in Fig. 14.8). By the definition of the stable manifold, the model evolution will then bring the trajectory of the control variable towards the UPO itself [1]. If our choice of a choke point in space is appropriate, the entire 3d model solution will follow the control variable and settle on the UPO as well.

376

14 Controlling chaos in a highdimensional continuous spatiotemporal model

(b)

2

4 6 8 period (years)

10

Figure 14.7 (a) Log of number of near-returns as function of the period p. Each peak corresponds to an UPO, and the peak at p = 4.3 years is the one stabilized here, (b) A segment of phase space trajectory during a typical standard run, showing the trajectory switching between the two UPOs of panels (c) and (d). The three axes are (Kw(t - 2r),Kw(t - r),Kw(t)). (c) A 3d reconstructed delay coordinate phase space plot of the near returns forming the p = 4.3 years UPO that is stabilized in the controlled run. Larger balls along the UPO denote smaller GNN and thus more controllable phase space points (see text). The most controllable phase space point, with the smallest GATJV, where the control correction is actually applied, is marked by "x". (d) An UPO with a period of p = 7.83 years.

We now derive the expression for the control correction in terms of the known location of the phase space point before the correction is applied, X(£), as well as the known stable and unstable eigenvectors of the map M. Suppose that the linear map M, evaluated at the control point, has Ns stable eigenvalues whose eigenvectors span the stable manifold. Let 5 be an N x Ns rectangular matrix composed of these Ns stable eigenvectors. Let X(t) be the phase space location of the model trajectory at time t, defined with the origin at the control point along the controlled UPO. The phase space location in the stable manifold to which we wish to bring the model trajectory can be written as 5 a where a is some Ns x 1 coefficient vector. The control correction can only be applied to the present time Kelvin amplitude Kw(t) = X]y(t), so that the phase space trajectory can only be corrected in the direction of a unit vector X;v along the iVth axis in phase space. The phase space location after the application of the control perturbation 8XN is, therefore, X(£) + XJV<5X/V. We are interested in the control correction SXN for

14-5 Controllability of delay-coordinate phase space points along an unstable periodic orbit 377

which the distance of the corrected phase space location to the stable manifold, d = ||£a — (X(t) -f X/v<5X/v)||, vanishes. The amplitude of the correction, SX^, is accordingly obtained by solving the equations requiring that the square of the distance d2 to the stable manifold is minimized, dd2/da dd2/d6XN

= 0 = 0.

(14.7) (14.8)

Solving these Ns + 1 equations for the coefficient vector a and for the requested control correction amplitude 6XN wefinallyfind SXN = (XN - X%GX)/(XlGXN

- 1),

(14.9)

where G = S(STS)-1ST. Given a time series from any spatiotemporal system, this simple control law may always be derived without additional knowledge of the dynamics. In the present case, we used N = 3 and found that typically there is one unstable eigenvalue of M (whose value, representing the amplification over a full orbit around the UPO, typically varies around 1.5), one neutral (value close to 1, and whose eigenvector points along the UPO) and one stable eigenvalue (< 1), so that we set Ns = 2. The control correction (14.9) is applied to the control variable Xpj = Kw(t) when the phase space trajectory is within a small specified radius from the control point in phase space. But before displaying the results of a chaos-control experiment, we need to discuss the choice of a control point in phase space along the UPO to be stabilized.

14.5 Controllability of delay-coordinate phase space points along an unstable periodic orbit One of our more generally applicable results here is a procedure for choosing the phase space points along a given UPO at which control may be applied. As explained above, our control correction is always applied in the direction of XN in the reconstructed phase space because this is the direction corresponding to the present-time control variable. If, for some control point, this direction is parallel to the stable manifold (Fig. 14.8), the control perturbations alongXiv cannot bring the phase space trajectory away from the unstable manifold. Such an uncontrollable situation happens when the direction of XN is parallel to that of the stable manifold, which means that the unit-length vector XN may be written as a linear combination of the stable manifold eigenvectors, XJV = 5a, where a is a coefficient vector. Multiplying both sides of this equation on the left by ST and then by (STS)~1 we obtain (STS)~1STXN = a. Multiplying again on the left of each side by S, we find S(STS)~1STXN = SSL = XN. Finally, multiplying by Xjy and using XJ^XN = 1, we find for an uncontrollable phase space point, NN

= X% X%GXN =

= 1.

(14.10)

378

14 Controlling chaos in a highdimensional continuous spatiotemporal model

stable manifold stable manifold unstable manifold

KU!(t-r)

Kw(t-2r)

Figure 14.8 A schematic drawing of an uncontrollable situation in which point Xi(£) cannot be brought towards the stable manifold using a control perturbation in the direction of Kw(t). In contrast, X2(£) is near a controllable point on the UPO, because a perturbation in the direction of XN — Kw{t) can bring X2(£) towards the stable manifold along the thin dash arrow starting

at X2(t). This uncontrollability condition leads, according to (14.9), to an infinite amplitude correction SXN- Likewise, a smaller GNN implies a smaller perturbation SXN required to bring X(t) to the stable manifold, and thus a better phase space point to apply the control. Fig. 14.7c shows how the variation of GNN along the stabilized UPO may be used to choose an appropriate control point in phase space. This controllability condition may be extended to non-delay coordinates as discussed in section 14.7 below. We note that the above controllability condition only addresses one aspect of controllability, i.e. the direction of realizable perturbations vs the direction of the stable and unstable manifolds. Given many controllable points along the UPO to be controlled based on the condition that GNN is as small as possible, one may need to apply additional considerations for the choice of the optimal control point. For example, one may bring into account the location of possible control points relative to the location of other UPOs that need to be avoided, or the amplitude of the local Lyapunov exponents of the stable and unstable manifold at different possible control points, etc.

14.6 Results Let us examine the results of the application of the above control procedure to our El Nino model. During the model integration, the control correction SXN is calculated using (14.9). The Kelvin wave amplitude at the western boundary of the

14-6 Results

379

Pacific is corrected by SXjy only when the model trajectory in phase space nears the control point, and only when 5XN is smaller than a pre-specified threshold. Fig. 14.9 shows the model solution with and without control, demonstrating that the procedure outlined here indeed works most efficiently for this complex El Niiio model.

-60

Figure 14.9

-40

-20 0 Amplitude

20

120E 19K

90*

Longitude

(a) Left curve: a time series of the Kelvin wave amplitude on the western boundary of the Pacific Ocean, to which the control is applied from year 150 to 200. Right curve: the magnitude of the applied control perturbation (shifted to the right by 30 units so it would not overlap the other time series), (b) A plot of the equatorial sea surface temperature as a function of longitude and time during the same run. The controlled, periodic behavior during years 150-200 represents a full-domain oscillation of a complex spatial structure and temporal evolution.

Before the control procedure is turned on at time t = 150 years, the model is in its chaotic regime. After the control is turned on, the model trajectory in phase space approaches the control point on the UPO to be stabilized at year 153154 and a control corrections are applied twice, yet do not succeed in stabilizing the system on the UPO. Later, after year 160, the model again approaches the control point in phase space, and this time the control correction manages to trap the model evolution on the UPO. From that point onward to time t = 200 years, the control correction is applied every about 4.3 years, and the model evolution

380

14 Controlling chaos in a highdimensional continuous spatiotemporal model

is perfectly periodic, as can be seen in both the time series of the Kelvin wave amplitude and the plot of the sea surface temperature along the entire equator. Once the control procedure is turned off at year 200, the model rapidly returns to its chaotic behavior. A closer look at the control perturbation and model response during the control period is given in Fig. 14.10. One can see that the control perturbations are applied to the Kelvin wave amplitude at the western boundary of the Pacific Ocean when this wave amplitude is at its minimum. The control perturbations start when the sea surface warming in the east Pacific is at its peak, and are applied over a few months until near the end of the warming period. 172

(a) 170 •

<

168

V* •

<

i

164

162

I

160 -60 -40 -20 0 20 Amplitude

120E 150E

180 150* 120W 90W

Longitude -2 -1.5 -1 -OS 0 O5

11X1

Figure 14.10 Same as Fig. 14.9, zooming over a shorter time interval We could not attribute any physical meaning to the timing of the perturbations. The timing of the control perturbations relative to the evolution of the controlled El Nino events is a consequence of the choice of the control point in phase space. This choice, in turn, is dictated by the limitations of realizable movements in phase space to the direction of the present-time Kelvin wave axis, as discussed above. There are some interesting implications of the successful control of chaos in this model to the understanding of El Nino's dynamics. Previous works debated whether the aperiodicity in the El Nino model used here is due to low-order chaos [19, 20], or due to "noise" expressed in the model as high frequency, small spatial-scale air-sea interactions in the western Pacific. This "noise" can be seen in

14-7 Using non-delay coordinates for phase space reconstruction

381

Fig. 14.9a as an intermittent high frequency signal at years 130-135, for example [25]. The existence of unstable manifolds of the UPOs and the successful control of chaos in this El Nino model are a clear demonstration that the aperiodicity in this model is due to low-order chaos (whether El Nino events in the actual equatorial Pacific are aperiodic due to chaos or noise is still under debate). Given that we have demonstrated that the high-frequency small-scale signal seen in the model is not the cause of the aperiodicity of the model El Nino events, we can proceed to speculate on the source of this signal. Each unstable spatiotemporal UPO in this system is, of course, characterized by both different temporal evolution and different spatial patterns. This leads to an especially interesting possibility that the small-scale, high-frequency "noise" in spatiotemporal systems (such as seen in the Western Pacific in this model) may be a result of the low-order chaotic behavior, due to the large-scale spatial fields readjusting when jumping from one UPO to another.

14.7 Using non-delay coordinates for phase space reconstruction Delay coordinates are useful when one has access only to a single parameter within the observed or simulated system. Often, however, one can measure more than one parameter, and then a more complete set of observations may be used to reconstruct the phase space picture in a way that is more reliable than using delay coordinates based on a single measured quantity. The use of non-delay coordinates has implications both on the reconstruction of the UPOs, and on the controllability condition derived above. As an example, we have repeated the calculation of the number of near-returns as function of a period using both delay coordinates and non-delay coordinates (see Fig. 14.7a). The non-delay coordinates phase space reconstruction was done using physically significant measures of the state of the El Nino cycle, that are expected to provide mutually independent information based on our understanding of the El Nino mechanism. More specifically, we have used as the reconstructed phase space coordinates the following physical variables: Kelvin wave amplitude in western Pacific; the NINO3 average sea surface temperature index (Fig. 14.2); the thermocline depth averaged over the same area as the NINO3 index; the thermocline depth averaged over the west equatorial Pacific; the thermocline depth averaged over the central Pacific south of the equator; and the thermocline depth averaged over the central Pacific north of the equator. Note that when using delay coordinates one is limited to a number of coordinates TV such that the delay time Nr is not much larger than the decorrelation time of the system. In the El Nino model considered here, this means a practical limit of about iV = 3 to N = 4 with a coordinate delay time of one year. When using non-delay coordinates, this limit on the number of coordinates does not exist, and one may choose as many independent measures of the system state as the phase space coordinates as are available.

382

Controlling Chaos in a Highdimensional Continuous Spatiotemporal Model

The calculation of number of near returns using both phase space reconstructions was done for both the weakly chaotic perpetual July model run (Fig. 14.6) that was controlled above and for the more strongly chaotic seasonal model run (Fig. 14.4). The results for the number of near returns as function of period are shown in Fig. 14.11. It is very clear that the non-delay coordinates provide a smoother and cleaner picture of the peaks in these plots and thus of the unstable periodic orbits of the model. In particular, using non-delay coordinates (Fig. 14.11b) one can see in the seasonal model the annual harmonics that are expected to be UPOs of the model, while they are hardly distinguishable in the delay coordinates calculation for the same model run (Fig. 14.11a). Our controllability condition (a) seasonal, delay coordinates

5 10 15 20 (c) July, delay coordinates

10 15 period (years)

(b) seasonal, non-delay coordinates

5 10 15 20 (d) July, non-delay coordinates

10 15 period (years)

Figure 14.11 log of number of near returns vs period: (a) seasonal model (strongly

chaotic), delay coordinates, (b) seasonal model, non-delay coordinates, (c) perpetual July model (weakly chaotic), delay coordinates, (b) perpetual July model, non-delay coordinates, (14.10) may also have a different form when non-delay coordinates are used. It is possible, for example, that several parameters of the system may be measured in order to reconstruct the phase space trajectory and the UPOs, but that only a single parameter is accessible to control perturbations. In this case, this control parameter takes the place of Xjy in our derivation, and the above control law and controllability condition are unchanged. If, however, the number of parameters accessible for applying control perturbations is less than the number of parameters used for phase space reconstruction, yet is larger than one, then our above derivation for both the control law (14.9) and the controllability condition (14.10) needs to be generalized. The conceptual basis behind the derivation, however, is unchanged.

14-8 Conclusions

383

14.8 Conclusions

We have presented a procedure for controlling chaos in a continuous spatiotemporal system, and have demonstrated it by controlling the low-order weakly chaotic behavior of a realistic El Nino model that is used in actual prediction of El Nino events in the Equatorial Pacific ocean and atmosphere. Rather than applying the control correction to a global adjustable parameter, the present method applies the perturbations directly to a dynamical degree of freedom of the model, as was done for simpler systems in [13]. In complex spatiotemporal systems, this requires a careful choice of the right dynamical variable to which to apply the corrections, as well as the spatial location in which this should be done. These two choices are most crucial, yet cannot be done according to some generic algorithm, but rather must be based on a good understanding of the relevant dynamics. The control method proposed here is inherently continuous in its treatment of space, time and phase space, and is not built around a projection of the dynamics to a discrete map. Thus a continuous UPO is first identified in a reconstructed phase space, and then a point on the UPO at which perturbations are to be applied is chosen. We have found that this point on the UPO cannot be chosen arbitrarily and have formulated a criterion for this choice, based on the direction of stable manifolds of the UPO relative to the direction of realizable control corrections in phase space. While most of the work here was based on delay-coordinate phase space reconstruction, we have also considered a non delay-coordinate phase space reconstruction, and demonstrated that when such a reconstruction is feasible (when more than one measurable parameter exists in the system), it has some advantages over the delay-coordinate approach. By controlling chaos in a realistic El Nino model we were able to gain some useful insights regarding El Nino's chaos. There is also an interesting lesson regarding spatiotemporal systems in general. Unlike fully developed turbulence, low-order chaotic behavior in a spatiotemporal system with many degrees of freedom typically involves the active participation of only the larger spatial scales and slower temporal scales of the system. We have seen, however, that the low-order chaos in our spatiotemporal model involved some high-frequency, small scale signal as well. Generally, each UPO is characterized by a unique large spatial scale structure, and slow temporal evolution. We have speculated that high frequency, small scale signals in such systems may be a result of low-order behavior, due to the large-scale spatial fields readjusting when the system jumps from one UPO to another. The successful application of the chaos control method presented here to a complex PDE El Nino model is a clear demonstration of the robustness and potential of the method. In addition, the results presented here may also contribute to the important problem of understanding and predicting El Nino events in the Equatorial Pacific.

384

References

Acknowledgments I thank Steve Zebiak and Marc Cane for helping with the use of their model, and J. York for his suggestion to examine the non-delay coordinate reconstruction. This research was partially supported by the Israel-US Binational Science Foundation.

References [1] E. Ott, C. Grebogi, and J. Yorke, Phys. Rev. Lett. 64, 1196 (1990). [2] T. Shinbrot, Advances in Physics, 44 73-111, (1995). [3] L. W. Ditto, M. L. Spano, J. F. Lindner, Physica D 86, 198-211, (1995). [4] I. Aranson, H. Levine, L. Tsimring, Phys. Rev. Lett, 72 2561-4 (1994). [5] D. Auerbach, Phys. Rev. Lett. 72, 1184, (1994). [6] C. Lourenco, M. Hougardy, A. Babloyantz, Phys. Rev. E, 52, 1528-1532, (1995). [7] A. Babloyantz, C. Lourenco, J. A. Sepulchre, Physica D, 86, 274-283, (1995). [8] Y. Braiman, J. F. Lindner, W. L. Ditto, Nature, 378, 465-467, (1995). [9] G. Hu, Z. Qu, K. He, Int. J. Bifurcation and Chaos, 5, 901-936, (1995). [10] M. Kushibe, Y. Liu, J. Ohtsubo, Phys. Rev. E, 53, 4502-4508, (1996). [11] M. Ding, W. Yang, V. In, W. L. Ditto, M. L. Spanno, B. Gluckman, Phys. Rev. E. 53 4334-4344, (1996). [12] E. Tziperman, H. Scher, S. Zebiak and M. A. Cane: Physical Review Letters 79, 6, 1034-1037. (1997) [13] A. Garfinkel, M. L. Spano, W. L. Ditto, J. N. Weiss, Science, 257, 1230-1235, (1992). [14] U. Dressier and G. Nitsche, Phys. Rev. Lett 68, 1-4 (1992). [15] S. G. Philander, El Nino, La Nina, and the Southern Oscillations (Academic Press, San Diego, California, (1990). [16] Bottomley M., C. K. Folland, J. Hsiung, R. E. Newell, and D. E. Parker. Global ocean surface temperature atlas "GOSTA". Meteorological Office, Bracknell, UK and the Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA. 20 pp and 313 plates. (1990).

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[17] Chang P., B. Wang., T. Li, L. Ji., Geophys. Res. Let, 21, 2817-2820, (1994). [18] Jin, F-F, D. Neelin and M. Ghil, Science, 264, 70-72, (1994). [19] Tziperman E., L. Stone, M. A. Cane and H. Jarosh, Science, 264, 72-74, (1994). [20] Tziperman, E., M. A. Cane and S. Zebiak, J. Atmos. Sci., 52, 293-306, (1995). [21] Zebiak, S. E. and M. A. Cane, Mon. Wea. Rev. 115, 2262 (1987). [22] M. A. Cane, S. E. Zebiak, and S. C. Dolan, Nature, 321, 827 (1986). D. Chen, S. E. Zebiak, A. J. Busalacchi and M. A. Cane. Science, (1995). [23] M. J. Suarez, and P. S. Schopf, J. Atmos. Sci. 45, 3283 (1988); N. E. Graham and W. B. White, Science 240, 1293 (1988); Battisti, D. S., J. Atmos. Sci., 45, 2889, (1989). [24] Gill, A. E. Atmosphere-ocean dynamics, Academic Press, Orlando, (1982). [25] Mantua, J. N. and D. S. Battisti J. Climate, 8, 2897-2927, (1995). [26] Neelin, J. D., D. S. Battisti, A. C. Hirst, F.-F. Jin, Y. Wakata, T. Yamagata, S. Zebiak. Special Joint issue of J. Geophys. Res. Atmospheres and J. Geophys. Res. Oceans. In Press. (1998). [27] Rasmusson, E., and T. Carpenter, Mon. Wea. Rev., 110, 354-384. (1982). [28] Bak, P., T. Bohr and M. H. Jensen, Physica Scripta, T9, 50-58. (1985). [29] Tziperman, E., M. A. Cane, S. Zebiak, Y. Xue, B. Blumenthal, In press, J. Climate, (1998).

15 Controlling Production Lines L. A. Bunimovich Southeast Applied Analysis Center & School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

15.1 Introduction A classical assembly line consists of some number m of work stations with n workers which are working on this line. Traditionally workers are assigned fixed work stations. We will consider the case when only one worker can work on any work station at any moment of time. In other words, two workers cannot work simultaneously on the same machine. Such situation is typical for a manufacturing which doesn't require some high skills, e.g., in the apparel industry. Then the station with the greatest work content determines the production rate. In the work-sharing manufacturing naturally m > n, i.e., a number of workers doesn't exceed a number of machines. Indeed, some workers must be idle otherwise. Traditionally each worker is assigned a fixed work station or several work stations. Thus each worker gets responsible for his own zone. Practically unavoidably such production line becomes unbalanced because workers get unequal shares of an entire work required to produce an item. In fact, this partition of entire work is forced by the existing configuration of work stations in the production line. (Recall that two workers are not permitted to work on the same machine if the fixed work stations are assigned to each worker.) Such unequal distribution of work leads to the relatively low output of a production line while some parts may be produced at much higher rate. Indeed, imagine that some worker is assigned to the relatively simple in operation machine which sews, say, sleeves, while another worker is assigned to the machine which sews collars which require more time. Then a lot of sleeves will be left somewhere in the middle of the line at the end of the working while only relatively few shirts are produced at this line during this time. Therefore a manager must come up with some idea how to resolve this problem, i.e., how to start the next working day, how to avoid the same situation in the future, etc. The standard approach to this problem is to develop a work content model. It must contain the very detailed description of all the operations performed in the production line under study, the times required to perform these operations, their variations, etc. Thus the work content model is always very complicated, practically nonmanageable and unavoidably inaccurate. Besides the zone manufacturing is extremely nonflexible. Each worker performs exactly the same sequence of operations on each item. Control of such production

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15 Controlling Production Lines

lines is centralized and the only way to change the production rate is to redistribute the tasks and tools over different stations. It is on one hand extremely expensive and, on the other hand, may require a long time for workers to get used to the new configuration of a production line. A flexibility of the production systems is especially important when products have short life-cycles. Such situation is, peculiar, for instance, to the apparel industry because of the seasonalities of products. The new idea how to increase flexibility of production has been recently introduced in the apparel industry by Aisin Seiki Co. Ltd., a subsidiary of Toyota, and named the "Toyota Sewn Products Management System," or TSS, which is a registered trademark of Aisin Seiki Co. Ltd. It is marketed in the western hemisphere by TSS Americas, Inc. In the TSS-line each worker follows some simple rule and the control of the system is essentially decentralized. The advertising literature says that TSS has been used successfully in the manufacturing of furniture, shoes, fish nets and of many others of sewn products. A work (evolution) of TSS production lines can be naturally represented by a dynamical system [1, 2] where different workers move as the particles along the production line and interact via (essentially nonhomogeneous and nonisotropic) repulsive potential. This potential just represents the restriction that two workers cannot share any work station on the production line at any moment of time. The order of workers on a TSS-line remains fixed. Therefore, the natural problem is to find such sequencing of workers on a line which corresponds to the maximal production rate. It is the central problem that will be discussed in this paper. Any fixed sequencing of workers on the TSS production line can be modeled by a dynamical system. Thus our problem, as the problem of a manager of such line, is to pick an optimal dynamical system out of the collection of n! dynamical systems which correspond to the different sequencing of workers on the line. The ideas and methods of the theory of dynamical systems allowed to suggest the effective approach to this problem [1, 2]. A production rate has been increased at the various industrial sites in the apparel industry where this approach has been implemented. Moreover, essentially the same approach has been effectively applied for the management of work of pickers in the warehouses [3, 5]. At the first sight, it seems to be strange. Indeed, the workers on the production line in the apparel industry are sewn (at least during the same season) the identical items like shirts, slacks, etc., while in the warehouses any picker always gets a new order which contains a new (random) collection of items (and of amounts for each item) to be picked from the shelves. Still the ideas of the theory of dynamical systems allowed to help to optimize as well as those intrinsically stochastic systems. Our approach formally has nothing to do with the standard approach to the control of chaos (like e.g., OGY algorithm) which is discussed in the other chapters of this book. However, psychologically those approaches are quite similar. In both cases we want to stabilize a system and to avoid it to be involved in a (chaotic, irregular) motion.

15.2 TSS Production Lines and Their Model

389

It explains, in particular, why the approach we discuss here is so effective in the applications. Indeed, we cannot always analyze completely what is going on with the system in all cases. (It is worthwhile to mention that the production lines in the apparel industry are often operated by a few workers (i.e., n is small). For instance, the lines producing T-shirts are operated by two workers, the lines producing ladies slacks by 4 workers, the lines producing hockey parts by 11 workers and the ones producing fine leather travel bags are operated by 16 workers. Moreover, almost 50% of all production lines in the apparel industry are operated by a fewer than five workers. Thus the models of such lines are low-dimensional dynamical systems which sometimes can be investigated in details.) What we cannot analyze though refers usually to the situations where dynamics is rather complex. Such situations are unwanted in the manufacturing. Therefore, one doesn't need to analyze a dynamics completely but rather indicate how to stay in the region with simple and stable (robust) dynamics. Therefore, we believe that the theory of dynamical systems has got a great potential for the applications to the industrial problems. The ones we discuss in this paper are very simple and illustrative. It allows to meet for the TSS lines two constant challenges in manufacturing which are to remove work from the system and to balance remaining work among productive elements. The approach we discuss here gives the means to find such sequencing of workers which ensures a self-balanced dynamics of the line and a stable and (quasi) optimal partition of work among the workers.

15.2 TSS Production Lines and Their Model We describe now how the TSS production lines work. Consider a production line in which each item requires processing on the same sequence of m work stations. Here by an item we mean any product (for instance, a shirt, a coat, a leather bag etc.) for the sewing production lines. Any work station can process at most one item at a time, and exactly one worker is required to do that. We also assume that all items are identical. Therefore each item requires the same total processing time. In the TSS line each worker carries an item from station to station, processing it at each station until passing it off to a subsequent worker. We label each worker by a number, which varies from 1 t o n according to the sequence of workers on the line (in the direction of product flow). TSS protocol requires each worker to independently follow the following rule: Forward Part. Remain devoted to a single item, and process it on successive work stations (where at any station the worker with a bigger number has priority). If your item is taken over by your successor (or if you are the last worker and you complete processing the item), then relinquish the item and begin to follow the Backward Part. Backward Part. Walk back and take over the item of your predecessor (or, if you are the first worker, pick up raw materials to start a new item). Begin to

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15 Controlling Production Lines

follow a Forward Part. It is seen immediately that the TSS rule has at least two drawbacks comparing to the standard zone manufacturing. The first is that a worker can be blocked during the forward phase if he is ready to work at a machine but this machine is occupied by another worker. According to the TSS rule a blocked worker must wait for the work station to become available. Observe that each worker works all the time (at least theoretically) on a line operated under the rule of a zone manufacturing. The second drawback is that during the backward phase each worker interrupts his predecessor and takes over his work. (It is worthwhile to mention that a great deal of effort is invested to avoid delay in this. For instance, Aisin-Seiki sells work stations have been specially configured to support TSS-style manufacturing. Moreover, the line has a [/-shape to reduce walking time.) Nevertheless, TSS production lines in the apparel industry have proven to be effective. They not only allowed to simplify the management of the flow lines but even allowed to increase a production rate of such lines. Still the question on an optimal sequencing of workers along a line remains open. To address this issue we will discuss a mathematical model [1] of the TSS production lines. First of all, we normalize to one time unit the total processing time required for the processing of one item. Let the processing requirement at station number j , 1 < j < m, be q^ a fixed percentage of the total work content of the product. Thus Y^JLi Qj = 1- ^ n e c a n think of {qj}, 1 < j < m, as of the partition of the production line onto the work stations with respect to their work content. We also assume that then the last worker immediately (i.e., at the same instant of time) after finishing an item takes over from worker number n — 1, who takes over from worker n — 2 and so on until the first worker on the line. In other words, a worker number fc, 1 < k < n, takes over worker numberfc— 1 at the same moment when the last worker finishes an item, while the first worker brings at this moment of time a new item into the system. We say that the line resets at such an instant. It may happen that two or more workers are blocked at the same point (form a line in front of the same machine number j , 1 < j < m) at a moment of reset. Then according to the TSS rule the last worker (the one with a minimal index) leaves this line at the moment of the reset and takes over the work of his predecessor. At the same moment the worker that has blocked this line starts to work on the jth machine from the beginning. Each worker can be naturally characterized by his velocity Vi(z), where i is a number of a worker 1 < i < m, and z, 0 < z < 1, is a point at the unit segment. We will assume that Vi(z) is a continuous function for all 0 < z < 1 and i = 1,2,..., n. By Vi we mean here is a velocity of the ith worker, where he is not blocked. Otherwise his velocity at this point becomes zero. It is the very important feature of the TSS line model suggested in [1] that the workers are different. It has been assumed traditionally that the workers are indistinguishable [4, 11, 12]. However, the measurements performed by Bartholdi

15.2 TSS Production Lines and Their Model

391

q f + q 2 x, q,+q2+q3+q« 1

Fig.1 The phase space of a TSS line with two workers and five work stations.

and Eisenstein on the industrial sites had shown that it is not the case [1]. One can think of Vi(x) as of a function of a worker's skills, determination, etc. A state of our system at a moment t can be described by a vector of worker's positions x(t) = {x\{t),x2{t),... ,xn(t)), where 0 < x\{t) < x2(t) < • • • < xn(t) < 1. Therefore, one can think of the TSS line as of a system which consists of n particles on a unit closed interval. Each particle moves with its own velocity but sometimes this velocity may abruptly fall to zero when a particle is blocked. One can think of the last restriction as of a kind of a hard core potential which corresponds to the restrictions that two particles cannot occupy the same work station. We can formally write this restriction as follows: if ]CjLi Qj < x% < Ylj=i Qjy th e n x i - i ^ Ylj=i Qj- Therefore the phase space M of our system is a closed polyhedron which in turn is a subset of the n dimensional simplex (Fig.15.1). This dynamical system is the one with continuous time. However, it has a global Poincare section and therefore one can study instead a dynamical system with the discrete time. Indeed, the first worker instantly jumps to the origin after a completion of an item by the last worker. Hence, {xi(t) = 0} is the global Poincare section for our system. A time in the discrete version of the TSS model is measured as a number of completed items (or, equivalently, as a number of resets). It is important to mention that all TSS lines examined in [1] required just a few seconds for any worker to walk back and take over the work of his predecessor. Therefore, the assumption that the workers move backward with effectively infinite velocity seems to be reasonable. We will denote a number of resets (a discrete time) also by t. Let / be the

392

15 Controlling Production Lines

transformation defined by the TSS rule that maps the vector of worker positions after one reset to that after the subsequent reset. Then one can write x ('

+1

) = /( X W)

(15.2.1)

15.3 Dynamics of TSS Lines The explicit expression for / can be written. However, it is so cumbersome that it doesn't help the analysis. Therefore, we will start with the simplest variant of (15.2.1) which corresponds to the assumption that all functions vi{z), 1 < i < n, 0 < ^ < 1, are constants, i.e., Vi(z) = V{. Suppose that neither one of workers has been blocked during the time when the (t -f- l)th. item has been produced. Then the map (15.2.1) is linear and has the form

4)

4^ + ^ ( 1 - ^ )

(15-3.1)

where A; = 2,3,... ,n. It is convenient to introduce the new variables

af,=e_^£ vi

a ( t)= izfP

( 1 5 .3. 2 )

yn

where z = 1,2, ..., n — 1. These new variables are called allocations [1]. They have the meaning of the clock times which separate the workers after the tth iteration. Then (15.3.1) becomes in these coordinates

(15.3.3) where i = 2 , 3 , . . . , n. If blocking is allowed then the map (15.2.1) is piecewise linear because each worker can move only with a finite number (not exceeding n) of velocities. It has been confirmed in experiments that worker's velocities differ [1]. Because of that the natural problem arises how to sequence them in order to produce a maximal production rate. (In case of constant velocities the maximal production rate equals Y^i=i Vi-) ft *s n o t always possible because of blocking since a worker which is blocked doesn't contribute to the production. The visits to the industrial sites revealed that everywhere where given some thoughts to sequencing the workers on the TSS lines [1]. At some sites the management operated by trial and error. At one site the management put the slowest workers at the last position in each line with the idea that the work there is in a sense easier because it is mostly labeling and packaging finished items. At another site management initially assigned any new, slow worker to the middle of

15.3 Dynamics of TSS Lines

393

the line with the idea that he would improve quickly being squeezed between two more experienced (faster) workers. Some other ideas were implemented at different industrial sites (see [1]). However, the following idea seems to be very natural. Suppose that there are no stations at all and all workers just "run" along the line with constant velocities Vi, i = 1,2,..., n. Then obviously a blocking never occurs only in one (out of n\) sequencing, which is from slowest to fastest (worker). Therefore, this sequencing certainly deserves some special consideration. The corresponding analysis was performed by Bartholdi and Eisenstein [1] who came to the idea of this sequencing on the basis of extensive numerical simulations of the TSS lines. It is interesting to mention that the management at some visited industrial site almost came to the same sequencing by trial and error [1]. We have V\ < V2 < V3 < • • • < Vn for sequencing from slowest to fastest. Suppose for simplicity that velocities of all workers are different. Then our linear system (15.3.3) is described by a stochastic matrix, because its entries ^ ^ can be formally interpreted as transition probabilities. It is easy to see that this matrix satisfies to Markov's ergodic theorem (see e.g., [6]) which means that some its power contains only positive entries. Therefore the dynamical system (15.3.3) has a unique fixed point which is stable and moreover it is a global attr actor. It is easy to see that for any sequencing of workers x*

= yn=1 y

(15.3.4)

i = 2 , 3 , . . . , n is the fixed point of the dynamical system (15.3.1). The essential generalization of this result has been proven in [1]. Suppose that sup ( y.y\) < 1 sit every portion of work content (at any point on a production line). Then we say that worker j is faster than worker i. Observe that the velocities of the workers are not assumed to be constant. Theorem 15.3.1. [1] For any TSS line, if the workers are sequenced from slowest to fastest, then any orbit of workers' positions {x^ = f^(x^)} converges to the unique fixed point. Therefore, for any configuration of work stations on the TSS line there exist a fixed point which is a global attractor of the dynamical system (15.2.1), if workers are sequenced from slowest to fastest. This result shows that a TSS line where workers are sequenced from slowest to fastest is a self-balanced (self-organized) system. Moreover, it can be shown [1] that TSS line never demonstrates an anomalous behavior if the workers are sequenced from slowest to fastest. By an anomalous behavior we mean a decrease of the production rate of a production line which occurs if some workers are added to the line or if some

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15 Controlling Production Lines

workers increase their skills (velocities). If the workers are not sequenced from slowest to fastest then both such situations may happen. Consider for instance the TSS line with processing times q = (|,^,^) and two workers with constant velocities V = (2,2). Obviously, this line achieves the maximal possible production rate of 4. Insert an additional worker of velocity one between the two original workers. Then the production rate will decrease to | . Consider now the same TSS line but with three workers with velocities V = (2,1,1). The production rate of this line again is the maximal one and equals 4. Suppose that the last worker doubles his velocity. Then, similarly to the previous example, the production rate decreases from 4 to | . In both cases the production rate decreases because, after the natural "improvements," the first worker becomes idle, while the slowest second worker is blocked during the part of each iteration. One may get the impression that the TSS line sequenced from slowest to fastest is always optimal. However, it is not the case. The following example demonstrates what may happen: Consider the TSS line with processing times q = (|, | , | , | ) and three workers with the velocities V = (3,1,2). For this line x* = (0, | , | ) is the stable fixed point. The production rate at x* equals 6. Suppose now that for the same TSS line the workers are sequenced from slowest to fastest. Then the fixed point is x = (0, | , | ) but the production rate at x is ^ . Therefore, the optimal sequencing for this TSS line is not from slowest to fastest. Observe, however that for the sequencing V = (3,2,1) there exists another stable fixed point x\ — (0, ^, ^). The production rate at x\ equals 4 because the fastest worker gets blocked for a long time. It has been claimed in [1] that the workers sequenced from slowest to fastest on the TSS line always generate the maximal production rate if there is no "bottleneck" station, where by "bottleneck" station one means the work station with relatively big work content. The natural recommendation is to put at such station the fastest worker. Our example shows, however, that the situation is actually more complicated. Indeed, the last station on the line corresponds to the maximal work content that is equal to | . However, this TSS doesn't achieve the maximal possible production rate if the fastest worker works there. Moreover, it is exactly the case when workers are sequenced from slowest to fastest and which doesn't demonstrate the maximal production rate. The explanation why it happens is simple. The point is that the first two work stations on the line actually form the "superworkstation" that can be considered as a "bottleneck" station. So we see that the situation is rather complicated and there is no universally optimal sequencing of the workers. Still, it is possible to offer a reasonable algorithm of control of TSS production lines. To do that let us first return to the case when all workers have constant velocities and there is no work stations on the line (15.3.1), (15.3.3). Observe, however, that this case rather corresponds to the presence of a "work station" at any point z, 0 < z < 1, of a production line. Indeed, the most effective blocking occurs when there are only a few work stations on the line. For instance, some workers are always

15.3 Dynamics of TSS Lines

395

blocked if there are fewer work stations than workers on a line. (If one thinks of a TSS production line as of a system of particles then the lengths (the work contents) of work stations correspond to the ranges of the corresponding binary potentials. Hence, if a number of machines becomes smaller than the ranges of the potentials grow.) Therefore a TSS line without machines represents actually a singular limit of a line with only one machine. Still it is worthwhile to analyze this case. A relatively comprehensive study of such "machineless" TSS lines has been conducted in [2] for 2- and 3-worker lines. Observe, that blocking can occur in this case as well. Indeed, if a faster worker can reach a slower one who is ahead of him then a faster worker becomes constantly blocked by the predecessor and starts to move with his velocity. As the result, at any instant of time all workers are partitioned onto the normal (single) workers and 'superworkers" (superparticles) where each "superworker" represents a group of "glued together" k workers, 1 < k < n, moving with the velocity of a worker which is the closest to the end of the line. The maximal possible number of such superworkers is determined by the order of workers on the line. For instance, "superworkers" never appear if the workers are sequenced from slowest to fastest. However, any other sequencing of workers allows the existence of superworkers. Suppose that the map / has a fixed point x* (a*). Suppose also that blocking doesn't occur in some neighborhood of the point x* (a*). Then / is linear at x* (a*) and can be represented by the relations (15.3.1) (or (15.3.3)). The characteristic polynomial of (15.3.3) at the fixed point has the form p(A)

= A- 1 + %±A- 2

+ •••+ £A + £

(15.3.5)

Therefore, the necessary condition for the stability of a fixed point x* is Vi < Vn

(15.3.6)

i.e., the velocity of the first worker must be less than the velocity of the last worker. This condition is not sufficient though. It is worthwhile to mention that the assumption that blocking is not permitted in some neighborhood of a fixed point x* is essential. For instance this condition is violated in the example considered before with V = (3,1,2), i.e., V\ < Vn where the fixed point x* is stable. Consider a two-worker production line. Its dynamics is given via the following formula

= min {£(i,4'>),: This dynamics can be easily analyzed [2]. If V\ < V2 then all orbits converge to the fixed point, with the maximal production rate V\ + V2. If V\ = V2 then there is the one-parametric family of period two neutral orbits, {x^\ 1 - #2° }» w*th the maximal production rate V\ +V2.

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15 Controlling Production Lines

If Vi > Vi then all orbits converge to the same period two orbit {1,0} with production rate 2Vi < V\ + V2. The dynamics of a 3-worker line is much more complicated. The corresponding map is given via the relations

•*2

The map (15.3.8) can be specified as \X2

xf\xf

+ r2(i - xf)

if ri (i - xf) <xf+r2(l-xf)
r2(l - s^).*^ + r2(l - xf) if *« + ra(l - xf) < (1,1)

(15.3.9) where r» = ^/V 3 , i = 1,2. Obviously, the partition of the phase space into regions where the dynamical function has different forms corresponds to the different types of blocking. For instance, in the first region defined by (15.3.9) no workers are blocked, only first worker is blocked in the second region, both the first and the second workers become blocked in the third region and only second worker becomes blocked in the last region. The bifurcation diagram for the three worker line is depicted in Fig.15.2 [2]. The region with the globally stablefixedpoint I (Vi/ £ ? = 1 VJ, (Vi + V2)/ £i=i Vi) \ is given via the relations V1/V3 < 1 (the necessary condition (15.3.6)) and V2/V3 < V1/V3 + 1. The globally stable period two cycle {(0,^/(1 + n)), (r x /(l + r x ), 1)} exists at the rest of the region {V1/V3 < 1}. The complement of this region is partitioned into two parts. Thefirstsubregion {V1/V3 > 1,V2/V3 > 1} corresponds to the existence of a period three trajectory {(1,1), (0,1), (0,0)} which is a global attractor. The last, fourth, subregion {V1/V3 > 1,V2/V3 > 1} contains infinitely many stable periodic trajectories with periods K > 3. The nature of their appearance is quite clear. Indeed, the slowest worker in this region is the intermediate one and the first worker the fastest. Therefore the second worker can move from an iteration to another iteration by arbitrary small steps. Computer simulations revealed [2] that in this region can coexist stable periodic trajectories of different periods.

15.3 Dynamics of TSS Lines

397

V,/V2 Period 2

Period 3

Period greater than 3

1

Fig.2 Bifurcation diagram of a 3-worker line. In the shaded area there exists a globally stable fixed point.

We were unable to conduct a comprehensive analysis of the dynamics in the fourth region. However, it is the region to be avoided in the applications to the real TSS lines. Therefore, our analysis does provide enough informations for such applications. Indeed, the management needs only the information how to stay in a region with a simple stable (robust) behavior. The interesting mathematical question though is whether or not some orbits in the fourth region demonstrate a chaotic behavior. We conjecture that it is not the case even though some computer simulations were halted after several days of searching without having detected a cycle. Some cycles of periods greater than 20,000 were found in these simulations [2]. The production line achieves the maximal possible production rate Yli-i Vi m the first region. In all other regions the rate of production is lower. Denote Vm\n, Knax and VJnid the minimal, maximal and intermediate velocities of the workers respectively. The production rate in the region 2 equals 2(l^nin + Knid) < ]C*=i ^ The rate of production equals 3Vm[n in the region 3. Finally, in the region 4 the eventual production rate of an orbit depends on a limit cycle which attracts it. Therefore the line always achieves the maximal possible production rate if the fastest worker is last. The line achieves the smallest production rate 3Vmin if the slowest worker is the last. Finally, the sequence (fastest, slowest, mid) may correspond to the highest production rate (region 1) but it may display also distinctive behavior (region 4) and have a lower production rate. Let now formulate the main requirements to the desired ("ideal") state of a production line.

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15 Controlling Production Lines

i) The desired state of the production line must be robust, i.e., it should be insensitive to the perturbations. In other words we are looking for an attractor of the corresponding dynamical system. ii) The production rate at this robust state should be maximal. The following important (but extremely simple statement) shows that any desired state of a TSS line without work stations (i.e., in fact, of a "TSS line with a continuum of work stations") is a stable fixed point. Main Lemma. Any desired state of a TSS line with arbitrary number of workers but without work stations is a stable fixed point. Proof. Let A be a stable invariant subset of a phase M space of the corresponding TSS line. Suppose also that for all points x G A there is no blocking at any point of the corresponding trajectory {f^(x)}, k — 0,1,2,..., (i.e., A is a desired state). Then A is a stable fixed point of the TSS map / . Moreover, if such desired state exists it is unique. Proof. There is no blocking at any point of the invariant set A according to the conditions of Main Lemma. Therefore the restriction of the TSS map on A is the linear map (which is explicitly given by the relations (15.3.1)). It is well known though (see e.g., [7]) that a linear map cannot have any stable invariant set besides a fixed point. Moreover, a linear map cannot have more than one stable fixed point. The Main Lemma suggests that it makes sense for the management of TSS lines to try such sequencings of workers for which the corresponding dynamical model (15.3.1) has a stable fixed point. Suppose we are given a collection (brigade) of n workers. In our model such brigade is always associated with a n-dimensional vector of their velocities. Let there be several sequencings of workers such that the corresponding system (15.3.1) ((15.3.3)) has a stable fixed point. The natural question is which one of those points is the best state for a TSS line. There are two properties that could be added to i), ii) as the properties of a desired state. iii) The basin of attraction for the desired state must be as big as possible. iv) The rate of convergence to the desired state must be as big as possible. The bifurcation diagram of the three workers line (Fig.15.2) shows that three (out of 6 possible) sequencing have globally attracting fixed points. Let us consider then the condition iv). A rate of convergence at a fixed point is characterized by the maximal (in modulus) root of a characteristic polynomial at this point. Consider the polynomial (15.3.5) and denote A = max(|Ai|, |A21,..., I An_i |). Then the smaller value of A < 1 corresponds to the higher stability of a fixed point. We will discuss now how the permutations of workers (i.e., of the velocities V\, V2,..., Vn) change a value of A. In view of (15.3.6) we can also assume that V\ < Vn. Any permutation can be broken into cycles. Further, each cyclic permutation we can obtain by consecutive inversions of two elements. It is convenient to assume that workers were initially

15.4 & Self-Organized Order Picking System

399

sequenced from slowest to fastest. Recall that in this case there exists a globally attracting fixed point. Suppose that we have interchanged the workers with numbers k and r, k > r. Then the polynomial —a\k + a\r will be added to the polynomial (15.3.6), where a > 0. Observe that all coefficients of the polynomial (15.3.6) are positive. Therefore, all its roots at a stable fixed point must belong to the interval (—1,0). Suppose that n is an even number. Then, it is easy to verify that a rate of stability at a fixed point can be increased only if r is odd. Analogously, r must be even if the polynomial p(X) has an even degree, i.e., a number of workers is odd. Such simple rules may help the management to find an optimal sequence of workers on a TSS line. They can also essentially reduce a number of trials. It is worthwhile to mention that the most stable fixed point for three- and fourworker TSS lines always corresponds to the sequencing from slowest to fastest. This fact can be easily verified with the help of just formulated odd-even rule.

15.4 A Self-Organized Order Picking System for a Warehouse In this section we discuss briefly another work-sharing lines that are peculiar for the warehouses. Although the dynamics of such lines is intrinsically stochastic still the same ideas as in the analysis of TSS lines can help their management [3, 5]. The warehouses frequently replenish stock-keeping units (sku's) in small amounts. Therefore sku's must be picked in less-than-caseload amounts. Typically sku's are picked from a flowrack which is divided into sections. Within each section are shelves and individual items of each sku are picked from the closest case. The sections of flowrack are arranged in aisles and a conveyor runs down each aisle. For simplicity we will consider a single-aisle of flowrack. Usually each aisle in a warehouse operates independently. Each customer brings an order which is a list of sku's together with quantities to be picked. The first picker takes the next order, opens a box, and slides it along the passive lane of the conveyor as he moves down the aisle picking the sku's for that order. At some point the order is left for the next worker to continue picking, while the first picker returns to the start for the next order. When an order is complete, the last picker pushes the filled boxes onto the powered portion of the conveyor, which takes them to the packing and shipping department. Observe that this system looks strikingly similar to the described before production lines. Indeed each picker certainly can work at any place along the aisle and pickers cannot pass each other. However, there is the very important difference there because each order contains a random collection of different sku's as well as different (also random) quantities to be picked. The management of warehouses faces the same problem of an optimal sequencing of pickers along the aisle. Again, the traditional approach to this problem

400

15 Controlling Production Lines

is a zone-picking. In this case zones are fixed, based on some model of work content. However, in practice, it requires a constant readjustment to maintain balance. Some warehouses have tried to improve balance of work by recomputing zones several times during the day. Some others have invested a lot to build sophisticated models of work content (see the detailed discussion in [3, 5]). The analysis of TSS lines suggests the idea to abolish zones, which are expensive to construct and unavoidably nonaccurate. Instead, the protocol analogous to TSS allows to adapt "dynamical zones" to the statistics of the order stream. We describe now the corresponding model in detail. A storage is divided into m positions ("work stations") at which a worker stands while picking. Thus the sku's are partitioned into m disjoint sets. These positions are numbered so that any picks from position j , 1 < j < m, will be completed before any picks from the subsequent positions. An order is a random vector N = (JVi,..., ATm), where Ni, 1 < i < m, represents the amount of standard work content to pick from location i for this order. Each Ni is a positive random variable. We assume that all orders are independent and identically distributed. However, the Ni could be dependent within the same order. These assumptions are relevant to the practice (see [3, 5]). The orders are to be picked by n, n < m workers. Each h 1 < i < n-> worker is characterized by a pick velocity v%. Let X = (X2,X3,... ,X n ), {X\ — 0), be the positions of the workers immediately after reseting and M (a phase space) be the collection of all possible vectors X. Denote by X^) = (X2 , . . . , Xn ) a random vector of worker positions at the moment after the reset, when worker number i takes over the order number t+n — i. Theorem 15.4.1. [3] Probability distribution of the vector ( X ^ , N ' ^ ) converges (in variance) to the equilibrium distribution as t -> 00. Any realization of X^^ depends upon the initial positions of the workers However, the equilibrium distribution doesn't depend upon X^0). It rather depends upon the distribution of orders and the velocities and the sequence of the workers. The proof of Theorem 15.4.1 follows from the fact that the sequence of random vectors (X'*',N^) forms an irreducible nonperiodic stationary Markov chain. Therefore Markov's ergodic theorem on the existence and uniqueness of an equilibrium distribution can be applied [6]. Usually a number of sku's in a warehouse is very big (about tens of thousands). Therefore m > n and our model of picker lines in a warehouse can be well approximated by the model of TSS production line without work stations (i.e., with a work station sitting at any point of the line) which has been considered in the previous section. Observe now that orders are random. Therefore any picker regardless of his number may receive in an order a lot of items to pick close to the position in the aisle where he is now or a rather few of such items. Therefore a velocity Vi of each picker with which he is moving along the aisle while picking is also a random variable. In other words, Vi is the function of the sequence of random

15.5 Optimizing Performance

401

orders N ^ . However, we can still compare naturally the velocities of the different pickers. Indeed, worker i is faster than worker j if i can pick any order (alone) faster than j does (alone also). Intuitively, it seems to be clear that sequencing from slowest to fastest will maximize the average pick rates (i.e., the production rate) of the workers. Indeed, in the TSS line without work stations neither worker is ever blocked if they are sequenced from slowest to fastest. In the warehouses pickers still can be blocked because of the variation of orders. However, a blocking must be less likely if pickers are sequenced from slowest to fastest. Computer experiments support this idea. The most important is that the implementation of this approach at some warehouses allowed to essentially increase the production rate of picker brigades [5].

15.5 Optimizing Performance The sequencing from slowest to fastest produces in many cases the best results. Especially, it always satisfies the criteria i) and iii) [1]. Therefore, the management may want always to try it first at the industrial sites. It is also important that one doesn't need to have the exact measurements of worker's velocities to maintain such sequence. Indeed, it is enough to know only the relations between the velocities of different workers (who is faster). However, the sequencing from slowest to fastest may not produce the highest production rate. Therefore, the management may try to adjust a sequence of workers on the production line to the existing partition of the line into work stations (according to their work contents). Such sequence of workers may demonstrate a higher production rate than the one where the workers are sequenced from slowest to fastest (see e.g., the example in Section 3). Such adjustment can be done via the following procedure. Suppose we are given a TSS production line q = (gi,#2j • •. ><7m) and a sequence of workers V = (Vi,V2,. • • ,Vn), m > n. Then we have to find such ratio V^ : Vi2 : ••• : VJn, 1 < ij < n, 1 < j < ^> which is the closest to the one of the ratios

Besides one should verify whether or not the TSS line without work stations and where workers are sequenced as V^, V$ 2 ,..., Vin has a stable fixed point. Such procedure can be easily implemented if workers' velocities are known with some accuracy.

15.6 Concluding Remarks The implementation of the simple dynamical model of the TSS line has given the striking results at the industrial sites [1, 5]. Why has it happened? The answer is

402

References

rather obvious. The point is that the assumptions of the mathematical model were first experimentally verified [1], It has been confirmed that the actual processing times for a single worker at any work station on a TSS line have a very narrow distribution. Therefore, workers velocities may be assumed to be constant. On the other hand, workers vary significantly in speed. Therefore, a model of TSS lines with distinguishable workers is relevant. Also, the workers can usually be ranked according to their velocities [1]. It should be mentioned that the first computer simulations of the TSS lines were conducted in [11]. However, no general conclusions about TSS were reached. In this paper, as well as in the two following papers [4, 12], all workers were assumed to be identical. Also, the processing times were assumed to be random in [10] and a notion of balance adopted in [10, 12] was actually a probabilistic (on average) balance of a production line. This notion of balance is weaker than the one considered in the deterministic approach [1, 2]. Also, these papers and the advertising literature for TSS use the term "self-balance" to mean local adjustments between adjacent workers. On contrary, the dynamical approach [1, 2] allows to achieve a global balance (global stability) of TSS lines. The modern theory of dynamical systems provides various tools for the global study of the time evolution of a system. This approach proved to be efficient as well for the analysis of some intrinsically stochastic systems as e.g., the brigades of pickers in a warehouse. The crucial point was to come up with the adequate model of distinguishable pickers [3, 5]. Again, pickers were traditionally assumed to be identical [9]. We believe that the theory of dynamical systems, as well as, generally the nonlinear science, has got a lot of potential for the successful applications to the control of production lines. Especially important in this respect would be to develop effective algorithm to control the re-entrant production lines [8], where the items visit some of the work stations more than once in their route through the line. Such re-entrant manufacturing systems are used e.g., in semiconductor processing.

Acknowledgments. I am very grateful to J. Bartholdi (School of Industrial and Systems Engineering, Georgia Institute of Technology) and D. Eisenstein (Graduate School of Business, University of Chicago) for many valuable discussions. This work was partially supported by the NSF Grant #DMS-9630637.

References [1] J. J. Bartholdi, III and D. D. Eisenstein, Production lines that balance themselves, Operations Research 44: 21-33 (1996).

References

403

[2] J. J. Bartholdi, III, L. A. Bunimovich and D. D. Eisenstein, Dynamics of 2- and 3-workers "bucket brigade" production lines, Operations Research (to appear). [3] J. J. Bartholdi, III, L. A. Bunimovich and D. D. Eisenstein, Bucket brigades: a self-balancing order-picking system for a warehouse (in preparation). [4] D. Bishak, Performance of manufacturing module with moving workers, Dept. of Business Administration, Univ. Alaska Fairbanks, Alaska, manuscript 99775-1070 (1994). [5] D. Eisenstein, A self-balanced order-picking system for a warehouse: A case study, Preprint, 1998. [6] W. Feller, An Introduction to Probability Theory and Its Applications, v.l, Wiley & Sons, NY, 1959. [7] A. B. Katok and B. Hasselblatt, An Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, NY, 1995. [8] P. R. Kumar, Re-entrant lines, Queueing Systems: Theory and Applications 13: 87-110 (1993). [9] J. W. O'Brian, Pharmaceutical distribution improved order picking, Proc. of the 1986 Fall Industrial Eng-g Conf., Inst. of Ind. Eng-g, pp. 476-480, 1986. [10] J. Ostolaza, J. McClain and J. Thomas, The use of dynamic (statedependent), assembly line balancing to improve throughout, J. Mfg. Oper. Mgt. 3: 105-133 (1990). [11] B. J. Schroer, J. Wang and M. G. Ziemke, A look at TSS through simulations, Bobbin Magazine, July, 114-119 (1991). [12] E. Zavadlav, J. McClain and J. Thomas, Self-buffering, self-balancing, selfflushing production lines, Graduate School of Business, Cornell University, Ithaca, NY, manuscript 14853 (1994).

16 Chaos Control in Biological Networks A. Babloyantz Service de Chimie-Physique and Center for Non-Linear Phenomena and Complex Systems, Universite Libre de Bruxelles C.P. 231, Boulevard du Triomphe, B- 1050 Bruxelles

16.1 Introduction The presence of spatio-temporal phenomena have been shown in a wide variety of systems ranging from physical systems to biological tissue. These systems are described by a large number of degrees of freedom. Turbulent behavior or spatiotemporal chaos is a common propriety seen in such systems. We define spatiotemporal chaos as the state of the system exhibiting some degree of low dimensional behavior with typical correlation lengths of the order of the size of the system or smaller. The aim of this paper is to demonstrate the possibility of stabilizing the unstable periodic orbits (UPO's) embedded in spatio-temporal chaotic attractors described by moderately large and very large number of coupled non-linear differential equations describing biological phenomena. On the other hand as we shall show, most relevant biological processes require more or less long time delays in the process of communication between cells in a biological tissue. The delays also appear in biochemical regulatory processes. Therefore, there is a need to extend the control algorithms to networks described by delay differential equations (DDE's). Once the existence of spatio-temporal chaos in a biological system is assumed, one may ask whether the stabilization of unstable periodic orbits (UPO's) of these systems is possible and if affirmative, what will be their physiological role if any. There are several techniques for the stabilization of UPO's which are well documented in the literature and also in the present volume. In this paper we shall use three different techniques. Ott, Grebogi and York (OGY) [1],[2] method is extedend to stabilize UPO's in a spatio-temporal chaotic system made of N x N oscillating units. The continuous self-delayed feedback method [3] is used for stabilizing a one variable delay differential equation. The continuous self-delayed feedback method was modified to make the stabilization of UPO's from a network of DDE's possible [4]. As biological neural networks require very large number of units described by delay differential equations, we used a technique in which the system is studied in Fourier space and the control is performed on Fourier modes [5]. The biological examples treated here are mainly related to brain activity. The aim is to put forward the idea that if the brain tissue exhibits spatio-temporal chaos, then the

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16 Chaos Control in Biological Networks

stabilization of UPO's of model cerebral cortex may account for the phenomena of attentiveness seen in human and animal behavior [6]. Moreover we show that the mechanism we propose for such activity could be tested by constructing an electrical analogue model of a network of interconnected oscillatory devices. In section two, the method of self-delayed feedback control is used to stabilize orbits in a one variable DDE. Section three is devoted to the control of a network of oscillators. In section four, we present a model of "attentiveness" as seen in cerebral cortex. In section five, the control is achieved in a network of delay differential equations modelling cerebral cortex. Finally section six discusses the control of infinite dimensional systems in Fourier space.

16.2 Control of a delay differential equation As a first example of a biologically plausible model we consider the Mackey-Glass equation [7] which is a classical example of a DDE arising from physiological modeling and illustrates the stabilization of UPO's, with self-delayed feedback control method. The equation takes the form (30xT

dt

(16.1)

-yx

where xT — x(t — r) . The parameters are 7 = 0.1, /3o = 0.2 , and n = 10 . Equation

10 200.0

Figure 16.1 Dependance of (D2(t)) on the parameter T, for the Mackey-Glass equation. o = 7x 10~3. (16.1) possesses a fixed point XF = V(A> - 7)/7 and a trivial fixed point x = 0. Let us consider the solution xp = 1 and increase the delay r . For r = 4.7, the fixed point looses its stability via a Hopf bifurcation and stable periodic oscillations appear. Through a further sequence of period doubling, beyond r = 16.8 a chaotic regime is seen. In the following, we study the chaotic regime for r = 23. UPO's of Eq.(16.1) can be stabilized with a continuous self-delayed feedback method, well documented in the references [3]. The essence of the method is the addition of a

16.2 Control of a delay differential equation

407

perturbation e(t) to Eq.(16.1) which is proportional to D(t), the difference between the present value of the observable x and the value it had at a previous time T. If e0 > 0 is a saturating value ensuring that the perturbation remains small then

e(t) = i

- e 0 KD{t) < - e 0 KD(t) -eo < KD(t) e0 KD(t) > e0

(16.2)

e0

We monitor the probe function D{t) by varying K and T. The vanishing of (D2(t)), for a given pair (K, T), is an indication that there is an orbit with period T in the attractor, and also that this pair of parameters is adequate for stabilization of the orbit. We choose the autonomous version of continuous control [8]. For

10° 10 3

10u

to6

I 10" 3 0.00

0.02

0.04

0.06

0.08

0.10

frequency Figure 16.2 Power spectrum of the variable x of the Mackey-Glass equation, before and after stabilization.

the purpose of illustration, let us fix the value of K and let T vary. The probe function (D2(t)) thus depends on T alone. Figure 16.1 displays this dependence, as computed from Eq.(16.1) in the presence of perturbation (16.2). The values of T corresponding to the minima of Fig.16.1 are the best choices to use in e(t) in order to achieve stabilization with a fixed value of K, which in this case is K = 0.05. The

408

16 Chaos Control in Biological Networks

most pronounced minimum in Fig.16.1 is located at T = 58.75. Different values of K give other minima, or even intervals of the parameter T for which (D2(t)) is zero or has a very low value. These intervals are associated with the presence of fixed points. Using T = 58.75 in the control terms (16.2), the chaotic regime is indeed stabilized into an orbit of period 58.75. The stabilization procedure can also be evaluated by comparing the power spectra of the variable x(t), with and without control. This is shown in Fig.16.2. We could also stabilize the fixed point XF = 1, as well as an unstable cycle with a period of 17.78.

16,3 Control of chaos in a network of oscillators. Our first task in this section is to extend the existing control methods generally limited to a small number of degrees of freedom to a network with many interconnected units [10]. Each unit is an oscillating system. However as we shall see, the interconnected units generate spatio-temporal chaos in the network. To be as general as possible we have chosen an analogue to the complex Ginzburg-Landau equation for the description of the network [11].

16.3.1 The model Let us consider a square network of N x N oscillating units with N — 9. The equations describing the dynamics of the system are

^ f = Zjk - (1 + i/3)\Zjk\2Zjk + (1 +

ia)D£CiUmZ,m (j,k = l,...,N)

(16.3)

The constant parameters a,/3 and D are real. Each oscillator of the network is described by a complex variable Zjk and is labeled by a number jk. The matrix Cjkim is defined as the connectivity matrix of the network. Only first-neighbor diffusive interactions are considered. Zero-flux boundary conditions are considered. The parameters and the connectivity of the network are chosen such that, in the absence of control, the system exhibits chaotic activity. A way to achieve chaotic dynamics is to consider parameters for which the uniform oscillations of the network are unstable. The uniform oscillations of the network may be expressed analytically by Zjk (t) = exp (-i/3t + 0O) for allj, k

(16.4)

where 0o € [0,2?r] is an arbitrary phase. One can show that this solution of Eq.(16.4) loses its stability if the following condition is fulfilled :

This is analogous to the Benjamin-Feir instability condition, derived for continuous media. Chaotic activity is generated in a network of 9 x 9 oscillators by the following

16.3 Control of chaos in a network of oscillators.

409

procedure. For the values of the parameters a = -10 and 0 = 2 by decreasing D from the critical value D = 2.5 , the uniform oscillations lose their stability and various nonuniform dynamical regimes appear. We first observe various periodic regimes, then quasiperiodic dynamics arise. Finally the onset of chaos is seen for D = 1.9. In the following, we fix the value of D = 1.3. The presence of chaotic

200.0

Figure 16.3 Top: Time series of (|Z 2 |), over an arbitary period of time. Middle: Time series of (ReZ). Bottom: Power spectrum of (ReZ) (t). The brackets (> denote a space average over the entire network.

activity can be confirmed in a number of ways. Fig. 16.3 shows the time-series of two observables of the dynamics, which are clearly aperiodic, and the broadband power spectrum of one of these observables. The projection of a Poincare section of the system shows that no well-defined structure is present therefore the attractor is not of low dimension. Further estimates of the fractal dimension via the Grassberger-Procaccia algorithm [12] confirmed the high dimensionality of the attractor [10]. In the remainder of this section, we summarize the procedure of stabilizing UPO's of Eq.(3), which is described in detail in [10]. Where it is explained that dynamics is dis^cretized by using an adequate Poincare section. Because the variables are complex, the original continuous dynamics has dimension 9 x 9 x 2 = 162. With the help 6f the Sparrow method [13], based on the Poincare section , four UPOs could be (found. These are denoted d with i = 0,1,2,3. The Co orbit corresponds to the I trivial solution (4), where all the elements of the network display a common sinusoidal oscillation with period T . The remaining orbits all show

410

16 Chaos Control in Biological Networks

spatial structure. Fig.(16.4) displays a snapshot of each of the three corresponding spatio-temporal dynamics. Orbit C\ corresponds to a rotating wave of oscillatory activity around the central unit of the network. In this wave, the amplitude of the oscillator in the center of the network vanishes. This property is analogous to the presence of a "phase defect" which can generate a spiral wave in a continuous oscillatory medium. Orbit C2 corresponds to standing wave activity. At a given moment, the amplitude is higher on one side of the network and lower on the opposite side. The pattern is reversed periodically. In this orbit, the activity is antisymmetric with respect to the reflection around a median and constant along any direction parallel to that median. Because the network has a square geometry, a similar solution exists with a symmetry axis perpendicular to the one seen in Fig.(16.4). Orbit C2 also exhibits standing wave activity. The two corners of the same diagonal show high activity whereas the corners of the opposite diagonal have a low value, and the pattern is reversed with a given frequency. This orbit is invariant with respect to reflection around the two diagonal axis of the square. One can

ReCi

ImCi

ReC2

ReC3

lmC3

Figure 16.4 Snapshots of network activity corresponding to the unstable periodic orbits &, C2 and C3. For orbit &, arrows show the direction of rotation of the wave. visualize the procedure of stabilization of spatio-temporal UPO's in the following way: The spatio-temporal system is visualized as a unique temporal chaotic system and UPO's are stabilized. Translated in spatial domain, phase gradients form spatio-temporal structures. Orbits C\ to C3 have periods 13.66, 16.38 and 2.25,

16.4 Chaotic categorizer

411

respectively. The relative stability of the orbits is given by the Floquet exponents that have two positive real parts. In the OGY method, micropulses are delivered to the parameters of the system. Here, the procedure is modified by perturbing the variables instead. This modification does not affect the basic principles of the method. All four orbits Co to C3 could be stabilized following this procedure. Details of which could be found in [10] and [14].

16.4 Chaotic categorizer In the previous section we showed how UPO's could be stabilized in a chaotic network. Because of computational cost of using OGY method we only stabilized few orbits. It is obvious that such a high dimensional system as the one described by Eq.(16.3) must have an infinite number of unstable orbits. This must be more through for many biological neural networks described by DDE's. Therefore one may speculate that if such UPO's are stabilized for short times, they will confer cognitive capability to the network. The aim of this section is to propose a device based on control of spatio-temporal chaos which shows few cognitive properties attributed generally to the living brains. The cognitive capability that we want to show in this device are the onset of the state of "attentiveness", pattern recognition and motion detection. Our first task is to convince ourselves that spatio-temporal chaos arises in neural tissue and confirm the chaotic nature of the brain activity. Since the advent of electroencephalography, the procedure has been and still is a diagnostic tool. The information content of the recordings may unveil epilepsy, determine sleep stages, and so on. Such properties certainly cannot be attributed to random noise which by definition, has no information content. Thus electroencephalograms (EEG) are certainly not noise and must have deterministic components, exhibited, for example by the fact that comparable stages of brain waves are universal across individuals. This reasoning brings us to the following question : if the EEG stems from deterministic dynamics, what is its nature? In 1985 Babloyantz et al.[15], using the tools of non-linear time series analysis, could show that several behavioral states of the brain, as measured from EEG, exhibit deterministic chaos. This seminal paper gave rise to a vast literature which is still expanding and confirms the deterministic nature of the EEG. If one accepts that the EEG reveals deterministic chaotic dynamics, one may ask the significance, if any, of such a process for the cognitive functions of the brain. The EEG is a temporal measure that reflects some averaged electro-chemical behavior of millions of neurons beneath the scalp. Therefore, one must explain how the averaged interactions among neural populations generate different EEG's. In 1991, Destexhe and Babloyantz [16] put forth the hypothesis that the existence of temporal chaos in human brain measured from EEG recordings is an averaged property of underlying spatio-temporal chaotic activity of a large number of neurons. In order to substantiate their hypothesis, they constructed a simple

412

16 Chaos Control in Biological Networks

Figure 16.5 Chaotic categorizer. The pacemaker P sends micropulses to layer /, thus stabilizing one of the unstable periodic orbits. The oscillators of layers / and / / are connected with a one-to-one correspondence. The distribution of active links (solid arrows) and inactive ones (dashed arrows) is determined by a pattern to be processed. A response is measured from output layer II.

thalamocortical model which accounts for behavioral states of the cortex during sleep and arousal. On the other hand, the discovery of massive interconnections between thalamus and neocortex, as well as the ability of the thalamus to produce periodic oscillations even when disconnected from the cortex, led to the "thalamic pacemaker" hypothesis for the generation of brain rhythms. Thus is seems that the behavioral states of the brain are somehow related to an oscillatory or synchronized input to the cortical tissue. The cortical tissue was modeled by considering two dimensional networks of 80% excitatory and 20% inhibitory neurons. The dynamics was described in terms of the electrical analogue of the neuronal membrane. Two percent of the neurons of the model cortex received inputs from a thalamic oscillator described by a phenomenological model which was introduced by Rose and Hindmarsh [17]. This two-variable model exhibits limit cycles with various frequencies as the input current into the thalamus changes. The model of cortex alone exhibited spatio-temporal chaotic behavior. The spatially averaged behavior of the cortical network showed a low-amplitude, high-frequency irregular activity. However, as the neurons received input from the thalamus, a more synchronized behavior appeared. A high-frequency and highamplitude oscillatory thalamic activity induced in the cortex small synchronized patches of coherent activity. The average value over the network, the model EEG, showed high amplitude lower frequency and more regular activity as compared with the averaged value of the spatio-temporal chaotic state. As the input current into the thalamic oscillator was decreased, a low amplitude and low frequency oscillatory activity influenced cortical neurons. The cortical network in turn exhibited large patches of synchronized activity. The average value of this spatio-temporal activity was reminiscent of deep sleep EEG. The study of the dynamics of the spatially averaged behavior is also a way of assessing the coherence of the network,

16.4 Chaotic categorizer

413

making the comparison with EEG data possible. Thus in the thalamo-cortical model reported in [16], it was found that the average spatio-temporal network activity follows deterministic chaos. Moreover, the slower rhythm corresponds to the lowest correlation dimension whereas eyes closed EEG is of higher dimension. This is also the case for the measured a waves EEG, where the deep sleep is characterized by high-amplitude, slow waves of low correlation dimension. If we assume that the cerebral cortex is a spatio-temporal chaotic system subject to the synchronizing influence of the thalamus, we are left with the question of how such a system may process information. Building on the fact that the cerebral cortex is made of several interconnected layers, we present here a model which we call a chaotic categorizer and motion detector [18]. The chaotic categorizer is made of two interconnected layers (see Fig.(16.5)). Each layer separately comprises NxN oscillating elements. The elements of the two layers are connected in a one-to-one correspondence via links that are active and represent a given pattern only if an external stimulus activates the first layer. The categorizer processes input information from the external world, in a topographically oriented manner. The external input modulates the action of one neuronal layer over another layer, thus originating different dynamical regimes. The output of the system is some function of the resulting dynamics, and is therefore dependent on features of the external stimulus. However, for information processing to occur, the system must first be placed in an "attentive" state. This is in total agreement with a wealth of experiments reported in the psychophysical literature. In the model, an attentive state is achieved by stabilizing one of the UPOs out of the spatio-temporal chaotic dynamics of one of the layers. As we saw in the preceding section several different UPOs may be stabilized, thus originating different attentive states. Thus chaos can be viewed as a "reservoir" containing an infinite number of UPOs, therefore confering to the system an infinite number of possibilities. A "pacemaker" P sends micropulses only to the layer I as shown in Fig.(16.5). In the absence of external stimuli, the activities of the two layers are independent and both show spatio-temporal chaotic behavior. The device is described by the following differential equations, which if disconnected from each other are equivalent to Eq.(16.3)

^

= Wjk - (1 + i/3)\Wjk\2Wjk + (1 + +-yIjk(Zjh - Wjk),

ia)D£CjklmWlm (j, fc,/,m = 1, ...,7V)

(16.6)

The variables Zjk are the complex amplitudes describing the oscillators in the layer I, whereas the Wjk refer to the corresponding variables in layer II. The terms Pjk (t) , represent the influence of the pacemaker P on each oscillator of layer I, and the terms proportional to 7 account for the forcing of layer II by layer I. We consider only first-neighbor interactions. The boundary conditions are of the zero-flux.

414

16 Chaos Control in Biological Networks

The information to be processed is sent to the device via a binary matrix Ijk . If Ijk = 0 then the connection between elements jk of the two layers is nonexistent and we recover the network of section 3. However, if Ijk = 1 then the two layers are connected via elements jk, and 7 describes the strength of the binding. Moreover, in this model if Pjk{t) = 0 then7 = 0 . Thus, it is only when the pacemaker P is active, and therefore a periodic orbit has been stabilized in layer I, that there is entrainment of the second layer by the first layer. The entrainment can be total if all Ijk = 1 , and it is partial if only some of Ijk are nonzero. Moreover, the dynamics of the second layer is critically dependent on the parameter 7 . if 7 = 0 then the two layers have independent dynamics, whereas large values of 7 with a large number of "on" links representing incoming information, can synchronize the activities of the two layers. If Pjk{t) = 0,7 = 0,iV = 9, a = —10,/3 = 2 and D - 1.3 , then both layers follow spatio-temporal chaotic activity as described before. The dynamics shows the usual features of chaos, namely aperiodicity and extreme sensitivity to small perturbations, as well as broad-band power spectrum. In this model the dynamics of the first layer is not influenced by the second layer. Therefore we shall use the orbit Ci,C2 and C3 which were stabilized in the proceeding section from such a layer. The choice of the orbit will depend from the requirements of spatio-temporal symmetries and the task to be processed. Let us go back to our device as shown in Fig.(16.5), and describe how it can process information. The total input into the system is divided into two parts : the pacemaker P, which emits the appropriate micropulses, and the input representing the pattern to be processed. Thus, the information is captured on the first layer and on the links relating the latter to the second layer. The response of the second layer defines the output of the system. The pacemaker P, according to the nature of the information to be processed, stabilizes the first layer into one of the orbits described in Fig.(16.5) In analogy with the sensory processing in the brain, where the first act is to become attentive to an external input, we call these orbits the "attentive states" of the device. Now the device is ready for processing the input. The latter is imprinted in the links jk , which are nonzero only if they represent an active portion of the input. The attentive state, associated with a well defined spatio- temporal structure, entrains the output layer according to the number and location of "on" links in the device. Thus, in the second layer each input pattern generates its own specific spatio-temporal structure. From this structures, one must derive an output function characterizing its input. The dynamics of the output layer can remain chaotic, or be of a quasi-periodic or periodic nature. In order to discriminate between various inputs, we need to quantify the spatio-temporal activity of the response layer. As we are interested in the differential change in the network coherence of layer II as a result of the input, the evaluation of the global activity of that layer will be sufficient. We use most often the average value of the squared amplitude of the forced layer (|W|2) as the output function. Here, the brackets denote a space average over the entire network. We have also considered other quantities such as the mean value of the

16.4 Chaotic categorizer

415

real part of W. The time evolution of the cross-correlation function between the two layers was also computed. The chaotic categorizer of Fig.(16.5) can be used as a pattern recognition device as well as a motion detector. The ability of the device to perform a given task is a function of the attentive state that is generated by the device under the action of the input to be processed. Cognitive tasks could be grouped by their symmetries, into different possible classes. In general terms, with our categorizer the orbits C2 and C3 are suitable for pattern recognition and also for detection of linear motion. On the other hand, C\ is an orbit that leads to detection of clockwise and counterclockwise motion.

16.4.1 Static pattern discrimination We start with the device in the attentive state C2 • In this state, the stabilized orbit in the first layer shows a polarity that oscillates in time. At a given time one may see a high activity at the right-hand side of the network while the activity is at its lowest level at the left-hand side. The situation reverses periodically. In this attentive state, the device is presented with a bar that activates the nine middle links between the two layers and is parallel to the direction of polarity of the orbit C2 . When these links are "on" there follows an entrainment of layer 77 by layer I. The value of the space average (l^l 2 ) is a measure of this entrainment and is shown in Fig. 16.6 One sees a constant value of 0.68. In another experiment, the bar is presented again to the middle links of the network but perpendicularly to the polarity of C2 . The response of the system as seen in Fig. 16.6 is irregular, with high amplitude and a decreased mean value. If the bar is presented along the diagonal, the response is periodic with a time average of 0.53 (see also Fig.16.6). Therefore, our simple device when in the attentive state C2 can discriminate between different orientation on the plane.

16.4.2 Symbol recognition We consider two different patterns, + and x. Fig.(16.7) shows the output of layer II when these two patterns are presented to the system, when it is in the attentive state C2 . We see that the two patterns are discriminated by the system. The same is true for the attentive state C3. Let us now perform another simple experiment, that illustrates the role of symmetries in finding the adequate attentive states for a given task. We consider a new pair of patterns, N and Z. The first letter can be recovered by a 90° rotation of the second. From symmetry considerations and the result of experiments with single bars, we expect that the attentive state C2 will discriminate between patterns N and Z, whereas the state C3 will give the same answer for both patterns. Our simulations confirmed these conjectures. We present only the result of the N/Z discrimination with the attentive state €2- Figure 16.8 shows its output functions for the two letters.

416

16 Chaos Control in Biological Networks

0.65

0.45

sr 0.25

0.05

Figure 16.6 Space average of the squared amplitude, (|W| 2 ), measured from layer / / . Orbit Ci is stabilized in layer I. Three patterns are presented to the device: a bar activating the nine middle links and parrallel to the direction of polarity of C2 (dot-dashed line), a bar perpendicularly to the direction of polarity (solid line), and a bar activating nine links along one of the diagonals of the network (dotted line).

16.4.3 Motion detection The device of Fig.(16.5) is able to discriminate between clockwise and counterclockwise rotation. To this end we assume that the attentive state of the system is achieved by the stabilization of the first layer into orbit C\ , which shows a phase rotation of period T = 13.66 . As we have stated already, in this example the phase motion of the stabilized orbit is clockwise. Because of the phase rotation of the orbit, the attentive state C\ is able to discriminate between clockwise and counterclockwise motion of a pattern presented to the system. To see this, let us consider the motion of a small object that activates three links at a time. In this experiment^ = 30. As the object moves, at each step only one new link is activated and one of the previous "on" links is deactivated. Thus, the successive activation and deactivation of links represents a circular motion. Fig. 16.9 illustrates this motion. In our example, the rotation period of the object varies from T = 6 to T = 25. In a first experiment, when in the attentive state C\ , the device perceives the clockwise motion of the object which is imprinted in the links. The response of layer II is shown in Fig. 16.10 for a period of rotation of T = 18.96. The value of the space average of the squared amplitude (\W\2) = 0.65 is almost constant in time. However, at very fine resolution small amplitude oscillations are seen around this value. The same response is seen for all values of the rotation periods considered. Thus, in the attentive state C\ , the device is not sensitive to

16.4 Chaotic categorizer

0.65

0.45

417

-

•

- 1

V

v

-IT" v1

"Tf A

0.25

•

-

:\ :: : :: 0.05

iv

V -

120 60 time Figure 16.7 Responses (|W|2)(£) of the output layer in the presence of the patterns + and x. With orbit C2 the response is shown as a dot-dashed line for pattern 4- and a dotted line for pattern x. When orbit C3 is used, the response corresponding to the pattern + is shown as a dashed line, whereas the solid line is obtained with the pattern x.

the speed of clockwise motion of the object. Presently we reverse the direction of rotation of the object and keep all other conditions as above. The response of layer II to counterclockwise motion is very different and is sensitive to the rotation speed. For T = 6 the motion generates a chaotic response around a time averaged value of 0.5. As the speed decreases, the time behavior of the response becomes less and less chaotic and gradually a time-periodic output function appears. Figure 16.10 shows the responses associated with the counterclockwise motions of periods T = 12.48 and T = 18.96. The corresponding time-averaged values of (|W|2) are 0.54 and 0.57 , therefore are sensitive to the speed of rotation. If we restrict the range T to 12 < T < 25, we observe that the value of (|W|2) is an increasing function of T. Thus, in this range not only our device discriminates between clockwise and counterclockwise motion, but it also evaluates the speed of counterclockwise motion. For slow motions, T > 26, and very fast motions, T < 0.5, the response to clockwise and counterclockwise rotation is practically identical. A difference may be seen only in the fine structure of the (|W|2) output function. Thus, for these velocities the device is "blind" with respect to the direction of rotation. The responses in these ranges are similar to the response to clockwise rotation for 6 < T < 25 (see Fig. 16.10). For values of 0.5 < T < 6, the response of the system does not follow the smooth change that was described above. A static object could be considered as rotating with period T -> 00 and thus it is perceived in the same manner as other objects rotating with long periods, that is, T > 26.

418

16 Chaos Control in Biological Networks

0.65

0.45

0.25

0.05

Figure 16.8 The response (|W|2)(£) showing a discrimination between patterns N (Solid line) and Z (dotted line). Orbit C2 is stabilized in layer I.

16.5 Chaos control in biological neural networks The examples of section 4 showed that whenever UPO's are stabilized in a network they may be useful for modeling some aspects of cognitive processes. However as we stated already, most biological processes require time delays in the description of their dynamics. Therefore it is important to see if the existing control techniques could be extended to a network of DDE's. In this section we show how UPO's in a model cortex can be controlled with the technic of continuous self-delayed feedback control. For the same system we also propose a control technique applied in Fourier-space which shall be called modal control. We model the cerebral cortex as a network of excitatory and inhibitory neurons in interaction . The information transfer between neurons mediated by synapses requires considering a propagating delay which is a function of the distance between the interacting units as well as the thickness of the axones. Thus the description of cortical dynamics necessitates the use of DDEs. It has been shown that the following model may represent satisfactorily several behavioral states of the cortex [16].

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16.5 Chaos control in biological neural networks

ooooooooo ooooooooo ootoooooo ooooooooo ooooooooo

10

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12

Figure 16.9 Snapshots showing the successive positions occupied by the active units of a moving pattern. After frame 12, the motion continues at frame 1, originating a continuous loop. The result is a circular motion.

=

l{Yj

VL)

(Yj Et^JgFxiXkit

iyk = l,...,NexJJ

- rkj))

= l,...,Nin

(16.7)

Here X{ and Yj represent respectively the post-synaptic potential of the Nex excitatory and Nin inhibitory neurons. The constant synaptic weights ^ki^u^kj and UjV refer respectively to the excitatory-to-excitatory, inhibitory-to-excitatory, excitatory-to-inhibitory and inhibitory-to-inhibitory connections. The sigmoidal firing function is

The constants a are different for the firing functions of excitatory and inhibitory neurons. For simplicity we assume that the sum Q of the synaptic weights corresponding to the same type of interactions remains constant and r^ — r . Let us note that the delay terms renders the dynamics infinite dimensional. In a given range of parameter values, the network (16.7) may exhibit spatiotemporal chaotic activity which is a function of connectivity patterns and the number and the ratio of excitatory to inhibitory neurons. In the chaotic states, irregular patches of coherent chaotic activity appear and disappear in time (16). We start by considering a homogeneous square network with Nex = Nin = 9 and first-neighbor coupling. The delay has the constant value 1 2 Tij = r = 16 msec. We take all a/ * = 1.575 and u/ > = u^ = 1.25 for all synapses. Chaos is observed for all network size.

420

16 Chaos Control in Biological Networks

0.65

0.60

0.55

0.50 120

Figure 16.10 (|VF|2)(£) computed in the attentive state C\ for a clockwise (dashed line) and a counterclockwise rotating pattern (dotted line), with period of rotation T = 18.96. If the object rotates in the counterclockwise direction with a period T = 12.48 (solid line), a higher amplitude response with a lower time-average value is seen.

Equations (7) admit a periodic bulk solution Xi(t) = X(t) and Y{(t) = Y(i) , with a period of 105.3 msec. This is an unstable solution, embedded in the chaotic attractor of the dynamics. Presently the delay feedback control as described in [3] must be adapted to systems with many interacting units. The details of the method could be found in ref. [6]. Here we only summarize the salient points of the procedure. In order to stabilize the unstable periodic bulk solution, we add the perturbation terms eXi (t) and eYi (t) to the right-hand sides of the equations for the Xi and the YJ variables, respectively. The perturbations are defined similarly to Eqs. (2) and are of the form (16.9) where Xi denotes the average membrane potential of the four first excitatory neighbors of neuron Xj. The definition of the ey5 (t) is analogous, with the average taken over the four first inhibitory neighbors of Yj. Control of the unstable bulk oscillations is achieved by adopting the parameter values T = 105.3 msec and K = 4.6 msec"1 in the control terms. As a result, the activities of all excitatory neurons become mutually synchronized, and the same happens with the inhibitory population. The network dynamics becomes periodic in time, although presenting a complex form. Figure 16.11 illustrates the stabilized dynamics of two adjacent excitatory neurons.

16.6 Control in Fourier space

0.00

100.00

200.00

30000

400.00

421

500.00

t(msec)

0.00

100.00

200.00

30O00

40(100

500.00

Figure 16.11 (a) Simultaneous variation of two adjacent excitatory neurons when the network is in the chaotic regime. Nex = Nin = 9.(b) The same neurons as in (a), after stabilization of the network.

16.6 Control in Fourier space In the preceding sections we showed that the extension of OGY method to large networks described by ODE's becomes computationally very expensive. Similar difficulties is encountered when one tries to adapt the continuous delay feedback method to a large number of coupled DDE's. In this section we introduce a modal approach which is valid for the control of network of ODE's, DDE's as well as a continuous media described by partial differential equations. Again we consider a spatio-temporal chaotic network. In such networks correlations exist between units such that the network activity may be organized according to different spatial modes. These modes are nonlinearly coupled. Let us assumes that chaos arises from an instability in a small number of spatial

422

16 Chaos Control in Biological Networks

modes. The system evolves in a weakly nonlinear range and the activity distribution presents a phase turbulence regime with no amplitude defects. Our approach to control consists in an extension of the discrete-time control methods of the OGY type, after an expansion in an appropriate basis of space functions fa . However, the dimension of the phase-space of the dynamics is much higher in the cases that we study. For continuous systems and for networks of coupled DDE's it is infinite. An important property of the spatially extended systems that we consider is the existence of so called inertial manifolds, which may be described as lowerdimensional subsets of the phase-space to where the system evolves asymptotically. In particular, the inertial manifold contains the chaotic attractor. One can then hope to describe the system's state with limited number of degrees of freedom. This property is essential if modal control is considered. Let us consider a scalar function ^(x,t) that completely determines the state of the extended system. Here X and t are space and time variables respectively. The state vector obeys to the unspecified time-evolution equation *)

(16.10)

We expand and define an Euclidean phase-space spanned by the a^s , where the dynamics will be followed. Control of chaos will be performed in the Fourier space thus defined. As mentioned above, the system may be very-high or infinitedimensional, but we assume that the corresponding strange attractor can be spanned by few degrees of freedom. The present technique of control does not require the knowledge of the evolution equation of the system. This control method requires, nonetheless, that measurements of the coefficients ai , be performed. One hopes that in the vicinity of the target fixed points, monitoring the few first coefficients should suffice for the feedback control. The technique of the Poincare section is used once again to characterize the asymptotic dynamics. The details, of the method and the application to the control of PDE's could be found in Ref. (20). This modal technique is particularly suitable for stabilization of unstable orbits in models of cortical networks. Let us consider again a network where the dynamics is once more given by Eq.(7), but the neurons are arranged in a one-dimensional array with first-neighbor coupling. The populations are in the proportion Nex : Nin = 1. The parameters of Eq. (7) are as before, except r = 1.8 msec and cc>2 = 1-64. An instantaneous activity "profile" is defined as the spatial distribution of the membrane potential of the excitatory neurons, X\ . The index i is the spatial coordinate, ranging from 1 to Nex . Thus space is discrete. However, due to the presence of delay terms the phase-space of the dynamics is once again infinite. For the parameter values indicated and for large enough Nex,Nin , spatiotemporal chaos is observed. We fix Nex = ATin = 8 . The system has therefore infinidimensional. Although random, the spatial distribution of activity shows partial coherence. This suggest that the technique of modal control can be used to stabilize UPO's out of the chaotic attractor. We define the following coordinates in Fourier space,

16.6 Control in Fourier space

423

membrane p o t e n t i a l (mV) -30

time

neuron number

Figure 16.12 Space-time diagram with one complete period-two orbit of the network.

N

(16.11) N

(16.12)

7VN

N -1 = 1,2,....

<(*) sin t=i

N-l

(16.13) (16.14)

The dynamics is followed in the phase-space spanned by these coefficients. Other spatial modes might be considered, but sine and cosine modes are most adequate to describe this system. Several UPOs are identified in the space {Aj,Bj}. Control is performed via very small perturbations of the synaptic weight UJ^ • Coming back from Fourier space to physical space, each orbit displays its own spatio-temporal symmetries. The activity of each neuron is periodic in time, however one sees phase differences between the neurons. This fact is translated into complex spatio-temporal patterns of Fig.16.12. A more detailed discussion of the modal stabilization procedure, for this neuronal system, shall be given elsewhere.

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16 Chaos Control in Biological Networks

16.7 Discussion

The aim of this paper was to show that the control of chaotic systems may have some relevance for understanding of biological system in particular some aspects of brain dynamics. The study of cortical networks involve the control of spatially extended systems with many degree's of freedom. Moreover the description of the biological processes considered here necessitates time delays which in turn requires the control of delay differential equations. This systems are infinite dimensional. The existing control techniques were developed for chaotic system described with few degrees of freedom. We proposed here three different techniques which can be applied for the control of networks described by ODE's, DDE's or partial differential equations. The first of these alternatives discussed in Sec.3, achieves stabilization of chaotic networks of connected units. The variables describing each element of the network are all grouped into a "giant" state vector, and control of chaos is performed in the high-dimensional phase-space of this vector. A Poincare section is defined in the abstract phase-space, and control consists in targeting a number of fixed points in this Poincare section, each corresponding to an UPO of the original system. The control perturbations push the state vector onto the stable space of the fixed point, and thus can be viewed as a high-dimensional version of the OGY control. The system state is monitored by measuring the value of all the variables, and control actions consist in small kicks that are applied also to all of the variables. This is different from the original OGY method where control is applied to the parameters of the system. There are difficulties inherent to a direct extension of the OGY method. Namely, the generalization to very large networks would render the computational load unaffordable. We also proposed a variation of a continuous feedback control method for extended systems. This method possess greater robustness against random noise, due to the permanent character of control. The design of the controller is easier. One has also a form of distributed control, where all the units are individually acted upon. Moreover the control feedback can be specified with very simple rules, which are self-repetitive along the network. During control the operation of the system is totally unsupervised. However this strategy possesses its own difficulties. These may arise in the estimation of the optimum control weight K and control delay T. In addition, the present version cannot stabilize orbits with arbitrary symmetries. The continuous feedback control is not limited to networks of coupled ODEs, it is also suitable for coupled DDEs. Finally in modal control, small perturbations act upon a global parameter of the dynamics. One takes advantage of the fact that, typically, there are correlations between the variables of the system at different spatial locations. This strategy allows for a drastic reduction of the number of variables required to follow the dynamics.

References

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Acknowlegement I am grateful to Carlos Loureno for providing some of the material in this paper and also to N. Ellis for his assistance in the preparation of the manuscript.

References [1] Ott E, Grebogi C, Yorke J. 1990 a. Controlling chaos. Phys Review Letters 64 , 1196-1199. [2] Ott E. 1993 Chaos in Dynamical Systems. Cambridge University Press. [3] Pyragas K. 1992 Continuous control of chaos by self-controlling feedback. Phys Lett A. 170, 421-428. [4] Loureno C, Babloyantz A. 1994 Control of chaos in networks with delay: A model for synchronization of cortical tissue. Neural Computation 6, 1141114. [5] Loureno C, Hougardy M, Babloyantz A. 1995 Control of low-dimensional spatiotemporal chaos in Fourier space. Phys Rev E. 52, 1528-1532. [6] Babloyantz A, Loureno C. 1994. Computation with chaos: A paradigm for cortical activity. Proc Natl Acad Sci USA. 91, 9027-9031. [7] Mackey M, Glass L. 1977. Oscillation and chaos in physiological control systems. Science. 197, 287-290. [8] Babloyantz A, Loureno C, Sepulchre JA. 1995. Control of chaos in delay differential equations, in a network of oscillator and in a model cortex. Physica D. 86, 274-283. [9] Loureno C. 1997. Ph.D Theses. ULB, Brussels [10] Sepulchre J. A , Babloyantz A. 1993 Controlling chaos in a network of oscillators. Physical Review E. 48, 945-950. [11] Kuramoto Y. 1984. Chemical Oscillations, Waves and Turbulence Springer Series in Synergetics; Springer- Verlag, Berlin [12] Grassberger P, Proccacia I. 1983. Characterization of strange attractors. Physical Review Letters 50, 346-349. [13] Sparrow C. 1982. The Lorenz Equations: Bifurcations, Chaos and Strange Attractors Springer-Verlag. New-York. [14] Sepulchre J. A. 1993 PhD dissertation, September 1993 Universit Libre de Bruxelles.

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[15] Babloyantz A, Salazar J, Nicolis C. 1985. Evidence of chaotic dynamics of brain activity during the sleep cycle. Phys Lett A. I l l , 152-156. [16] Destexhe A, Babloyantz A. 1991. Pacemaker-induced coherence in cortical networks. Neural Computation. 3,145-154. [17] Rose R. M, and Hindmarsh, J. L. 1985. A model of a thalamic neuron. Proc.R. Soc. London Ser.B. 225, 161-193 [18] Churchland P, Sejkowski T. The Computational Brain. MIT Press, Cambridge MA. [19] Babloyantz A, Loureno C. 1996 Brain chaos and computation Int. J. Neur. Sys. 7, 492-516 [20] Loureno C, Babloyantz A. 1996 Control of spatiotemporal chaos in neuronal networks. Int. J. Neur. Sys. 7, 507-517

17 Chaos Control in Biological Systems M. L. Spano1 and W. L. Ditto2 1. NSWC, Carderock Laboratory, Code 684, 9500 MacArthur Blvd., W. Bethesda, MD 20817 2. Applied Chaos Laboratory, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332

17.1 Introduction Between regularity and randomness lies chaos. Our prevailing scientific paradigms interpret irregularity as randomness. When we see irregularity, we chauvinistically cling to randomness and disorder as explanations. Why should this be so? Why is it that when the ubiquitous irregularity of biological Systems is studied instant conclusions are drawn about randomness and the whole vast machinery of probability and statistics is belligerently applied? Recently we have begun to realize that irregularity is much richer than mere randomness can encompass. Thus we are brought to chaos. Sustained irregularity has always upset our notions of how the world should behave. Yet it seems to be the canonical behavior of biological Systems. One informal definition of chaos, sustained irregulär behavior, although descriptive, is too vague to define the rieh behavior of chaotic Systems. A more precise defining feature of chaotic Systems is their sensitivity to initial conditions [See, e.g., Baker and Gollub, 1990]. It is this definition which we will utilize for the characterization and control of chaotic Systems. Fleeting glimpses of order within disorder are quite common. We have all seen short Stretches of almost periodic behavior in otherwise irregulär Systems. A tantalizing example lies in the stock market, where many hope to reap windfall fortunes from analyzing short-term order and predicting the volatile market. But shortterm order is a profound, even defining, feature of chaotic Systems. To be explicit, chaotic Systems exhibit: (1) Sensitivity to initial conditions, where the behavior of the System can change dramatically in response to small perturbations in the system's parameters and/or initial values. This makes long term (but not short term!) prediction impossible. (2) Complex geometric structure(s) in the system's phase space (fractal objeets whose composition includes an infinite variety of unstable periodic behaviors) to which the system's behavior is attracted. The combination of sensitivity to initial conditions and complex geometry in State space can produce to the casual observer an appearance of randomness. However, upon closer inspection one glimpses a remarkable order. Thefleshof chaotic Systems adheres to a skeleton of infinite unstable periodic behaviors that seemingly come and go with no apparent pattern. That is, no apparent pattern until one looks with the tools of nonlinear dynamics. Armed with an understanding of unstable periodic Handbook of Chaos Control. Edited by Heinz G. Schuster Copyright © 1999 WILEY-VCH Verlag GmbH ISBN: 3-527-29436-8

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17 Chaos Control in Biological Systems

motions, chaotic systems take on a more understandable, and consequently controllable, character. It is this order that allows the successful short-term prediction of a system's behavior and the subsequent control which would be impossible in a totally random system. In the following we will apply chaos theory to specific biological systems.

17.2 Cardiac Dynamics 17.2.1 Introduction to ventricular fibrillation Sudden cardiac death is a major health problem that claims one in six lives. Nearly all instances of sudden cardiac death occur outside a hospital and the majority are due to an irregular and rapid heart rhythm termed ventricular fibrillation (VF). Heart attacks and primary heart muscle disease (cardiomyopathy) are the most common causes of VF. In VF, the main pumps of the heart, the ventricles, quiver in an irregular manner such that blood is not effectively pumped throughout the body. The only presently known treatment for VF is to pass an electrical signal with a large current through the heart muscle. This shock, if successful, effectively resets the heart back to a rhythm compatible with life. Without such treatment, sustained VF is always fatal.

17.2.2 Fibrillation as a dynamical state Applications of nonlinear dynamical techniques to VF have, until recently, yielded contradictory results, primarily due to the inadequacies of current techniques to resolve determinism in short and noisy data sets. Mechanistic elucidation of VF has been hampered by its rapidly changing and markedly heterogeneous electrophysiological nature, rendering waveforms obtained during VF challenging to quantitatively analyze. In light of recent successes with the identification of unstable periodic motion embedded in chaotic systems and control of these motions in physical and biological systems [Ditto et al., 1990; Ditto and Pecora, 1993; Ott and Spano, 1995; Pierson and Moss, 1995; So et al., 1996; So et al., 1997], new techniques to decide if experimentally obtained, irregular biological waveforms represent deterministic or stochastic (random) behavior abound. Previous quantitative measures for determinism, such as Fourier spectra, fractal dimension, and Lyapunov exponents, along with other statistical techniques, have proven uniformly inadequate for detecting determinism in these short, possibly nonstationary, time series. Elucidation of the presence of determinism during VF is important because it may make novel therapeutic and diagnostic strategies possible. However, whether VF represents a deterministic or stochastic process has been controversial [Guevara, 1981; Goldberger, 1986; Chialvo, 1990a; Kaplan and Cohen, 1990; Witkowski and Penkoske, 1990; Glass, 1990]. The presence of the infinite number of unstable periodic motions that comprise the skeleton of a chaotic system has recently been exploited experimentally. This corollary of a chaotic system's sensitivity to ini-

17.2 Cardiac Dynamics

429

tial conditions is the key to control of such systems [Ditto et al., 1990; Ditto and Pecora, 1993; Ott and Spano, 1995]. This form of stabilization is analogous to a baseball on a saddle. Control is achieved through movement of the saddle's position or adjustment of the ball's motion to keep the baseball constantly rolling back toward the unstable equilibrium point in the center of the saddle. This theoretical technique was originally pioneered by Ott, Grebogi and Yorke [Ott et al., 1990]. Their technique (and variations thereof) has been successfully applied to control chaos in the vibrations of a magnetoelastic ribbon [Ditto et al., 1990], electrical circuits [Hunt, 1991], the output of a solid state laser [Roy et al., 1992], chemical oscillations [Petrov et al., 1993; Parmananda et al., 1993], drug-induced arrhythmias of in vitro rabbit ventricle [Garfinkel et al., 1992], seizure-related population spiking of hippocampal slices [Schiff et al., 1994], and many other systems. These successes all share in common a proportional feedback control around unstable periodic state space trajectories.

17.2.3 Detection of deterministic dynamics in canine ventricular fibrillation Our experimental preparations consisted of open-chest, anesthetized dogs whose hearts were studied in vivo after VF was electrically induced [Witkowski et al., 1995]. Transmembrane cardiac current Im was measured from the ventricular epicardium without cell disruption [Witkowski et al., 1993]. Minimal signal filtering was specifically employed to avoid the pitfalls associated with filtered noise [Rapp et al., 1993]. The Im time series was then examined to detect activations (beats). The intervals between successive activations, A(i), form a related time series that is most useful in diagnosing and controlling chaos. We plotted the (i+l)th interval, A(i+1), versus the previous interval, A(i), in a Poincar map (where deterministic points typically are attracted to a geometric structure know as an attractor). Such a Poincare map provides a reduced view of the dynamics of the measured data. We searched for evidence of unstable periodic orbits, which appear as unstable fixed points in a Poincare map. These unstable fixed points have associated directions along which the system state point approaches (stable manifold) and recedes from (unstable manifold) the fixed point. A typical example of such a sequence in the Poincare map is demonstrated by following state points 23-29 in Figure 17.1a. From point 23 to point 25 the state of the system is drawn toward the unstable fixed point along the stable manifold. Points 26 through 29 demonstrate exponential divergence away from the fixed point along the unstable manifold. This pattern is repeated time and again throughout the experimental run. The solid lines in the figure denote the positions of the stable and unstable manifolds as determined by fitting to a number of such sequences. Figure 17. lb-d display similar behavior in other data sets. The resultant geometry is known as a flip saddle and is consistent with previous experimental results for in vitro rabbit hearts [Garfinkel et al., 1992]. The detection of UFPs is only the beginning in the search to understand and control the physical mechanisms that underlie VF. While the canine model of VF

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431

is widely employed to test promising new defibrillation devices, the relevance of the findings in our study to the clinically important disturbance of VF in humans requires additional studies. VF can be induced in virtually all human hearts. It is fascinating to speculate that a greater understanding of the nonlinear dynamical behavior of VF might lead to the possibility that VF may be controlled through chaos control techniques. For this to become a reality we really need to know not just the temporal dynamics but also the spatiotemporal dynamics which underlie VF. With that in mind we instituted a study to visualize the spatiotemporal electrical patterns long suspected of being the dynamical manifestation of fibrillation.

17.2.4 Imaging of the spatiotemporal evolution of ventricular fibrillation Rotors, electrophysiological structures that emit rotating spiral waves, occur in a variety of systems that all share with the heart the functional properties of excitability and refractoriness. These reentrant waves, seen in numerical solutions of simplified models of cardiac tissue [Holden, 1997] are believed to occur during ventricular tachycardias [Winfree, 1994; Panfilov and Holden, 1997]. The detection of such forms of reentry in fibrillating mammalian ventricles has been difficult [Gray et al, 1995; Lee et al., 1996; Rogers et al., 1996; Gray et al., 1998]. Here we show that in isolated perfused dog hearts, high spatial- and temporal-resolution optical transmembrane potential mapping can readily detect transiently erupting rotors during the early phase of ventricular fibrillation. This activity is characterized by a relatively high spatiotemporal cross correlation. During this early fibrillatory interval, frequent wavefront collisions and wavebreak generation [Pertsovet al., 1993] are also dominant features. Interestingly, this spatiotemporal pattern undergoes an evolution to a less highly spatially correlated mechanism devoid of the epicardial manifestations of rotors despite continued myocardial perfusion. In 1930 Carl Wiggers used a movie camera operating at 32 frames/sec to record the movements of the surface of the in situ heart in which VF was induced [Wiggers, 1930]. He described 4 stages: (1) an initial stage consisting of 2 to 8 rapidly activated peristaltic waves; (2) a subsequent convulsive incoordination stage that lasted 14-40 seconds, so named because "When the ventricles are held in the palm of the hand, a fluttering, undulatory, convulsive sensation is experienced" without the ability to generate any blood pressure; (3) and (4) are subsequent stages that reflect the progressive ischemia. In terms of clinical interventions the most significant of these stages is stage 2, when countermeasures can be instituted and sudden death aborted. Recently, using an electronic camera operating at 60 frames/sec, together with voltage sensitive dye staining of the heart, a single rapidly moving rotor which produced an electrocardiographic pattern in rabbit hearts that resembled fibrillation was described [Gray et al., 1995]. However, similar rotating waves in larger mammalian hearts are described as uncommon occurrences in canine hearts [Lee et al., 1996] and rare in porcine hearts [Rogers et al., 1996]. We have recorded the electrical activity from a limited epicardial area of the right and left ventricles in isolated, blood-perfused canine hearts. The anterior

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right ventricle and part of the left ventricle were compressed under an optical window to minimize motion artifacts. (No pharmacological agents were employed to reduce mechanical motion artifacts in the dynamics.) The measurement technique employs an image intensified charge coupled device (CCD) optical recording system [Witkowski et al., 1998b] imaging an epicardial surface stained with the voltage sensitive dye di-4-ANEPPS (Figure 17.2). An area of approximately 5.5 x 5.5 cm was imaged, which represents approximately 30 % of the epicardial surface (Figure 17.3). At the levels of illumination utilized, continuous recordings lasting 10-15 minutes could be realized with no detectable phototoxic damage. A 128 x 128 pixel, frame transfer CCD camera operating at 838 frames per second (1.19 ms/frame) was used. The analog video signal underwent 12-bit A/D conversion prior to transfer to a frame grabber. Fluorescent images were obtained with a temporal resolution of 1.2 msec and a spatial resolution of approximately 0.5 mm [Witkowski et al., 1998a; Witkowski et al., 1998b]. VF was induced with a single, critically timed electrical pulse. Both the early onset of VF (corresponding to Wiggers' stages 1 and 2) as well as sustained VF that lasted for more than 10 minutes in perfused hearts were imaged with this apparatus. The optical transmembrane potentials as well as their temporal derivatives were then viewed as movies. (Quicktime and AVI movie sequences from this study are available at the web site: http://www.physics.gatech.edu/chaos.) The data from 15 frames of optical images (Figure 17.4a) clearly shows the onset of spiral wave formation in our canine heart experiment. The initial waves triggered by the induction of VF consistently produced a reentrant cycle with a "figure-of-eight" morphology as in Chen et al., 1988. These are composed of two mirror image rotors that share a common reentrant pathway. As an example, this reentry might last for a total of 8 cycles before being abolished by wavefronts that collided in the area of this initial reentry. These collisions often resulted in the subsequent emergence of two oppositely directed, spatially discrete wavefronts with observable dangling ends [Witkowski et al., 1998b] as shown in Figure 17.4b. Each of the dangling ends of the emerging wavefronts is also called a wavebreak or phase singularity [Pertsov et al., 1993]. (The web-available movie files illustrate these features.) Thereafter, other rotors formed. All episodes of induced VF were self sustaining and terminated only when the heart was defibrillated. A completely different electrophysiological pattern was also observed when perfused VF had persisted for 10 minutes. During this "chronic" VF, no rotors were observed. The source mechanism is still probably reentry, but we believe that its geometric aspect has become more three-dimensional. This pattern was reproducible in that, after defibrillation and a recuperation interval of 10 minutes, the acute pattern of VF (Wiggers' stage 2) was once again the initial manifestation after VF induction.

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(a) Schematic representation of the experimental set-up. Two 1 kW stabilized xenon arc lamps were used to illuminate the epicardial cardiac surface under study with 100 mW/cm2 of quasimonochromatic green light. This focused fluorescent source was imaged after barrier filter rejection of reflected light components and retention of fluorescent components with wavelengths^590nm, as illustrated. The cooled fiber optically coupled image intensified CCD frame transfer camera system was operated at 1.19 msec/frame with 12 bits of dynamic range, (b) Time series from the sequential processing steps for a representative single pixel from 500 frames during ventricular fibrillation are shown. The effects (on signal to noise) of subsequent image processing steps are depicted; those included 9x9 gaussian spatial followed by 21 point median temporal filtering, and culminating with 5 point temporal derivative estimation with final clipping to maintain only the positive values (setting all negative values to 0). The optical calibration bar indicates a 2fluorescence. [Witkowski et al., 1998a]

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Figure 17.3 Photograph of formalin-fixed canine heart with the pacing electrodes marked by white headed pins, and the approximate area of epicardium which was imaged is outlined with white tape. [Witkowski et al., 1998a]

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Figure 17.4 (a) Rotors, the source structures that immediately surround the core of rotating spiral waves, occur in a variety of systems that all share with the heart the functional properties of excitability and refractoriness. Here we show that in isolated, perfused dog hearts, high spatial and temporal resolution optical transmembrane potential mapping can readily detect transiently erupting rotors during the early phase of ventricular fibrillation. This activity is characterized by a relatively high spatiotemporal cross correlation, (b) During this early fibrillatory interval frequent wavefront collisions and wavebreak generation are also dominant features.

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11 Chaos Control in Biological Systems

17.3 Control of Chaos in Cardiac Systems 17.3.1 Control of isolated cardiac tissue We have found that it is possible to control a chaotic cardiac arrhythmia using chaos control. Our cardiac preparation [Garfinkel et al., 1992] consisted of an isolated perfused portion of the interventricular septum from a rabbit heart as shown in Figure 17.5. The heart was stimulated by passing a 3 ms constant voltage pulse, typically 10-30 volts, at twice threshold between platinum electrodes embedded in the preparation. Electrical activity was monitored by recording monophasic action potentials with Ag-AgCl wires on the surface of the heart. Monophasic action potentials were digitized at 2 kHz and processed in real time by a computer to detect the activation time of each beat from the maximum of the first derivative of the voltage signal. Arrhythmias were induced by adding 2 to 5 mM ouabain with or without 2 to 10

Figure 17.5 Photograph of an isolated well-perfused portion of the interventricular septum from a rabbit heart, arterially perfused through the septal branch of the left coronary artery with a physiologically oxygenated Kreb's solution at 37°C.

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mM epinephrine to the arterial perfusate. The mechanism of ouabain/epinephrineinduced arrhythmias is probably a combination of triggered activity and nontriggered automaticity caused by progressive intracellular Ca2+ overload from Na+ pump inhibition and increased Ca2+ current. Typically the ouabain/epinephrine combination induced spontaneous beating, initially at a constant interbeat interval and then progressing to period-2 and higher order periodicity before developing a highly irregular aperiodic pattern of spontaneous activity. The duration of the aperiodic phase was variable, lasting up to several minutes before spontaneous electrical activity irreversibly ceased. The spontaneous activity induced by ouabain/epinephrine in this preparation showed a number of features symptomatic of chaos. Most importantly, in progressing from spontaneous beating at a fixed interbeat interval to highly aperiodic behavior, the arrhythmia passed through a series of transient stages that involved higher order periodicities. These features are illustrated in Figure 17.6 in which the nth interbeat interval (In) has been plotted against the previous interval (In-1) at various stages during ouabain/epinephrineinduced arrhythmias. As before, this Poincare return map allows us to view the dynamics of the system as a sequence of pairs of points (In, In-1), thus converting the continuous dynamics of our system to a map. On such a map, chaotic data (Figure 17.6e) can easily be distinguished from periodic data (Figure 17.6a-d). Additionally, the sequence of the data points on such a plot reveals the stable and unstable directions, knowledge of which is required to implement control (Figure 17.7). In this rabbit preparation, each control attempt consisted of a learning phase and a control phase. During the learning phase an unstable fixed point (UFP) was identified and characterized [Garfinkel et al., 1992]. During the subsequent control phase, the computer waited until a close approach to the UFP was detected (—In In-1— i e where e defines the control region and is a fraction of the total attractor size). The algorithm then initiated a control stimulus that moved the next point on the Poincare plot (as predicted by the local dynamics of the UFP) onto the stable manifold (i.e., the contracting direction) of the UFP, thereby allowing the natural dynamics of the system to subsequently draw the system state onto the UFP itself. Stimuli were administered on subsequent points to keep the system on the stable manifold. Thus the system state was continually contracted towards the UFP. This is shown schematically in Figure 17.8. If the current point on the Poincare plot strayed outside the control region, stimuli were then discontinued until the system state point reentered the control region. (It should be noted here that on-demand pacing is simply the special case of this algorithm where the stable manifold is horizontal.) An example of applying this algorithm to the rabbit heart is shown in Figure 17.9. Chaos control with this approach was complicated by the fact that in this experiment intervention was, of necessity, unidirectional. By delivering an electrical stimulus before the next spontaneous beat, the interbeat interval could be shortened, but it could not directly be lengthened. This is because a stimulus, which elicits a beat from the heart, shortens the interbeat interval between the previous spontaneous beat and the beat elicited by the stimulus. The effects of

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17 Chaos Control in Biological Systems

Figure 17.6 Time series of monophasic action potentials (right) and the Poincare maps of interbeat intervals (left) at various stages during arrhythmias induced by ouabain/epinephrine in typical rabbit septa. Typically the arrhythmia was initially characterized by spontaneous periodic beating at a constant interbeat interval (top), proceeded through higher order periodicities such as period-2, period-4, and period-5 (middle frames), and ending in a completely aperiodic pattern (bottom). Note that, in the Poincare map of the final stage, the points form an extended structure that is neither pointlike nor a set of points (i.e., is not periodic) and is not space-filling (i.e., is not random). This is indicative of chaos. [Garfinkel et al., 1992]

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this limitation are apparent in Figure 17.9, where the best control we could achieve was a period-3. Several observations should be made about the pattern of the stimuli delivered by the chaos control program. First, these stimuli did not simply overdrive the heart. Stimuli were delivered sporadically, not on every beat and never more than once in every three beats on average. In contrast, periodic pacing, in which stimuli were delivered at a fixed rate, was never effective at restoring a periodic rhythm and often made the original aperiodicity more marked. Non-chaos control, irregular pacing was similarly ineffective at converting chaotic to periodic behavior. Encouraged by both the evidence for UFPs in rabbit and canine hearts and by our success in controlling the rabbit tissue preparations, we decided upon the aggressive course of attempting to control chaos in fibrillating atria (upper chamber) of human hearts.

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17 Chaos Control in Biological Systems

Figure 17.8 Schematic of the chaos control technique. In is the current and In-1 the previous interval between activations. The central point at the intersection of the three lines represents an unstable fixed point (UFP). The stable (inward arrow) and unstable (outward arrow) manifolds are shown as calculated from multiple close returns to the unstable fixed point during the learning phase of the algorithm. Without intervention, the natural dynamics around the unstable fixed point carries the activation intervals from A to B to C, contracting inward along the stable manifold and expanding outward along the unstable manifold, as shown. Chaos control is implemented after a determination of these stable and unstable manifolds (or eigenvectors) and contraction and expansion rates along the manifolds (or eigenvalues), as determined by a data- derived least squares linear fit or model of the dynamics in the vicinity of the unstable fixed point. A point occurring at A is predicted to move to B, as determined by this model of the local dynamics. A stimulus is introduced into the high right atrium to force a premature activation that directs A onto the stable manifold at location B' instead of allowing the uncontrolled dynamics to proceed to B. The contraction along the stable direction then pulls the next activation interval closer to C, which is closer to the unstable fixed point. Thus the sequence of ABC is modified to AB'C, keeping the dynamics close to the unstable fixed point. This process is repeated in a feedback loop to stabilize the UFP. [Garfinkel et al., 1992]

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17.3.2 Control of atrial fibrillation in humans Atrial fibrillation is the most common arrhythmia requiring treatment intervention [Prystowsky et al., 1996]. The occurrence of atrial fibrillation increases with age, touching more than 5rapid and irregular ventricular rate as well as the loss of atrial mechanical function diminish overall cardiac performance and may cause palpitation, breathlessness, fatigue and lightheadedness. In addition to these disabilities, atrial fibrillation dramatically increases the risk of stroke and cardiovascular-related death [Kannel et al., 1982]. Evidence has suggested that biological activity, including the beating of myocytes in vitro [Chialvo et al., 1990a; Chialvo, 1990b], cardiac arrhythmias [Garfinkel et al., 1992; Witkowski et al., 1995; Hall et al., 1997], and brain hippocampal electrical bursting [Schiff et al., 1994; So et al., 1997] exhibit deterministic dynamical behavior. Chaos is the deterministic collection of a large number of unstable periodic motions. Such unstable behavior (including its associated local dynamics) forms the basis for various chaos control techniques. [For reviews of chaos con-

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trol, see Ditto and Pecora, 1993; Ott and Spano, 1995; and Lindner and Ditto, 1995; Christini and Collins, 1995; Christini and Collins, 1996; Pei and Moss, 1996.] Recent work on the control of chaos in low dimensions [Ott et al., 1990], high dimensions [So and Ott, 1995; Ding et al., 1996], and spatially extended systems [Petrov et al., 1996a; Petrov and Showalter, 1996b] in physical and biological systems has enabled the application of chaos control to human atrial fibrillation (AF). The human AF study was performed on 25 patients undergoing clinicallyindicated electrophysiological testing. The study was conducted under a protocol approved by the Human Research Committee at Emory University and all patients gave written, informed consent. A quadripolar electrode catheter with 5 mm interelectrode spacing was inserted via the femoral vein and advanced under fluoroscopic guidance to the anterolateral aspect of the right atrium as shown in Figure 17.10. The tip of the catheter was positioned to achieve a bipolar stimulation threshold of 2 mA at a 2 ms pulse width. Atrial fibrillation was induced using rapid pacing (50Hz) for 1 to 2 seconds. Local atrial activation was recorded from the proximal pair of electrodes (poles 3 and 4). The signal was amplified (with no filtering) and sent to the active control and passive recording computers, where it was digitized at 2 kHz and 5 kHz, respectively and activations (beats) were detected. Control stimuli were output from the computer and used to trigger a stimulus isolation unit that was connected to the distal poles (1 and 2) of the atrial electrode catheter. As before, we implemented the control algorithm outlined in Figure 17.8, with each control attempt having a learning phase and a control phase. The identification of an unstable fixed point [Garfinkel et al., 1992; Schiff et al., 1994] and its subsequent characterization [Pierson and Moss, 1995] were much improved by intervening advances in technique. Additionally, to demonstrate that we were indeed attempting control around an unstable fixed point rather than a noisy random point, we applied the So algorithm [So et al., 1996; So et al., 1997] to our data after the control runs (since this method is computationally too costly to implement in real time). It also detected the fixed point around which we had attempted control. This algorithm transforms the data such that in a suitable phase space points near an unstable fixed point are mapped onto the unstable fixed point position. Other points are mapped randomly over the attractor. Thus, in a 1-D histogram of the distribution of the transformed points, UPOs are observable as sharp peaks. In this case "near" refers to a region around the UPO that can be approximated by a linear map. (Higher order versions of this method have also been formulated.) This linear region is similar to the linear region used by Pierson et al. The results of the So transform method are displayed in Figure 17.11. The large peak (red) near an interbeat interval of 0.20 seconds denotes the period-1 UPO. The same transform applied to 100 surrogates is shown in blue. The peak exceeds the surrogate background by more than 40 standard deviations of the surrogate ensemble, as shown in the inset. Also note that, since this method maps points near the UPO onto the UPO position, the fact that the transformed data on either side of the peak fall below the surrogate average provides additional confirmation that we have correctly detected a UPO in this data.

17.3 Control of Chaos in Cardiac Systems

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A typical outcome of human atrial chaos control is shown in Figure 17.12. The outcomes of chaos control were categorized as follows: (1) Excellent Chaos Control was defined by successful capture (a capture is an activation within 15 ms of the application of a control stimulus) for at least 25 sequential intervals around a UFP. The mean of the controlled intervals was equal to or longer than the mean of the activation intervals of the spontaneous atrial fibrillation and the standard deviation from the mean was at least two times less than the standard deviation from the mean of the uncontrolled activation intervals. (2) Partial Chaos Control was defined as in (1) except with more frequent losses of control (lOchaos control were escapes from the control region) about the UFP. (3) Unsuccessful Chaos Control was defined as all other cases, including those with infrequent capture, lack of suitable UFP's, indiscernible dynamics about the UFP, and all other results. Out of

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the 25 patients in the study, excellent chaos control was achieved in 9/25 patients (36achieved in 10/25 patients (40patients (24incomplete activation detection as the cause for the frequent loss of control, rather than any failure of the chaos control algorithm. Additional reasons for loss of control included poor characterization of UFPs and rapid changes in the (uncontrolled) dynamics. To diagnose poor characterization of the UFPs, control was turned off and then reinitiated with a new learning phase. After the second learning phase was completed, a dramatically different UFP was found, thus calling into question the accuracy of the original UFP characterization. In contrast, during excellent chaos control we were able to discontinue chaos control and subsequently reacquire the same UFP, as shown in Figure 17.12. In these cases the UFP always had similar values for its position and its manifolds to those found previously. More significantly, the chaos control remained excellent. In the six unsuccessful control attempts, we were never able to both locate and control (for any significant length of time) a UFP with a mean

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17 Chaos Control in Biological Systems

cycle length at or above the uncontrolled mean. Since this algorithm is implemented in a 2-D Poincare section, it is useful to consider the results on that section. These are displayed in Figure 17.13. The top pane shows the distribution of the data before control was implemented for a typical case, while the bottom pane gives the distribution as a result of the control algorithm. The symmetry is important in that it indicates that the deviations around the control point are truly random and not the result of poor control technique or bad control parameters. It has been demonstrated [Allessie et al., 1991; Kirchhof et al., 1993] that during atrial fibrillation conventional fixed-rate pacing (stimulation at a constant cycle length) and on-demand pacing (a stimulation rate at which the intrinsic activation interval exceeds the programmed pacing interval) only entrains local activations in a narrow window of cycle lengths around the mean activation interval. Both techniques suffer from inconsistent capture when pacing at shorter and longer cycle lengths [Allessie et al., 1991; Kirchhof et al., 1993]. When pacing at intervals much shorter than the mean interval, either method only eliminates the long intervals, leaving the shorter ones unchanged. In contrast our results have demonstrated the effectiveness of chaos control for entraining the atrium at intervals equal to and significantly longer than the mean spontaneous interval with the ability to eliminate both short and long fluctuations about the mean interval. Thus it dramatically reduces the variation from the mean cycle length. It should be noted that chaos control functions quite differently from periodic or demand pacing by locating, characterizing and exploiting the natural dynamics around a UFP in the Poincare plot. In addition chaos control initiates a stimulus only when the system state comes near the UFP. The stimulus forces a predicted interval onto the stable manifold (the contracting direction) and allows the natural contraction along the stable manifold to pull the state point onto the UFP, thereby minimizing the number of control interventions needed. Chaos control uses control stimuli to exploit the natural contraction of the stable direction of a UFP rather than enforcing a rigid target point (interval). Thus we have shown that chaos control can be used to stabilize an unstable fixed point whose corresponding activation interval was equal to or significantly longer than the mean of the uncontrolled activation intervals during human AF. However several unresolved questions remain. First it is unclear what extent of the atrium is captured during control. A previous study on the regional entrainment of atrial fibrillation in dogs [Allessie et al., 1991] has shown capture of a region 4 cm in diameter. We are currently working on determining the spatial extent of such chaos control in animal experiments. Second, while sinus rhythm occasionally follows chaos, further studies will be required to determine a causal connection. Third, it is an open question, in lieu of chaos control-induced cardioversion, whether chaos control can lower the energy threshold required for defibrillation of the atrium. The atrial defibrillation threshold, even with newly developed endocardial leads, remains sufficiently high to result in stimulation of skeletal muscles and patient discomfort [Wickelgren, 1996]. Despite these uncertainties, chaos control in human

17.3 Control of Chaos in Cardiac Systems

447

Figure 17.13 Histogram of the data on a Poincare plot before (upper) and after (lower) implementation of chaos control of human atrial fibrillation. [Langberg et al., 1998]

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17 Chaos Control in Biological Systems

atrial fibrillation offers a promising alternative for altering the dynamics of the arrhythmia. This alternative requires much less energy than the existing high energy shock techniques, which are designed to overpower the dynamics of the atrium. Thus a better understanding of the dynamics of human atrial fibrillation and its response to chaos control techniques presents us with an intriguing new direction for the study and treatment of human fibrillation.

17.4 Control of Chaos in Brain Tissue Our success in controlling chaos cardiac systems led us to see if a similar strategy could control chaotic behavior in brain tissue [Schiff et al., 1994]. One of the hallmarks of the human epileptic brain during periods between seizures is the presence of brief bursts of focal neuronal activity known as interictal spikes. Often such spikes emanate from the same region of the brain from which the seizures are generated. Several types of in vitro brain slice preparations, usually after exposure to convulsant drugs that reduce neuronal inhibition, exhibit population burst-firing activity similar to the interictal spike. One of these preparations is the high potassium [K+] model, where slices from the hippocampus of the temporal lobe of the rat brain (a frequent site of epileptogenesis in the human) are exposed to artificial cerebrospinal fluid containing high [K-f ] which causes spontaneous bursts of synchronized neuronal activity which originate in a region known as the third part of the cornu ammonis or CA3 as shown in Figure 14. If one observes the timing of these bursts, clear evidence for unstable fixed points is seen in the return map. As reported we were able to regularize the timing of such bursts through intervention with stimuli delivered by micropipette with timing as dictated by the chaos control algorithm to put the system onto the stable direction. As shown in Figure 17.15, not only were we able to regularize the intervals between spikes, but we were also able through a chaos "anticontrol" strategy to make the intervals more chaotic. It is the latter which might serve a useful purpose in breaking up seizing activity through the prevention or eradication of pathological order in the timing of the spikes. This original anticontrol or chaos maintenance strategy [Schiff et al., 1994] has been further elaborated and expanded to high dimensions in the past few years [Yang et al., 1995; In et al., 1995; and In et al., 1998].

17.5 DC Field Interactions with Mammalian Neuronal Tissue It has long been known that electric fields affect neuronal excitability [Rushton, 1927; Katz and Schmitt, 1940; Terzoulo and Bullock, 1956]. When properly aligned, such fields can increase or decrease the excitability of neurons [Jefferys, 1981; Chan and Nicholson, 1986] in a manner similar to lowering or raising the threshold for initiation of nerve action potentials [Chan et al., 1988]. Although

17.5 DC Field Interactions with Mammalian Neuronal Tissue

449

Stimulation electrode Schaffer collateral fibers

Recording electrode

Pyramidal cells

Dentate granule cells

Figure 17.14 Schematic diagram of the transverse hippocampal slice and arrangement of recording electrodes. [Schiff et al., 1994]

there is a long history of attempting seizure control in humans with stimulation of the nervous system at sites remote from seizure foci [Cooper et al., 1976; Van Buren et al., 1978; Murphy et al., 1995], there seems to have been no attempt to apply what is known about electric fields in order to suppress seizures. Since previous in vitro experiments have suggested that current injected directly into tissue could suppress evoked [Kayyali and Durand, 1991] or spontaneous [Nakagawa and Durand, 1991] epileptiform activity in brain slices and that externally applied electric fields might suppress or enhance stimulus-evoked neuronal population spikes [Bawin et al., 1986a; Bawin et al., 1986b], we postulated that externally applied electric fields might suppress spontaneous seizure activity. With elevated potassium (8.5 mM) in the perfusate, hippocampal tissue slices exhibit two features which share physiological similarities with spontaneous seizure activity: in the CA3 region intermittent burst discharges arise similar to interictal epileptic spikes [Rutecki et al., 1985; Pedley and Traub, 1990] and in the CA1 region more prolonged electrographic seizure-like events are observed [Traynelis and Dingledine, 1988]. Using relatively small externally applied electric fields, aligned parallel to the large pyramidal neurons under study (CA1 or CA3), it has been demonstrated that both kinds of spontaneous epileptiform activity can be suppressed [Gluckman et al., 1996b]. Often, this effect is dramatic, with complete cessation of the activity under study. This effect is seen whether the networks are produced by cutting the slices transversely or longitudinally. These results suggest that noninvasive sup-

450

17 Chaos Control in Biological Systems

2.0 HAntiControl Cta|-|AntiControl Off|_lControl On

1.5

1.0

0.5

** • Control On

Control Off)

i

0.0

500

1000

i '

1500

' i

2000

Spike Number n Figure 17.15 Demonstration of chaos anticontrol and chaos control in a hippocampal slice of a rat brain exposed to artificial cerebrospinal fluid containing high [K-f ] and undergoing spontaneous chaotic population burstfiringor spiking. [Schiff et al., 1994]

pression of epileptic activity in vivo might be feasible. Brain slice experiments are particularly well suited for studying electric field interactions with neurons. The neurons remain in their native architecture and the neural layers can be identified and oriented visually within a perfusion chamber. Within the chamber, a well-defined electric field can be generated with electrode plates and the field can be mapped precisely. Thus the interaction between the field and the neurons under study can be specified with far more accuracy than is possible in vivo. Hippocampal tissue slices were prepared as described above [Figure 17.14]. Figure 17.17 illustrates the effect of DC field changes on spontaneous activity from slices cut either transversely (a, b) or longitudinally (c, d). In each pane of the figure, the electric field change, initiated at time t = 0, is indicated in the trace above a recording of the neuronal activity. Insets show details of the neuronal activity at an expanded time base, and schematics to the right indicate the orientation of the slices with respect to the direction of the electric field. In the schematics, the orientation of the soma and major basal dendrites of the neurons under study are shown. In the transversely cut slice in high potassium perfusate (Figure 17.17a), CA3 pyramidal cells generate compact bursts of activity whose synchrony shares similarities with interictal spikes [Pedley and Traub, 1990]. The baseline burst frequency was recorded with a superimposed - 30 mV/mm field, and immediately suppressed upon switching the polarity of the field to 4- 30 mV/mm. Without a superimposed

17.5 DC Field Interactions with Mammalian Neuronal Tissue

451

Figure 17.16 Photograph (as viewed from the side) and schematic of perfusion chamber (as viewed from above) for hippocampal slice preparation. The hippocampal slice rests just below the upper surface of the bath. An electric field is imposed by a potential between parallel Ag-AgCl plates submerged in the bath. [Gluckman et al., 1996a]

negative field, the CA3 pyramidal cells in this slice were spontaneously active (but with a slower burst frequency). In the transverse slice (Figure 17.17b), the population events in CA3 are propagated through the Schaffer collateral fibres to the CAl pyramidal cells, where similar bursts of activity are evoked, interspersed with more prolonged seizurelike events (Figure 17.17b inset). Note that with the field aligned with the CAl pyramidal cells in this fashion, seizure-like events are selectively eliminated while spike-like events, propagated from CA3 neurons perpendicular to the field, persist. If the hippocampus is cut longitudinally (Figure 17.17c,d), the pyramidal cells in CA3 and CAl are more uniformly aligned than in the more curved configuration in the transverse slice. In such longitudinal slices, the synchronous activity in

452

17 Chaos Control in Biological Systems

•10.0mV/mm

-10

1 a 12.5mVAnm

CA1

CA3

CA1

•12

-8

-4

0

4

8

12

16

Time (s)

Figure 17.17 Examples of suppression of spontaneous activity within hippocampal slices as a function of DC applied field. Results are shown in transverse slices for CA3 (a) and CA1 (b), and in longitudinal slices for CA3 (c) and CA1 (d). Insets show details of neuronal activity at expanded time base. Vertical calibration bars indicate 1 mV for main tracings, while vertical and horizontal calibration bars for insets indicate 1 mV and 500 msec respectively. Right sided Y- axes indicate externally applied field strength in mV/mm. Insets illustrate schematics of the field orientation, in relation to the anatomy of the slices, that resulted in suppression of the neuronal activity. The orientation of a soma and basal dendrite of the pyramidal cells being monitored is shown in each schematic inset. The electric field is the negative gradient of the potential, so the open arrows ( ) point from the negative to the positive electrode plate during the period of suppression. In all cases of suppression, the electric field was directed along the axis defined from basal dendrite to soma. As discussed in the text, this orientation and polarity of field would be expected to hyperpolarize the soma. [Gluckman et al., 1996b]

17.6 Summary

453

CA3 (Figure 17.17c) and CAl (Figure 17.17d) are readily suppressed with 25-30 mV/mm fields aligned parallel with the dendritic-somatic axis of the pyramidal cells. Finally, note that over several minutes the effect of the DC field abates, an effect at least in part due to gradual polarization of the Ag-AgC12 electrodes. But regardless of the region of the hippocampal slice selected, and irrespective of the type of slice (longitudinal or transverse), relatively small electric fields successfully interrupted the activity under study. It is known that much smaller fields are needed to modify the activity of neurons than to promote firing from rest [Terzoulo and Bullock, 1956]. The physics of these interactions has been well worked out in recent years [Chan et al., 1988; Trachina and Nicholson, 1986]. In essence, applying electric fields parallel to the dendritic-somatic axis will polarize the neurons, hyperpolarizing or depolarizing the transmembrane potential at the soma where action potentials are initiated. Such polarization effects are consistent with the increase or decrease in the seizurelike activity observed as a function of field orientation in these experiments. Several potential experimental applications of these results are of interest. Recall that, using current injected into tissue, it has been shown that chaos control techniques are applicable to these neural networks [Schiff et al., 1994]. In previous attempts to apply chaos control theory [Ott et al., 1990] to biological experiments [Garfinkel et al, 1992; Schiff et al., 1994], suprathreshold stimuli using current injection directly into tissue were required. This constituted a relatively high-energy control perturbation. In contrast, electricfieldsoffer a subthreshold, low energy interaction with excitable tissues, permitting more subtle control than was previously possible by delivering graduated perturbations through a system-wide parameter. In addition, such a scheme removes the problems associated with only being able to shorten intervals between events. Might electric fields be clinically applicable to human focal epilepsy? The barriers to the application of electric fields to suppress well-defined epileptic foci seem more technical than theoretical. Although all "non-polarizable" electrodes will polarize to some degree with constant DC currents, the application of such electric fields might not have to be prolonged in order to achieve suppression of a focus. Although the sulcal geometry of the human brain is convoluted, the surface of the gyri contain pyramidal cells all in a roughly perpendicular arrangement to the brain surface. Further work will determine whether suppression of activity in regions of cortex parallel to an applied field will be sufficient to interfere with seizure generation and propagation.

17.6 Summary In less than a decade since the original chaos control experiments, the experimental applications of chaos control have exploded in number. Advances in the lore of chaos control, especially extensions to higher dimensional systems, the development of the maintenance of chaos, and the beginnings of techniques for controlling

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17 Chaos Control in Biological Systems

spatially extended systems, have enabled chaos control to be applied to reasonably complex systems, including many medical systems of clinical interest. These range from atrial fibrillation, for which devices incorporating chaos control appear to be quite viable, through ventricular fibrillation, which awaits practical extensions of chaos control theory to spatiotemporal systems, all the way to epilepsy, which may benefit from techniques for maintaining chaos. Closely related to chaos control is the idea of chaos suppression. While it has not been possible to discuss in this paper all the various techniques developed for chaos suppression, one exciting application is the use of electric fields to suppress the (chaotic) epileptiform activity of the hippocampus. We have found that while, the techniques of nonlinear dynamics are applicable to many different biological systems, it is also true that these selfsame biological systems can teach us much about the nonlinear dynamics of complex systems. For example, a nonlinear technique developed for biological systems (maintenance of chaos) has been subsequently applied with great success to physical systems. Thus there is a synergy between our efforts to apply the techniques of nonlinear dynamics to biology and our understanding of nonlinear dynamics itself.

References Allessie, M., Kirchhof, C, Scheffer, G. J., Chorro, F., and Brugada, J., Circ. 84, 1689 (1991). Baker, G. L., and Gollub, J. P., Chaotic Dynamics: An Introduction (Cambridge University Press, Cambridge, 1990). Bawin, S. M., Sheppard, A. R., Mahoney, M. D., Abu-Assal, M., and Adey, W. R., Brain Res. 362, 350 (1986a). Bawin, S. M., Abu-Assal, M. L., Sheppard, A. R., Mahoney, M. D., and Adey, W. R., Brain Res. 399, 194 (1986b). Chan, C.Y., and Nicholson, C, J. Physiol. (Lond.) 371, 89 (1986). Chan, C.Y., Houndsgaard, J., and Nicholson, C, J. Physiol. (Lond.) 402, 751 (1988). Chen, P.-S., Wolf, P. D., Dixon, E. G., Danieley, N. D., Frazier, D. W., Smith, W. M., Ideker, R. E., Circ. Res. 62, 1191 (1988). Chialvo, D. R., and Gilmour, R. F., Jr., and Jalife, J., Nature 343, pp. 653, (1990a). Chialvo, D. R., Nature 330, 749 (1990b). Christini D. J., and Collins, J. J., Phys. Rev. E 52, 5806 (1995). Christini D. J., and Collins, J. J., Phys. Rev. E 53, R49 (1996). Cooper, I. S., Amin, L, Riklan, M., Waltz, J. M., and Poon, T. P., Arch. Neurol. 33, 559 (1976). Ding, M., Yang, Y., In, V., Ditto, W. L., Spano, M. L., and Gluckman, B., Phys. Rev. E 53, 4334 (1996). Ditto, W. L., Rauseo, S. N., and Spano, M. L., Phys. Rev. Lett. 257, 3211 (1990). Ditto, W. L. and Pecora, L. M., Scientific American 269, No. 2, 78 (1993). Garfinkel, A., Spano, M. L., Ditto, W. L., and Weiss, J. N., Science 257, 1230 (1992).

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Glass, L., J. Cardiovas. Electrophys. 1, 481 (1990). Gluckman, B. J., Netoff, T. I., Neel, E. J., Ditto, W. L., Spano M. L., and Schiff, S. J., Phys. Rev. Lett. 77, 4098 (1996a). Gluckman, B. J., Neel, E. J., Netoff, T. L, Ditto, W. L., Spano M. L., and Schiff, S. J., J. Neurophys. 96, 4202 (1996b). Goldberger, A. L., Bhargava, V., West, B. J., and Mandell, A. J., Physica D 19, 282 (1986). Gray, R. A., Jalife, J., Panfilov, A. V., Baxter, W. T., Cabo, C, Davidenko, J. M., and Pertsov, A. M., Science 270, 1222 (1995). Gray, R. A., Pertsov, A. M., and Jalife, J., Nature 392, 75 (1998). Guevara, M. R., Glass, L., and Schrier, A., Science 214, 1350 (1981). Hall, K., Christini, D. J., Tremblay, M., Collins, J. J., Glass, L., and Billete J., Phys. Rev. Lett. 78, 4518 (1997). Holden, A. V., Nature 387, 655 (1997). Hunt, E. R., Phys. Rev. Lett. 67, 53 (1991). In, V., Mahan, S. E., Ditto, W. L., and Spano, M. L., Phys. Rev. Lett. 74, 4420 (1995). In, V., Spano, M. L., and Ding, M., Phys. Rev. Lett. 80, 700 (1998). Jefferys, J. G. R., J. Physiol. (Lond.) 319, 143 (1981). Kannel, W. B., Abbott, R. D., Savage, D. D., and McNamara, P. M., New England Journal of Medicine 306, 1018 (1982). Kaplan, D. T., and Cohen, R. J., Circ. Res. 67, 886 (1990). Katz, B., and Schmitt, O. H., J. Physiol. (Lond.) 97, 471 (1940). Kayyali, H., and Durand, D., Exper. Neurol. 113, 249 (1991). Kirchhof, C, Chorro, F., Scheffer, G. J., Brugada, J., Konings, K., Zetelaki, Z., and Allessie, M., Circ. S^>, 736 (1993). Langberg, J. J., Bolmann, A., McTeague, K., Spano, M. L., In, V., Neff, J., Meadows, B., and Ditto, W. L., in preparation (1998). Lee, J. J., Kamjoo, K., Hough, D., Hwang, C, Fan, W., Fishbein, M. C, Bonometti, C, Ikeda, T., Karagueuzian, H. S., Chen, P. S., Circ. Res. 78, 660 (1996). Lindner J. L, and Ditto, W. L., Appl. Mech. Rev. 48, 795 (1995). Murphy, J. V., Hornig, G., and Schallert, G., Arch. Neurol. 52, 886 (1995). Nakagawa, M., and Durand, D., Brain Res. 567, 241 (1991). Nicolis, C, Tellus 34, 1 (1982). Ott, E., Grebogi, C, and Yorke, J. A., Phys. Rev. Lett. 64, 1196 (1990). Ott, E., and Spano, M. L., Physics Today 48, 34 (May, 1995). Panfilov, A. V. and Holden, A. V. , eds., Computational Biology of the Heart (Wiley, Chichester, 1997). Parmananda, P., Sherard, P., Rollins, R. W., and Dewald, H. D., Phys. Rev. E 47, R3003 (1993). Pedley, T. A., and Traub, R. D., in Current Practice of Clinical Electroencephalography, Second Edition (eds Daly, D. D., and Pedley, T. A.) pp. 107 (Raven Press, New York, 1990). Pei, X., and Moss, F., Nature 379, 618 (1996).

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Pertsov, A. M., Davidenko, J. M., Salomonsz, R., Baxter, W. T., and Jalife, J., Circ. Res. 72, 631 (1993). Petrov, V., Gaspar, V., Masere, J., and Showalter, K., Nature 361, 240 (1993). Petrov, V., Schatz, M. F., Muehlner, K. A., VanHook, S. J., McCormick, W. D., Swift, J. B., and Swinney, H. L., Phys. Rev. Lett. 77, 3779 (1996a). Petrov, P., and Showalter, K., Phys. Rev. Lett. 76, 3312 (1996b). Pierson, D., and Moss, F., Phys. Rev. Lett. 75, 731 (1995). Prystowsky, E. N., Benson, W. J., Fuster, V., Hart, R. G., Kay, G. N., Myerburg, R. J., Naccarelli, G. V., and Wyse, G., Circ. 93, pp. 1262 (1996). Rapp, P. E., Albano, A. M., Schah, T. I., and Farwell, L.A. Phys. Rev. E 47, 2289 (1993). Rogers, J. M., Huang, J., KenKnight, B. H., Smith, W. M., and Ideker, R. E., Circ. 94(8), 1-48 (1996). Roy, R., Murphy, T. W., Jr., Maier, T. D., Gills, Z., and Hunt, E. R., Phys. Rev. Lett. 68, 1259 (1992). Rushton, W. A. H., J. Physiol. (Lond.) 63, 357 (1927). Rutecki, P.A., Lebeda, F.J., and Johnston, D., J. Neurophys. 54, 1363 (1985). Schiff, S. J., Jerger, K., Duong, D. H., Taeun, C, Spano, M. L., and Ditto, W. L., Nature 370, 615 (1994). So, P., and Ott, E., Phys. Rev. E 51, 2955 (1995). So, P., Ott, E., Schiff, S. J., Kaplan, D. T., Sauer, T., and Grebogi, C, Phys. Rev. Lett. 76, 4705 (1996). So, P., Ott, E., Sauer, T., Gluckman, B. J., Grebogi, C, and Schiff, S. J., Phys. Rev. E 55, 5398 (1997). Terzoulo, C. A., and Bullock, T. H., Proc. Nat. Acad. Sci. 42, 687 (1956). Trachina, D., and Nicholson, C, Biophys. J. 50, 1139 (1986). Traynelis, A. F., and Dingledine, R., J. Neurophys. 59, 259 (1988). Van Buren, J. M., Wood, J. H., Oakley, J., and Hambrecht, F. J., Neurosurgery 48, 407 (1978). Wickelgren, I., Science 272, 668 (1996). Wiggers, C. J., Am. Heart J. 5, 351 (1930). Winfree, A. T., Science 266, 1003 (1994). Witkowski, F. X., and Penkoske, P. A., in Mathematical Approaches to Cardiac Arrhythmias, J. Jalife, ed. (New York Academy of Sciences, New York, 1990), pp. 219. Witkowski, F. X., Kavanagh, K. M., Penkoske P. A., Plonsey, R., Circ. Res., 72, 424 (1993). Witkowski, F. X., Kavanagh, K. M., Penkoske, P. A., Plonsey, R., Spano, M. L, Ditto, W. L., and Kaplan, D. T., Phys. Rev. Lett. 75, 1230 (1995). Witkowski, F. X., Leon, L. J., Penkoske, P. A., Giles, W. R., Spano, M. L., Ditto, W. L.,and Winfree, A. T., Nature 392, pp. 78 (1998a). Witkowski, F. X., Leon, L. J., Penkoske, P. A., Clark, R. B., Spano, M. L., Ditto W. L., and Giles, W. R., Chaos 8, 94 (1998b). Yang, W., Ding, M., Mandell, A., and Ott, E., Phys. Rev. E 51, 102 (1995).

Experimental Control of Chaos

18

Experimental Control of Chaos in Electronic Circuits

G. A. Johnson, M. Locher and E. R. Hunt Department of Physics and Astronomy, Condensed Matter and Surface Sciences Program, Ohio University, Athens, OH 45701

18.1 Introduction As the existence of this Handbook suggests, the control of chaotic behavior as a subfield of Nonlinear Dynamics has generated considerable interest and activity in the present decade. Arguably the primary motivation behind the current surge of research in this area is the foreseen applications of efficiently taming systems that otherwise undergo irregular, unpredictable dynamics. Since the world is inherently nonlinear, this class of systems is both large and diverse. Already, areas of promising applications include controlling chaotic behavior in biological systems [1], varieties of lasers [2, 3], chemical systems [4, 5] and mechanical systems including combustion engines [6] and chaotic, fatigue-inducing vibrations in helicopters [7]. The nature of chaos itself makes electronic circuits the ideal testing ground for developing chaos control techniques. The idea behind the control strategies is to gently nudge the system in a manner such that it follows the trajectory of a natural orbit of the system, but one that is unstable. The instabilities in chaotic systems are generally one of just a few types, hence different systems may exhibit similar behavior, and in fact have been shown to display universal properties. Exploiting the similarities in the dynamics of entire classes of systems, the results of simple and inexpensive experiments can be reasonably extended to apply to more realworld applications. A number of existing well-studied electronic circuits have been shown to generate chaos and display the variety of instabilities that are of interest in the chaos control field. In this light, these systems lend themselves well to the task at hand, as they can be built from readily available, inexpensive components and provide simple - yet physical - representations of a whole classes of dynamic systems. The aim of this chapter is to provide the reader with an introduction to a few of the well known and often-used chaotic circuits and the strategies used to control them. For the most part, the chapter follows a chronological development as the systems under study become more complex in nature, ending with the control of a spatiotemporal chaotic system consisting of 32 chaotic subcircuits. Since several excellent and thorough reviews of the many methods for controlling chaos exist in the current literature [8], this chapter is not meant to provide such a summary. Instead, we focus on the development and application of the occasional proportional

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18 Experimental Control of Chaos in Electronic Circuits

feedback, or OPF, control technique, specifically as applied to electronic circuits. In particular, we discuss the control of three different types of instabilities: saddle points, spirals, and convectively unstable states in spatially extended, open-flow systems. Most of the current work in the field is based on the pioneering work of Ott, Grebogi, and Yorke [9] (OGY) in 1990, and the experimental work of Ditto, Rauseo, and Spano [10] in 1991, in which a small feedback signal is generated to perturb a system parameter in a manner that stabilizes an unstable periodic orbit embedded in a chaotic attractor. The main challenge of this work is to preserve the integrity of the system by making only tiny parameter adjustments so that the only real change in the system is its periodicity. In other words, the location of the fixed point of the stabilized orbit is to be the same as that of the unstable fixed point. To do this, the OGY scheme employs a feedback signal which may be expressed as /xn = S • (£n — £F)> where £n typically represents a two-dimensional state vector of the system at the nth piercing of the Poincare plane, and £p the fixed-point location of a periodic orbit. The vector a contains geometrical information in the neighborhood of the fixed point which can be extracted from time-series data. This feedback signal is applied only when the system visits the near vicinity of the fixed point to ensure only small perturbations and valid linear approximations. While the OGY method was devised to be a general control strategy for saddle points in low-dimensional chaotic systems, different types of instabilities have given rise to a number of variations of OGY. Originally developed for highly dissipative chaotic systems, the OPF error signal has the form fin = a(Xn — Xs) where X is a single scalar variable and Xs the set point. Under ideal conditions the set point, a movable reference point, coincides with the fixed point Xp. Similar methods for dissipative systems include that of Bielawski et al. [3] where X n _i replaces Xs, and the recursive method of Rollins et al [4] that includes the OPF term in addition to a second term proportional to /i n _i. For somewhat different systems with spiral fixed points, other strategies were developed such as that of Schuster, Niebur, and the present authors [11]. Another group led by Schwartz [12] has applied variations of the OPF method to the problem of fixed points with multiple unstable directions. These techniques all use discrete-time variables and are well understood in terms of one-dimensional first return maps. For the continuous-time systems discussed in this chapter, we discretize the continuous signals by taking the peaks of the voltage and current traces to be the chaotic variables.

18.2 The OPF Method The OPF control strategy is best described in terms of the first return map X n + i vs. Xn where X is a measurable system variable. Ideally, a system parameter exists that is easy to alter in a time much shorter than one mapping period of the system. A necessary condition of the OPF method is that changes in the parameter result in the return map shifting (at least in part) along the diagonal. A simple relationship can be found between a small parameter change Ap and the horizontal

18.2 The OPF Method

461

displacement, 5X, of the map near the fixed point. For small parameter changes, the relationship is taken to be approximately linear, such that SX = KAp

(18.1)

where K is a system-dependent constant. Provided that the system responds quickly to parameter changes, the control scheme can be implemented to manipulate Xn+\ using the measured value of Xn. Assuming at least a nearly one-dimensional map, we can express Xn+\ in terms of a mapping function F: Xn+1=F(Xn,p).

(18.2)

Prom this simple expression the strategy becomes clear: measure X n , change p to get the desired X n + i , and repeat as necessary. To stabilize the period-1 fixed point (the intersection of the map with the diagonal), AX is measured with respect to the fixed point XF, and the perturbation: Ap = aAX = a{Xn - XF)

(18.3)

is applied for one iteration of the map. Since Eq. 18.1 is valid only for small Ap, no feedback is applied if the magnitude of AX is greater than some threshold value referred to as the control window. To demonstrate, consider the return map of Fig. 18.1, which is a sketch of the map for the diode resonator circuit - which is to be described in more detail later in this chapter. The dissipative nature of the circuit is evident in the thin-line quality of the map, indicating that the magnitude of the stable eigenvalue is very large. Two separate maps are sketched to indicate the dynamics at two different parameter values, demonstrating the necessary movement along the diagonal as the parameter is shifted. Zooming in on the period-1 fixed point, it is seen that the measured variable In (a current) on the lower map is a distance A / from the fixed point IF of the lower map. However, the same value of In on the upper map has maps into an / n + i equal to the value of the fixed point of the lower map (our goal state). Again with this sketch, the strategy becomes clear: measure the variable when it is near the fixed point, shift the parameter such that the system resides on the proper temporary map (the upper in this case), the following map iteration will put the system at IF and the feedback signal at zero. Another way of looking at the control process is to consider the slope of the return map of the system about the fixed point as in Fig. 18.2. If the magnitude of the slope exceeds 1, the point is unstable and iterations starting near the fixed point will fall progressively farther away. In the linear region about the fixed point, the mapping may be expressed in terms of the slope, m, so that A X n + 1 = mAXn.

(18.4)

The addition of the feedback term changes the effective magnitude of AXn by the amount /cAp, so that the we can write = m(AXn - KAP)

(18.5)

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18 Experimental Control of Chaos in Electronic Circuits

n+1

n Figure 18.1 To stabilize the period-1 fixed point, IF, we measure In to get A/ so that

the control parameter can be adjusted to move the system to the upper map where the same In maps into IF of the lower map.

and express an effective slope (the slope with control on) as (18.6) so that optimal control, meff = 0 (every point in the control window maps to the fixed point), is given by Ap = AXn/K. With this picture, we can also calculate the range of the feedback perturbation amplitude which successfully stabilizes the fixed point, namely when |ra e //| < 1, or equivalently, AXn(\m\

-

(18.7)

Summarizing the method with the aid of Fig. 18.2: a) control is initiated as the current is measured at point A, b) the parameter is shifted by an amount Ap, moving the map a distance ttAp, and the system to point A*, c) the next iteration goes to J3* instead of to B and the same procedure can be repeated, this time with the map shifting down (not shown). While not optimized, control renders the fixed point stable in this case because B* is closer to the fixed point than is A, a

18.2 The OPF Method

X n+1

463

m(AX-

Xn Figure 18.2 The slope picture of the OPF stabilization technique. AXn is measured and the corresponding parameter shift moves the map by the amount nAp. Without feedback |AX n +i| is seen to be larger than |AX n |, because \m\ > 1 and AIn+i = mAXn. With feedback AXn+i = m(AXn — «Ap), and the effective slope is given by m e // = m(l — K,Ap/AXn).

condition that will bring each iterate successively closer to XpA note of caution is required at this point. While this control picture for period-1 appears robust, Rollins et al. [4] showed that the diagonal movement of the return map is not always sufficient to implement OPF control. In some cases the magnitude of the shift of the map due to the parameter change depends on the previous shift of the parameter. Hence in their recursive proportional feedback (RPF) scheme, which was employed to control chaotic oscillations in a electrochemical cell, a correction to the OPF was added to take the previous parameter shift into account. When controlling a system on an orbit other than period-1 with the OPF method, a guess is made at the location of the fixed point (referred to as the set-point) and a small perturbation is made any time the system trajectory visits the control window of the set-point. Ideally, the perturbations generally modify the control parameter just a few percent of the original setting. When the set-point coincides with the fixed point of the stabilized orbit, the corrections approach zero. Experimentally, the correction signal can be reduced to a level that is indistinguishable from the typical noise on an oscilloscope trace. For high-period orbits, each perturbation is applied for one drive cycle, and generally more than one perturbation is applied as the longer trajectories may pierce the control window multiple

464

18 Experimental Control of Chaos in Electronic Circuits

times. Again, in this and similar control schemes, a detailed knowledge of the location of the fixed points or system equations is nonessential. Upon implementing control, if no orbit becomes stabilized, the feedback gain can be adjusted or inverted; the window width can be modified; or the set-point can be shifted. In effect, we let the system locate the unstable fixed points of the high-period orbits through a trial-and-error search. The stability of high-period orbits is determined by the product of the local slopes at each point of the return map that is pierced by the orbit. For instance, a period-5 orbit is stable if the magnitude of the product of the five slopes in the vicinity of the five fixed points is less than 1. By locating a control window about one of the fixed points an unstable high-period orbit, the slope in that region of the map can be reduced nearly to zero, hence the product of the slopes can be greatly reduced - thereby stabilizing the orbit. While some of these orbits will be shown to last for dozens of cycles, many of these high-period orbits are not true unstable states of the system. Most likely, the orbits are combinations of a number of orbits that may become mixed as a result of the perturbations that do not approach zero in these cases. The correction pulses slightly deform the region of the attractor within the control window, thereby creating new orbit paths which, depending on the control parameters, may or may not be stable.

18.2.1 Circuit Implementation The circuit which generates the error signal in the OPF scheme is shown in Fig. 18.3. The chaotic signal is input to the circuit as a voltage signal, V*n. The initial input signal is referenced to the set-point voltage controlled by the variable resistance at the first op amp. This difference signal, V{n — Vset, is sent to the window comparator (two LM319 comparators) to determine when the chaotic signal visits the control window of the set-point. The synchronizing signal input is delayed and shaped so that the output of the lower left one-shot (74123) is a narrow square pulse centered about the peaks of the input signal, so that its signal acts as a peak indicator. If the difference signal is in the window and the peak indicator is high, the second one-shot is triggered to send the pictured pulses to the sample/hold and the gate. The difference signal (at its peak) is held by the sample/hold on the negative transition of the pulse shown. The held voltage is then passed through the gate when the gate pulse shown is high, otherwise the gate output is ground. The final stage is the gain control supplied by the op amp inverter with variable gain. The signal from the inverter is sent to the modulation input of the control parameter.

18.2.2 Controlling the Diode Resonator The first system to be controlled using the OPF circuit was the diode resonator [13]. The system consists of a p-n junction diode in series with an inductor (10-100 mH), as in Fig. 18.4, and is typically driven sinusoidally between 50 and 100 kHz.

GATE

18.2 The OPF Method

TIMIN

S/H

o

Figure 18.3 The OPF control circuit. See text for a description of its operation.

465

466

18 Experimental Control of Chaos in Electronic Circuits

The current through the diode undergoes the period-doubling route to chaos as the drive voltage is increased. A current to voltage converter is employed to view the current, and the peak forward current is taken to be a discrete chaotic variable. The mechanism giving rise to chaos is well understood [14] and is attributed to

OPF CIRCUIT Figure 18.4 The diode resonator and feedback systems. The current through the circuit consisting of the inductor and diode is converted to a voltage then put through the OPF circuit. The resulting feedback signal is fed to the external modulation input of the signal generator.

the non-ideal nature of the p-n junction. When the forward current through the conducting diode reaches zero, the diode does not shut off immediately as in an ideal diode. Instead, the diode continues to conduct for a short period known as the reverse recovery time, the time required for the minority charge carriers to diffuse back across the junction. When this well-known property of p-n junction diodes was included in a model of the diode resonator circuit, it was demonstrated that the circuit gives rise to the observed chaotic behavior. To control periodic orbits in the diode resonator system, we make use of the system response to changes in drive voltage, and take / n +i = F(In,V). The two maps of Fig. 18.1 represent the system at two slightly different drive voltage amplitudes, demonstrating the desired movement along the diagonal as discussed previously. The feedback signal is applied to the external amplitude modulation

18.2 The OPF Method

0.6

-

467

\

A

v. 0.4 n+l

0.2 -

an 0.0

•

0.2

0.4

.

i

0.6

Figure 18.5 The return map of the system with feedback gain just below the level necessary to stabilize the period-1 fixed point.

input of the signal generator as in Fig. 18.4. In Fig. 18.1 it is shown how to reach the fixed-point in a single iteration. The feedback system is somewhat forgiving, however, in that if the feedback gain a is near to but not precisely at the right value, the system does not reach the period-1 fixed point in one step, yet the point becomes attracting so that each iteration brings the system closer than the last, as in Fig. 18.2. To illustrate, Fig. 18.5 shows what happens when the gain is just below the critical level necessary to stabilize the period-1 fixed point. The location of the control window is evident as the slope of the attractor is flattened in this region. The points entering the window slowly step outward away from the center, then leave the window for a number of iterations before repeating the sequence. The density of points in the window and the slope indicate that the gain is quite near its minimum value sufficient to stabilize the fixed point. In controlling low-period orbits, a single correction per orbit is sufficient in order to keep the orbit stable, and the correction can usually be reduced to the noise level. However, the nature of chaos itself limits the length of these singlecorrection orbits, and for longer orbits multiple corrections are often required. On average, a correction is applied every 6 or 7 cycles in the diode resonator system. This number varies with different systems due to noise levels and the nature of the system orbits. The somewhat unexpected result of the diode resonator experiment was the stabilization of high-period orbits of lengths up to 28 drive cycles. The control of these orbits is essentially achieved by a trial-and-error search. The control window position and width, as well as feedback gain, can be varied until orbits

468

18 Experimental Control of Chaos in Electronic Circuits

0.55

Figure 18.6 A period-13 orbit in the diode resonator circuit. The upper figure is the return map and the lower figure shows the time series data of the current through the diode and the error signal required to stabilize the orbit.

become stable. Two typical high-period orbits are shown in Figs. 18.6 and 18.7

18.3 Controlling Coupled Diode Resonators Because the OPF method appears tailored to controlling low-dimensional, highly dissipative systems with a single unstable direction, the versatility of the technique hinges upon its extension to systems of different nature. Therefore we now shift our attention to stabilizing non-saddle type fixed points. In particular, we extend and modify the OPF method to control a system comprised of two coupled diode resonators. The system provides unstable spiral fixed point, i.e., a complex conjugate pair of eigenvalues with a real part greater than 1. The coupled diode resonator system [15] is a parallel combination of two diode resonators in series with a resistor, which provides a coupling between the currents through the two diodes, as in Fig. 18.8. Taking the two currents as system variables, we observe that the system period doubles once, then undergoes a Hopf bifurcation into a quasiperiodic state as the drive voltage is increased. Contin-

18.3 Controlling Coupled Diode Resonators

469

0.60

0.00

0.00

0.20

0.40

0.60

In

I -

Figure 18.7 A period-15 orbit stabilized with 3 corrections.

uing, the quasiperiodicity gives way to mode-locked states, often a second Hopf bifurcation, and finally chaos. To view the dynamics of the system, we produce a display of the Poincare section, i.e., the peak currents through one branch versus the peak currents through the other, as shown in Fig. 18.9. The section for this period-doubled quasiperiodic system appears as two circles immediately following the Hopf bifurcation. A smooth rotation of the points making up the circles occurs in the quasiperiodic regime giving way to stationary points in the phase-locked regime. As the critical line is approached, the circles deform via folds, signaling the break-up of the torus and the transition to chaos. As described elsewhere [16 - 20], a number of systems which undergo a Hopf bifurcation have been stabilized in steady states by two different methods, a modified OPF method and derivative feedback. In these cases, the systems are autonomous and the state preceding the Hopf bifurcation is a non-oscillating state of steady light intensity or a constant current or voltage. Here we demonstrate that the spontaneous oscillation is indeed controllable with the unmodified OPF control scheme [21], suppressing the Hopf bifurcation while retaining a dynamic period-2 state. To control the double resonator circuit, we take either of the two currents to be the control variable and apply the OPF method outlined in control of the single diode resonator. The perturbation in this system is again applied in the

470

18 Experimental Control of Chaos in Electronic Circuits

Figure 18.8 The double diode resonator circuit.

form of amplitude modulation of the signal generator. This state can be stabilized in either the quasiperiodic or chaotic regimes. The double exposure photograph in Fig. 18.10 shows the stabilized orbit embedded in the chaotic regime. With proper positioning of the set-point, this state is stable with a feedback signal which approaches zero down to the normal noise level.

18.3.1 On Higher Dimensional Control The results here leave the question open: when will the method fail? To further explore this question, the work of Schuster and Niebur combined with another experiment by the present authors show that the ability to control higher dimensional systems depends only on a) experimenters' ability to sample enough system variables and b) a single detail about the nature of the system response to parametric perturbations. The theoretical work of Schuster and Niebur outlines the necessary and sufficient conditions for controlling an arbitrarily high-dimensional system. The reader is encouraged to see Ref. [11] for a detailed development. Essentially, it is shown that perturbations delivered to a single parameter r in the form, rn =e-

(xF

-xn),

(18.8)

when |xp - x n | is smaller than some threshold (i.e., the system enters a small control window), are sufficient to stabilize the fixed point xV for a properly chosen set of strengths €i(i = l,...,d) for d dimensions. The vector x n = (xi,x2, ...,xd) represents a complete sampling of the system under study. Interestingly, this proof also leads to the result that in e-space (which the authors call parameter space) the region in which control is achieved has d + 1 corners, e.g., a triangular area for d = 2. Furthermore, the locations of these corners can

18.3 Controlling Coupled Diode Resonators

471

Figure 18.9 The dynamics of the coupled diode resonator circuit. As the drive voltage is increased the system period-doubles (top), Hopf bifurcates to a quasiperiodic attractor (middle), then eventually becomes chaotic (bottom).

be used to calculate the local Liapunov exponents of the fixed point. In light of these results, we set up our experiment to verify these predictions about the shape of the control region and the magnitude of the exponents. In the new experiment, to control the fixed-point current, I F , the perturbations delivered to the amplitude modulation of the drive voltage have the form SVn = (ci,c2) • (IF - In)

(18.9)

where I n = (7i,/2) n , as labeled in Fig. 18.8. Because the system has perioddoubled prior to the Hopf bifurcation, we sample the system every other cycle, reducing the Poincare section to a single circle. Likewise, we apply the control signal for a period of two drive cycles for the most robust control. Experimentally varying the feedback strength vector, ?, we see that the region of control is indeed

472

18 Experimental Control of Chaos in Electronic Circuits

Figure 18.10 The pre-Hopf period-2 state is shown stabilized in the chaotic regime in this double exposure photograph of h vs h-

triangular in parameter space, as predicted by theory, shown in Fig. 18.11. Note also that in agreement with our previous experiment, if t^ = 0, control can still be achieved over a range of t\. To further probe the theory, we can also check the local Liapunov exponent experimentally and as calculated by the theory - using the corners obtained in Fig. 18.11. Experimentally we measure the rate at which the system spirals away from the fixed point as the control signal is switched off. Figure 18.12 shows the deviation from the fixed point as a function of time, when the control is released at t = 0. We find the indicated slope to be 0.16 +/- 0.02 per two drive cycles, which corresponds to Re(A) = 1.45. After about 30 time steps (60 drive cycles), the system settles onto a mode-locked orbit of period-14 with an associated winding number of ~ . Therefore, looking at one of the two circles of the Poincare section gives us Im(A) = 2TT X f. The ratio Im(A)/Re(A) = 7.3 can be obtained from the corners in parameter space, and agree quite well at 7.1. The second condition that determines the controllability of high-dimensional systems has to do with how the fixed point moves as the parameter is shifted. Control will it fail if the movement is orthogonal to even one of the unstable directions. As yet, it is unseen just how limiting this restriction might be, because the control of these types of systems is just now beginning to be studied more intensely. Perhaps the greater obstacle will be sampling a sufficient number of

18.4 Controlling Spatiotemporal Chaos

473

Figure 18.11 The controllability region in e-space maps into a 3-cornered area as predicted by the theory of Schuster et al. The system is controlled in its period-2 state at all points inside the region bound by the plotted points.

variables as the systems become more complex.

18.4 Controlling Spatiotemporal Chaos The successful control of low-dimensional chaos in real systems has naturally piqued an interest in the controllability of spatially extended chaotic systems. With possible valuable applications on the horizon, the initial work in this field has begun. At first glance, the behavior of physical systems which exhibit spatiotemporal chaos, such as a turbulent fluid flows, might be expected to exhibit similarities to simple low-dimensional chaotic systems. Yet as one digs deeper, it is clear that the concepts associated with low-dimensional chaotic systems are themselves unable to cope with spatially extended systems. Still, low-dimensional chaotic systems have been employed in the study of turbulence in the following way. Systems comprised of a large number of low-dimensional chaotic systems coupled in various configurations provide spatially discrete, yet extended, systems which may exhibit

474

18 Experimental Control of Chaos in Electronic Circuits

0.100 —i

0.010 -

0.001 —

10

20 30 40 time (two drive periods)

50

60

Figure 18.12 Experimental determination of the local Liapunov exponent for the coupled diode resonator system.

temporally complex behavior and have spatial disorder due to the interaction between the chaotic system elements. Because there exists a certain degree of universality in the behavior of the classes of systems, modeling real physical, chemical, and biological systems exhibiting spatiotemporal dynamics with much simpler systems of the same class can provide a means of understanding the less-surmountable systems. This modeling has taken a number of forms ranging from partial differential equations (PDEs) to discrete models such as coupled maps and cellular automata - discrete in time, space and field. The amount of quantitative information lost in the discretization of models is somewhat of an open question [22], however, the universal properties are presumed to remain insensitive to the simplifications. The choice of model and relevance of the observations to the class of real systems is crucial in the study of spatiotemporal chaos.

I8.4 Controlling Spatiotemporal Chaos

475

18.4.1 Open Flow Systems We will concentrate here on the class of spatiotemporal systems known as open flows, that is systems with a single preferred direction in which to propagate information. Often-cited examples of physical open flows include fluid flow on a flat plate, channel flow, fluid flow in a pipe, and rising smoke. Perhaps the simplest system to display the properties of the open flow class is the 1-dimensional lattice of coupled logistic maps, as first introduced by Kaneko [23]. This particular coupling configuration was taken to hold as a simple model of fluid flow experiments in which the dynamics downstream from a laminar source exhibit complex dynamics as induced by small disturbances originating in the laminar regime. Kaneko's work focuses on the extreme asymmetrically coupled lattice that can be expressed as

where e = [0,1), i = space, n = time, and f(x) = rx{\ — x). The result is a system which, under certain parameter settings, exhibits phenomena, such as convective instability and spatial period doubling, similar to that observed experimentally in fluidflowsystems. The term convective instability is used to describe states in a system in which a perturbation grows in time and moves in space, such that the perturbation is temporary in a stationary reference frame. The main consequence of convective instability is that spatially homogeneous states in extended systems of identical elements are allowed only in the periodic parameter regimes of the single, uncoupled system element. For example, this means that increasing the parameter past the first period-doubling bifurcation of the logistic map renders the period-1 solution unstable throughout the system of coupled maps. Even anchoring the first site to the unstable fixed point in this case will fail to stabilize the solution beyond a limited region in space. Because the first site will never be exactly on the fixed point (because of round-off error), the system is provided with an ever-present, finite perturbation which grows exponentially in space, destroying any spatial coherence or stationary structure. In this light, it is easy to envision the effects of noise amplitudes typically associated with real physical systems. In systems with convective instabilities, the dynamics far from the source will be dominated by so-called noise-sustained structure, macroscopic fluctuations generated purely by stochastic processes that become amplified in space. Noise-sustained structures have been found in a number of systems with effectively only one spatial dimension; in both experiment [24-27] and numerical simulations [28-30]. Another property of openflowsystems is the phenomenon of spatial period doubling. A convectively unstable state of period 2m(m = 0,1,2...) becomes unstable by giving way to a period 2 m + 1 state as the spatial index is increased. After a number of spatial bifurcations, the dynamics far downstream become highly complex, apparently chaotic. Typically, the more noise in the system, the fewer bifurcations are observed before the onset of complex behavior. The whole scenario, from

476

18 Experimental Control of Chaos in Electronic Circuits

a period-1 to chaos as the spatial index increases, is considered analogous to a fluid transforming from laminar to turbulent flow. While the connection may appear loose, in fact, spatial period doubling is observed in a number of fluid flow experiments, see for example Gollub et al. [31] or Schatz et al. [24]. In these experiments, a single spatial bifurcation from a period-1 to period-2 is observed, prior to a turbulent state.

18.4.2 The Diode Resonator Open Flow System Our experimental system consists of 32 coupled diode resonator circuits driven by a sinusoidal source at a frequency of 70 kHz. The individual circuits are comprised of the series combination of a 30 mH inductance and a General Instruments 852 silicon diode. The output of the signal generator is applied to a bus from which 32 inverting amplifiers take their input signal, and in turn drive the individual diode resonator circuits. The diodes are matched based on their bifurcation sequences such that they all go chaotic at nearly the same drive voltage. The voltage across the diode, V^, is one of the chaotic variables. Placing a buffer and resistor between neighboring diodes, as shown in Fig. 18.13, provides a one-way coupling proportional to the difference (VV - Vdt~1) where i represents the spatial index. The coupling strength is determined by the coupling resistor, ReThe circuit at site-z receives an injection (or drain) of current whenever either the iih or the i-lth diode (or both) is in the nonconducting phase. When neighboring diodes are both conducting, the voltage drops are nearly equal so that only a negligibly small current flows through Re-

Figure 18.13 The openflowcircuit consisting of 32 diode resonator circuits. The unidirectional coupling is provided by a buffer and a resistor, Re- The current allowed toflowto the right is given by (V^"1— VJJRQ 1 .

18.4 Controlling Spatiotemporal Chaos

477

Dynamics of the Open Flow System The behavior of the coupled systems in the parameter range in which the first diode resonator is in the chaotic regime is shown in Fig. 18.14a. The figure shows several snapshots of the system by plotting the peak currents through all the diodes at simultaneous measurements. The coupling between sites, which is given by R = 18.2 kft in this case, has a synchronizing effect which slaves the first several sites to the dynamics of the first, as indicated by the flat regions at low site-numbers. The chaotic nature of the individual circuits works against the coupling effects and eventually destroys the coherent (flat) region and produces complex, highdimensional dynamics downstream. Keeping the drive voltage the same, if we fix the first diode resonator to the unstable period-1 fixed point, the following few sites assume the period-1 orbit, then give way to a period-2 orbit for a number of sites, and eventually periodic behavior is destroyed for high site-number. Upon close inspection, the dynamics at the 32nd site are virtually identical for the either boundary condition at the 1st site. The phenomenon of spatial period doubling, as shown in Fig. 18.14b, provides a level of similarity with the fluid flow experiments referenced above. We demonstrate the convective instability of in the system with the following method. While driven in the chaotic regime, the first site is controlled on the period-1fixedpoint. The coupling synchronizes a number of sites depending on the coupling strength. For a coupling resistance of 18.2 kfi, roughly eight sites remain in a period-1 state. Now we perturb the second site for one drive cycle with a small square pulse and observe the effect downstream. (We apply the perturbation to the second site because the first site is under the influence of the control circuit.) Figure 18.15 shows the time-averaged current peaks at sites 2-7 while applying a well-separated series of square pulses. The perturbation is barely perceptible at site-2, but is dramatically amplified by the time it reaches site-7. In the figure the maximum of the disturbance moves to the right as the site-number increases, revealing the velocity of the reference frame in which the system is linearly unstable.

18.4.3 Control A Short History Because of their simplicity, numerical systems consisting of coupled map systems have provided a convenient testing ground for pioneering work in the area of spatiotemporal chaos control. Auerbach [32] first demonstrated the stabilizing of convectively unstable periodic states in an open flow model using modified lowdimensional feedback techniques. By employing feedback based on local information to sites distributed periodically in space, spatial period doubling may be suppressed and coherence maintained throughout the length of the lattice of quadratic maps, f(x) = 1 - ax2. The resulting states are temporally periodic and spatially uniform. In her procedure Auerbach stabilizes the first map, which induces a limited region of coherence, before giving way to spatial period doubling. By applying a control signal to a site in the coherent region, shortly before the point of insta-

478

18 Experimental Control of Chaos in Electronic Circuits

0.4.-

-0.4 -

30

Figure 18.14 The spatial picture for different boundary conditions in the experimental openflowsystem, a) The first circuit is allowed to evolve chaotically, and b) the first circuit is controlled on the period-1fixedpoint. In either case, the dynamics far downstream are virtually identical.

bility, the noise amplification of the system is negated. Prom this second control point in space, a second coherent region is formed and the process can be repeated until the entire system is spatially uniform. Meanwhile, Gang and Zhilin [33] have stabilized very different states that are periodic in time and space in an otherwise chaotic coupled map lattice. In this case the system possesses symmetric coupling and periodic boundary conditions - hence no preferred direction of propagation. Similar to the Auerbach method, a number of controllers are used, distributed throughout the spatial axis. Methods have also been successfully developed to control or suppress chaos in spatially continuous media described by nonlinear, one-dimensional partial differential equations. Gang and Kaifen [34] have applied two methods to control a chaotic, drift-wave equation with applications in plasma physics. They transform the equation into a set of twenty-six ordinary differential equations, then force, or

18.4 Controlling Spatiotemporal Chaos

479

•a

Site 2 10

20 time (cycles)

30

40

Figure 18.15 At t = 0 a perturbation is delivered to the second diode resonator while the first is controlled in a period-1 state. The convectively unstable nature of the system is evident as the perturbation grows in amplitude as it moves from site-2 to site-7.

pin, one to a reference state. They do this at a single point in x-space or a single point in k-space, which they refer to as pinning and monochromatic wave-injection, respectively. The state which they stabilize is a traveling wave in a closed ring. Aranson et al. [35] have also demonstrated a single-point control strategy for a PDE system. They apply their control scheme to a variation of the Ginzburg-Landau equation with periodic boundary conditions. The control is sufficient to stabilize extended systems of limited size. Their one-dimensional system is partially stabilized by pinning a point in x-space to a constant amplitude. The presence of this stationary 'hole' has a stabilizing effect in both directions on a finite region of the surrounding space. As well as being comprised of physical components, the system of coupled diode resonators represents an intermediate class of discreteness. The spatial realm is clearly discrete, while the local dynamics and the coupling are continuous. Also in

480

18 Experimental Control of Chaos in Electronic Circuits

contrast to the above numerical experiments, we present in this chapter, methods for controlling the spatial and temporal dynamics of openflowsystems of unlimited size with a single control point in space [36]. Control of the Open Flow System By employing periodic boundary conditions in our system, i.e. coupling the last site to the first, we see behavior that appears quite different from the open flow system. In fact, it is apparent that there is a quasiperiodic nature induced in the individual diode resonators. At higher drive amplitudes the maps can reveal very rich, high dimensional dynamics and chaos. We also find windows of temporal periodicity with lengths from 5 to 100s of drive periods. Spatially, these states form wave-like patterns that have wavelengths to fit the lattice size. Changing the lattice size reveals similar states, often with different temporal periods and spatial wavelength. The important result is that we have established the existence of stable, spatially inhomogenous states in a system that differs from the open flow system only by its boundary conditions. This observation provides the basis of the control strategy of the open flow system, as we expect that the same states are stable in the open flow system given the proper conditions at the first site. Therefore, our strategy for stabilizing the open flow system is to mimic the dynamics of the mode-locked states of the closed loop at the first site of the open flow. We do this one of two ways: 1) by applying the OPF chaos control technique to site-1, or 2) by simulating a 'site-0' with a wave generator, i.e. applying a variable frequency sine wave through a resistor to the first diode. Typically, once a well-chosen periodic orbit at first diode resonator is established, the system connects to the stable spatiotemporal state through a number of transient sites. Making use of the OPF controller, we apply the strategy above and find that controlling certain periodic orbits at site-1 results in stable spatial waveforms which extend throughout the length of the lattice of diode resonators. An example is given in Fig. 18.16 in which the first site has been stabilized into a period-5 orbit. The spatial structure of the period-5 state is characterized by a wavelength of approximately 8.3 sites so that there is an average phase shift of about 43 degrees between sites at a given time. The return map of a single diode resonator shows that the temporal winding number of the individual elements of the system is 2/5, i.e. the attractor viewed as a circle is traversed twice in five drive cycles. These quantities dictate the evolution of the system illustrated by the successive iterates shown in the figure. We also are able to stabilize the system with the injection of a sine wave of appropriate frequency. By applying the sine wave to the system, the periodic voltage simulates the voltage across the diode of an additional diode resonator, or 'site-0.' Some of the extremely high-period (> 100) locked states observed in the closed loop can be reproduced with this method. For example, in the closed-loop system with R = 8.2 kfi, we observe a locked state of period-31 with a winding number of 11/31 and a spatial wavelength of 16 sites. Returning to the open flow system, we inject into the first site a sine

18-4 Controlling Spatiotemporal Chaos

481

I

o

8.

30

Figure 18.16 The spatial waves associated with a controlled period-5 at site-1 in the diode resonator system. After a short spatial transient, this spatiotemporal state is characterized by a wavelength of approximately 8.3 sites and a temporal winding number of 2/5. Every site is in a period-5 orbit, so the subsequent iterates plotted repeat everyfivecycles.

wave of frequency equal to the product of the winding number (.3548...) and 70 kHz (24.838... kHz). All sites in the lattice assume a period-31 orbit, and again a few sites connect the sine wave at site-0 to the previously observed stable state downstream. This example is illustrated in Fig. 18.17. The topmost return map is the period-31 locked state as seen at the 13th lattice site. The middle map shows the same site when the system is injected with 24,838 Hz, again resulting in a stable period-31 orbit at every site. Finally, the bottom return map illustrates the chaotic state which results under the same conditions in the open flow without the sinewave input. The only critical control parameter in this case is the frequency of the sine wave in relation to the 70 kHz drive. When the ratio of the frequencies is rational and within the range of observed winding numbers, the periodic states will nearly always stabilize.

482

18 Experimental Control of Chaos in Electronic Circuits

0.6 ••

m

-

t •

0.3 -

•

t

#

1

• m

•

Loop

0.0 0.6 t

* •

•

I

-

•

•

0.3 n+1

t

Sine

0.0 0.6

" Vf2:

0.3

•

Open.

0.0 0.0

0.3 I

0.6

Figure 18.17 The 'site-0' (Sine) method in stabilizes a period-31 orbit which exists in the closed-loop system (Loop). The same parameters in the uncontrolled open flow results in a highly chaotic state (Open).

18.5 Conclusions

483

Properties of the Stabilized States While the focus of this section is on the control of the openflowsystem, the nature of the spatial waveforms should be briefly addressed as it is crucial to the stability of the encountered states. The stabilized spatial waveforms have two immediately striking properties: 1) long-range stability and 2) insensitivity to noise. In the experimental system, the waveforms are stable throughout the 32-element system. This is in sharp contrast to the very short coherence length of the coherent period-1 state. We attribute these properties to the presence of traveling phase kinks [37] which make up the wave-like patterns in the open flow system. As is well known in the case of symmetric coupling [38], coupled map lattices demonstrate stable spatial domains, bordered by stationary kink/antikink pairs, in the otherwise chaotic regime. The kinks are defined essentially as the border between groups, or domains of lattice elements that are 180° degrees out of phase in the sense that adjacent groups are typically in or near a period-2 state of opposite phase. Depending on the coupling, the kink widths can spread across a number of system elements, and when packed together form the observed wave-like patterns. In simulations it is easy to observe that from this stable, symmetric state, the kinks can be made to propagate by making the coupling stronger in one direction (which becomes the direction of motion). While in our open flow system of unidirectionally coupled diode resonators the kinks tend to quickly propagate out of the lattice, the proper controlled orbits at the first site serve as a constant source of kinks. Roughly speaking, this condition is met when the orbit periodically visits the vicinity of both period-2 fixed points in an odd number of cycles. Because the kinks themselves are stable and move a constant speed, if the kinks are created with sufficient frequency, the system consists solely of stable structure and hence becomes temporally periodic everywhere. An example, nicely demonstrating the underlying period-2 behavior, is presented in Fig. 18.18. In the open flow system with R = 18.2 kfi, a period-9 orbit is controlled via OPF at the first site. The lines connect points of simultaneously sampled currents through the individual circuits. The kinks are evident in the step-like nature of the spatial dimension as the stable structure sweeps through the system.

18.5 Conclusions The stabilization of different types of unstable, regular behavior in a sampling of different electronic systems was presented. The circuits presented represent virtually all of the types of dynamics associated with low-dimensional chaos. It was demonstrated that the feedback techniques presented were capable of stabilizing unstable fixed points as well as high-period orbits which in some cases may be created by the mixing of a number of orbits. Similarly, it was demonstrated that fixed points which have no stable direction may be controlled with a single parameter,

484

18 Experimental Control of Chaos in Electronic Circuits

o

Figure 18.18 The period-9 stable state. The fiat regions between the kinks reveal the period-2-like behavior as seen in the closed-loop systems.

supporting recent investigations into this subject. With these and further explorations in controlling the dynamics of electronic circuits, the limits of the presented control techniques may be examined, allowing the appropriate application of these findings to more physically complex and real-world systems. The second half of this work consists of the study of open flow systems, and represents the first experimental control of this class of spatiotemporal chaos. The open flow is modeled by coupling a large number of diode resonators in a unidirectional chain or lattice. While the individual elements of the system exhibit high-dimensional dynamics dominated by fluctuations 'upstream' which grow exponentially as they pass through the system, by applying control techniques to the first element, the entire system may become periodic. Because it is expected that the different classes of spatiotemporal systems have some universal properties, it is possible that similar, simple-to-apply control techniques may someday prove useful in related flow systems.

References

485

References [1] Garfinkle, A., Spano, M. L., Ditto, W. L., and Weiss, J. N., Science 257, 1230 (1992). [2] Roy, R., Murphy, T. W., Maier, T. D., Gills, Z., and Hunt, E. R.,Phys. Rev. Lett 68, 1259 (1992). [3] Bielawski, S., Derozier, D., and Glorieux, P., Phys. Rev. A47, 2492 (1993). [4] Rollins, R. W., Parmananda, P., and Sherard, P., Phys. Rev. E47, 780 (1993). [5] Petrov, V., Gaspar, V., Masere, J., and Showalter, K., Nature 361 240 (1993). [6] Rhode, M. A., R. W. Rollins, Markworth, A. J. Edwards, K. D., Nguyen, K., Daw, C. S. and Thomas, J. F., J. Appl Phys. 78 (4), 2224 (1995). [7] Hill, W. S., private communications. [8] For example, see Ditto, W. L., and Pecora, L. M., Scientific American Aug. 1993, 78; Shinbrot, T., Advances in Physics 44, 73 (1995); Hunt, E. R., and Johnson, G., IEEE Spectrum Nov. 1993, 32. [9] Ott, E., Grebogi, C, and Yorke, J. A., Phys. Rev. Lett. 64, 1196 (1990). [10] Ditto, W. L., Rauseo, S. N., and Spano, M. L., Phys. Rev. Lett. 65, 3211 (1990). [11] Schuster, H. G., Niebur, E., Hunt, E. R., Johnson, G. A., and Locher, M., Phys. Rev. Lett. 76, 400 (1996). [12] Carr, T. W., and Schwartz, I. B., Phys. Rev. E50, 3410 (1994). [13] Hunt, E. R., Phys. Rev. Lett. 67, 1953 (1991). [14] Rollins, R. W., and Hunt, E. R., Phys. Rev. Lett. 49, 1295 (1982). [15] Su, Z., Rollins, R. W., and Hunt, E. R., Phys. Rev. A40, 2689, 2698 (1989). [16] Gills, Z., Iwata, C, Roy, R., Schwartz, I. B., and Triandaf, I., Phys. Rev. Lett. 69, 3169 (1992). [17] Parmananda, P., Rhode, A., Johnson, G. A., Rollins, R. W., Dewald, H. D., and Markworth, A. J., Phys. Rev. E 49, 5007 (1994). [18] Bielawski, S., Bouazaoui, M., Derozier, D., and Glorieux, P.,Phys. Rev. A47, 3276 (1993). [19] Johnson, G. A., and Hunt, E. R., IEEE Trans. Circ. Syst. 40, 833 (1993). [20] Johnson, G. A., and Hunt, E. R., J. Circuits Syst. Comput. 3 109 (1993).

486

References

[21] Hunt, E. R., and Johnson, G. A., Proc. 2nd Experimental Chaos Conference (World Scientific, Singapore, 1995). [22] Cross, M. C , and Hohenberg, P. C , Rev. Mod. Phys. 65, 950 (1993). [23] Kaneko, K., Phys. Lett. 111A, 321 (1985). [24] Liu, J., and Gollub, J. P., Phys. Rev. Lett. 70, 2289 (1993). [25] Schatz, M. F., Tagg, R. P., Swinney, H. L., Fischer, P. F., and Patera, A. T., Phys. Rev. Lett. 66, 1579 (1991). [26] Babcock, K. L., Ahlers, G., and Cannell, D. S., Phys. Rev. Lett 67, 3388 (1991). [27] Tsameret, A., and Steinberg, V., Phys. Rev. Lett 67, 3392 (1991). [28] Deissler, R. J., J. Stat Phys. 40, 371 (1985). [29] Deissler, R. J., Phys. Lett 100A, 451 (1984). [30] Deissler, R. J., and Kaneko, K., Phys. Lett 119A, 397 (1987). [31] Liu, J., and Gollub, J. P., Phys. Rev. Lett 70, 2289 (1993). [32] Auerbach, D., Phys. Rev. Lett. 72, 1184 (1994). [33] Gang, H., and Zhilin, Q., Phys. Rev. Lett. 72, 68 (1994). [34] Gang, H., and Kaifen, H., Phys. Rev. Lett. 71, 3794 (1993). [35] Aranson, I., Levine, H., and Tsimring, L., Phys. Rev. Lett. 72, 2561 (1994). [36] Johnson, G. A., Locher, M., and Hunt, E. R., Phys. Rev. E51, R1625 (1995). [37] Dodd, R. K., Eilbeck, J. C , Gibbon, J. D., and Morris, H. C , Solitons and Nonlinear Wave Equations, (Academic Press, London, 1982). [38] Jackson, E. A., Perspectives of nonlinear dynamics, (Cambridge University Press, New York, 1991).

19 Controlling Laser Chaos P. Glorieux Laboratoire de Physique des Lasers, Atomes et Molecules, Centre d'Etudes et de Recherches Lasers et Applications, Universite des Sciences et Techniques de Lille, 59655 Villeneuve d'Ascq Cedex (France)

19.1 Introduction As far as applications are concerned, lasers and more generally optics appear as a very attractive field in which the concept of controlling chaos could find its full relevance. Lasers indeed are known to present spontaneous instabilities which sometimes reduce and even cancel the field of their possible applications [1]. In this respect, it is not surprising that physicists quickly tried to apply the suggestions of Ott, Grebogi and Yorke (OGY) to overcome the drawbacks of some optical systems whose chaotic pulsations make useless for applications [3]. Besides the OGY main stream for control of chaos via unstable periodic orbits (UPO), there have also been several demonstrations of controlling laser chaotic dynamics by the use of other kinds of small perturbations. For instance, it has long been known that parametric modulation could stabilise unstable states and advantage can also be taken from this approach to control laser dynamics. In this review, we want to show how the seminal ideas of OGY and other techniques of controlling chaos have been implemented in experimental devices and discuss to what extent they could lead to better device performances. We present here the achievements of the different investigations pursued on lasers and similar systems and restrict this review to experimental approaches. The nature of the control method or of the kind of lasers to which it has been applied may be used as classification criteria but it is surprising to discover that almost all studies were carried out on the so-called class B lasers and that almost each laser has been stabilised by a different control technique. Therefore the above discussion becomes irrelevant and the only sensible classification is that which separates feedback from non feedback techniques[5]. In the former the correction signal is designed by using some variable information extracted from the dynamics of the system, contrary to the nonfeedback techniques in which a periodic modulation is applied to a suitable control parameter to suppress the undesirable behaviors. Sometimes the distinction between invasive and noninvasive methods of control of chaos is made. Indeed the former change the stationary states and their stability while the later do not alter the stationary solutions but only their stability. Experiments in which parametric noise is added may also be regarded as a particular case of nonfeedback techniques while synchronisation of chaos may be considered as a generalised feedback technique since chaotic information is ex-

488

19 Controlling laser chaos

tracted from one (sub-) system and sent to another one and possibly vice-versa. These synchronisation techniques are not discussed here since their goal is rather to coherently drive chaotic systems rather than to suppress chaos. Recent developments in this domain have also recently been achieved both numerically and experimentally. Before going further into the presentation it may be interesting for non specialists to get acquainted with what we call here class B lasers, their typical observables and control parameters and some of their technical properties such as typical time scales and dimensionality[1]. The laser models presented in the first section are not required to implement the control technique but they are necessary to understand some of the limits or specificities of the methods discussed here. All these elements collected in the first part of this review may be skipped by the laser specialist. The two following parts are devoted to feedback and nonfeedback realisations of control of chaotic lasers and applications of chaos control in lasers are reviewed in the last section.

19.2 Class B lasers 19.2.1 The singlemode class B laser A laser is an optical oscillator in which coherent radiation is generated by stimulated emission of radiation from quantum systems such as electrons, atoms or molecules generally described as the "atomic medium" and considered in a first step as two-level systems. Therefore laser dynamical variables are separated into field and "atomic" variables. For the electric field, the simplest situation is that of the singlemode laser in which only one field frequency may develop in the laser cavity, therefore reducing the complexity of the system and leading more easily to stable operation, which is the goal of most laser designers. Most laser models use the mean field limit in which the electric field is assumed to be uniform inside the optical cavity. Then it may be described by a single variable in a monomode laser. In absence of external injection, there is a phase invariance for the field equation and it is sufficient to consider only the laser intensity M. Note also that in most experiments the polarisation of the laser is fixed, therefore changing the problem of vectorial dynamics into a much simpler one which deals only with scalar variables. The atomic variables involved in a laser are the population inversion N and the electric polarisation. This atomic polarisation acts as a source term for the laser fields. In class B lasers, it relaxes towards its equilibrium much faster than the other variables and is usually adiabatically eliminated from models of these lasers. In the simplest model of a singlemode class B laser, the only relevant variables are the laser intensity M (electromagnetic field variable) and the population inversion density TV (atomic variable). These variables are coupled by the stimulated emission process. Under these conditions, they are described by the so-called "rate equations", i.e. a set of two ordinary differential equations using the notations of

19.2 Class B lasers

489

[1]

dN —

=

-7{l(N-N0)-(30BMN

(19.2)

where 2K is the inverse lifetime of the photons inside the cavity and 7y that of the population inversion. No is the population inversion in absence of laser action. This parameter characterizes the efficiency of the pump process which provides energy to the laser.Throughout derivations of the different models of lasers may be found in[l]. Systeml9.1 can readily be put in an adimensional form, e.g. ^ dr dn — dr

= =

-Gm(l-n)

(19.3)

, _x A-n(m+l)

where the time is expressed in units of 77 1 . A model as simple as (19.1) is efficient to describe such lasers as the singlemode CO2 and YAG:Nd3+ lasers. Semiconductor lasers, which also belong to class B are better described in a model including the intensity dependent linewidth materialised by the linewidth enhancement factor a as discussed below. Other refinements have been introduced to achieve a more reliable description of laser dynamics for instance to take into account the many rotational levels of vibrational manifolds coupled to the lasing levels in a CO2 laser (see below laser with saturable absorber or with feedback). Table 19.1 summarizes and compares parameter values of the most common class B lasers. Tc = 1/2K is the photon lifetime inside the cavity which scales equations (19.1) and T\ = 7i71is the population inversion lifetime. Note that the ratio 7 of these two times is always much smaller than unity and is often used as a small parameter in the analytical theories of lasers based on perturbation techniques and multiple timescale analysis. In fact the most relevant timescale of this problem is that given by the relaxation oscillations at frequency UJR. It is equal to \ / | | (A — 1) in the model (19.1) or y/G (A — 1) in the adimensional form(19.3). As any two-dimensional system, the set of ordinary differential equations (19.3) does not exhibit chaos or spontaneous instabilities. However, it is easily checked that it displays damped relaxation oscillations at frequency UR. The damping equal toy/GA/2 strongly depends on the laser under consideration. The small damping observed in CO2 and YAG:Nd lasers suggests that they are very sensitive to weak perturbations at frequencies near UJR. When external modulation is used, chaos is most easily obtained by parametric modulation at UR or one of its subharmonics. Controlling chaos requires to act on the sytem at this time scale, making some methods easy to implement e.g. in fiber lasers and unapplicable to semiconductor lasers whose eigenfrequencies are 105 times larger. Chaos only appears when additional degrees of freedom are added, either by

490

19 Controlling laser chaos

X(fj,m)

co2

YAG.Nd fiber.-Nd semicond

10.6 1.06 1.06 Q

b

10~ -10~' 10~7 - 10~8 10~6 - 10"7

io- n -io- 1 2

Tito 3

4

10~ - 10~ 2.10"4 2.10' 4 10"9

7 =T C /Ti 10"d - 10"5 10" 3 - 10"5 -2.10" 3 10"2 - 10~3

uR(10b Hz) .1-.3 .1-.5 .01-.03 ~10 3

Table 19.1 Characteristics of usual lasers

including other optical elements e.g. a saturable absorber inside the laser cavity or by external action on one parameter especially at frequencies near UR. Chaos also appears in autonomous class B lasers in presence of optical feedback [28]. In multimode lasers, the competition between the modes may also be responsible for instabilities and chaos. All these devices are presented in the next section. The so-called "NMR laser" [8, 9] which is based on different physics and technology will not be considered here.When external modulation is used, chaos is most easily obtained by parametric modulation at LOR or one of its subharmonics. Controlling chaos requires to act on the sytem at this time scale making some methods easy to implement on fiber lasers and unapplicable to semiconductor lasers.

19.2.2 Class B lasers with modulated parameters The pump modulated laser has been historically the first in which chaos was observed unambiguously [13]. Several parameters are experimentally accessible for applying the modulation, they include pump power [13, 14, 17], the cavity loss [15, 16] or detuning [16, 17]. Modulating the pump acts on parameter iVo in eq.(19.1) while that of the losses changes tt. Modulation of the cavity optical length amounts to that of an effective pump parameter. The relative sensitivity of the different modulation techniques was discussed in [14, 15] but practically technical considerations often impose the parameter to be modulated. Chaos in such lasers is very well characterized. Its correlation dimension was measured to be slightly larger than 2 using the Grassberger-Procaccia algorithm[18] [19], as a consequence chaos should not be too difficult to control in this system. More significantly in the context of controlling chaos, many unstable periodic orbits were found under a wide variety of control parameters and a full topological analysis of the chaotic attractor has been performed using the template method of Gilmore et al. [20]. Recently such a study has also been carried out on the Nd:YAG laser in presence of pump modulation and the evolution of the topological

19.2 Class B lasers

491

organization of the UPOs versus the modulation frequency has been unfolded. The Nd-doped fiber laser provides another kind of class B lasers. In most experiments in which chaos was controlled, the laser was allowed to oscillate in two orthogonal linear polarisations and is described by two coupled sets of rate equations [17]. Moreover because spontaneous emission is better confined in a fiber, the field equations should include additional source terms. A model which was shown to provide an excellent description of this fiber laser was given in[17] and reads dmi/dr drii/dr

— (rii + f3rij - 1) + a(rii + (3rij) = [ni(r) - (1 +ra*+ (3mj)rii]

with i = 1 or 2 and j — 3 - i. Chaos has been controlled using the derivative feedback and Occasional Proportional Feedback methods for the fiber lasers with modulated pump and the delayed feedback method for the CO2 laser with modulated losses. This laser is also ideally suited for checking the nonfeedback techniques as will be shown in the following. The accessible parameters are specific of each laser : the loss i.e. Tc, controlled through a voltage applied to an intracavity electrooptic modulator for the CO2 laser and the pump i.e. No for the fiber and the Nd:YAG lasers. Note the different actions of these parameters and their very different efficiencies because the latter is multiplied by a factor which is typically of the order of 10" 3 .

19.2.3 CO2 laser with electronic feedback. In this laser instabilities are induced by a feedback signal which drives the losses with a damping rate comparable to the population decay rate[21]. This feedback signal is proportional to the laser intensity and partly compensated by an DC offset. It is applied to an intracavity modulator. A very accurate modelization of this laser was given on the basis of a four level model of the active medium and proved its efficiency in a variety of experimental situations[22]. Meucci et al showed that for a wide range of parameters the model may be cast into a simple form: dxx/dt dx2/dt dx^jdt

= K[X2-(1 + FB)] = -T1x2 - 2nx2eXl + P' + x3 — -ax3 + hx2

(19.4)

with the notations given in[37].Note that in this formulation of the class B laser model, the x\ variable is proportional to the logarithm of the intensity. Chaos appears in this laser for specific sets of parameters and it may be controlled by acting in the feedback loop which is intrinsic to this device, therefore control is somehow natural in this system.

19.2.4 Class B lasers with saturable absorber The laser with a saturable absorber (LSA) has been known to present spontaneous instabilities for more than thirty years but it is relatively recently that it was shown

492

19 Controlling laser chaos

to exhibit chaos. As far as the model is concerned, the saturable absorber is usually described by including an additional variable in the rate equations to account for the absorber dynamics. However experiments on CO2 lasers with saturable absorbers show that the 2-level model does not provide a good description of this kind of LSA. In fact the active medium variables should include an additional variable to account for the CO2 population dynamics. This so-called "3-level model" of the CO2 laser gives an excellent agreement with the experimental findings although there could be discrepancies on the numerical values of the parameters to be introduced. Although a wide variety of chaotic behaviors has been obtained in the CO2 laser with saturable absorber, to our knowledge there has been no report of its control. On the other hand, synchronization of chaotic lasers or of a laser and a prerecorded chaotic signal were successful with CO2 lasers containing saturable absorbers[ll, 12].

19.2.5 Multimode class B lasers with intracavity second harmonic generation When N modes are active in the lasers, 2N additional degrees of freedom bring the possibility of richer dynamics. This is what occurs in the YAG laser with an intracavity doubler[23][24] in which chaos was obtained for a given rotational orientation of the KTP and YAG crystals. For an arbitrary number of modes, the coupled rate equations of such a system are dJ J- c

1,

(It

The adimensional model as given by Wu and Mandel for this laser reads[25] dl —^

( =

I Dm - km - 7 J m - 27

=

( Mm + J2 ^rnnln + 1 1 Dm + AnO

where Im (resp. Dm) the intensity (resp. population inversion) for the m-th mode. These are essentially a set of rate equations, one couple for each mode, including intermode coupling parameters (3mn through the saturation of gain and 7 is proportional to losses due to second harmonic generation. In this laser as for the standard Nd:YAG laser, the parameter used for control is the pump power which rules all the parameters Gm.

19.3 Feedback methods of controlling chaos

493

19.2.6 Class B lasers in presence of feedback Chaos also appears in autonomous class B lasers in presence of optical feedback. Both microchip [28] and semiconductor lasers are concerned by this kind of dynamics but the nature of the feedback is different for both. In the standard feedback as observed in the semiconductor laser, the reinjected field is coherent with the laser one, i.e. this field interferes with the intracavity field. In these conditions, feedback introduces a delay term in the field equation. The standard model for such a diode lader is the Lang-Kobayashi model which provides a basis for more detailed discussions including e.g. parameter modulation, multimode action, noise sources such as spontaneous emission,.. .[26]. It reads

Because of the feedback term E(t — r), this model is infinite dimensional. Note the presence of the linewidth enhancement factor a. Pieroux et al. introduced a minimal model that accounts for such lasers with feedback not only in the coherent case but also in the configurations discussed hereafter[27]. The corresponding equations are dor

i+

£ - *< »> In the microchip laser of Otsuka and Chern, the feedback is incoherent i.e. it has lost his phase reference and the intensities add up[28]. The relevant variable in the model is the intensity. In these conditions, the laser is described by dn

— dt

=

/

/

w

w — n — n(s + js{t — T))

It can be reduced to the same adimensional form as the preceding one by letting /(y, j) = {w + s0 [y + jy (t - 6)]} . Such a model may also been used to modelize the semiconductor laser with optolectronic feedback as shown in [2 7].

19.3 Feedback methods of controlling chaos 19.3.1 Basic ingredients of chaos control Let us now recall some basic elements of the OGY method to explain how it was changed in order to be more fitted to experiments.

494

19 Controlling laser chaos

Let X(t) be the vector which characterizes the state of the system and Y(t) the vector reconstructed from time delayed series of a single variable z(t) Y ( t ) = [z(t), z(t - T ) , z(t -2r),...z(t-

(de - 1) r)]

where de is the embedding dimension. The Poincare section in the reconstructed space is readily obtained by choosing e.g. a series of vectors such as one of their component, say n, is constant Yn(t)=z(t-(n-l)r) = C In practice care must be taken so that all the components of Y(t) are measured between two passages through the section plane. Let £ be the deviation of the trajectory from the goal UPO in the reconstructed space £n = Yn — and P be the first return application in the Poincare section

where ji stands for the parameters. The actual position of the trajectory depends on the value of the control parameter \x which is tuned in a small range near its reference value /io — p* < /x < //o + P* • The derivative of Xi? (and of Yp) with respect to ji characterizes the sensitivity of the UPO to changes of the control parameter

Stable e s and unstable eu directions together with the associated Floquet multipliers Xu and As are determined by standard techniques. Dynamics around the fixed point is then simply given by X n + i - XF(fjbn) = A [X n -

with A = ( V

in

n u

XF(fin)}

> ] in the {e u ,e s } basis. For small variations

A

s )

= Yn-YF = A (£n_i - Pn-lg) +Png

In the controlling process we allow the control parameter to vary. If it is changed to value Pn-i at time t n _i, the dynamics in the interval £n_i,£n will depend on the parameter value during this interval, namely p n _i.The deviation at the end of this interval is easily calculated in the linear approximation to be £ n+ i = A£n + pnh

19.3 Feedback methods of controlling chaos

495

With h = The OGY algorithm suggests to use a correction to the control parameter such that the successive £n converge towards £F(0). The correction is optimum when the next iterate falls on the unstable manifold of £p (corrected) and the stable manifold

Pn = T-eu.fn (hu = eM.h). In fact any correction of the kind pn — a(eu.^n) stabilizes the UPO if £p (0) is a stable point for the application

which requires — (1 4- Au) < ahu < (1 — Xu). Dressier and Nitsche proposed a variant of the OGY method suited for attractors reconstructed using the time delay method[4]. Because of the difficulties related to the extension of the dimension of the phase space, Dressier and Nitsche introduce a more complicated correction taking into account pn-i, the value of the control parameter for the previous Poincare section. We will see in the next section that other modifications are requires to obtain a technique suitable for experiments.

19.3.2 Experimental implementation of control Requirements for efficient control In this section we detail another modification of OGY which makes it better suited to experiments for two reasons : (i) usually the exact position of the UPO is unknown. As Yp is not predetermined, it should be automatically "found" by the correction algorithm, (ii) a method which achieves this allows tracking Yi?, i.e. measuring its variations as the control parameter is tuned because the system remains locked to the unstable orbit even when the parameters are slowly varied. To summarize, a correction that fits our goals should - converge to the fixed point in absence of correction, - automatically, - be robust to changes of the feedback parameter, - require only experimentally accessible information. To achieve this goal, one could use a feedback term proportional to Y n - Y n _i instead of Y n — Yp- According to our previous discussion of OGY, the optimum correction would be e u (Y n - Y n _i) since it alters only the multiplier associated with the unstable direction. The dynamics of the system in the vicinity of the fixed point of the Poincare section would be (19.5) with pn = a (Q - C_i) and £J = £n.eu.

496

19 Controlling laser chaos

Correction every other period Just like in the Dressler-Nitsche approach, there is a difficulty due to the increased dimensionality related to the discretization of the system through the time-delay reconstruction. Indeed the linear stability analysis of the system indictaes that stabilization is possible with feedback coefficients such as -1-AU , . 1 ahu < - — 2(1-A u ) i.e. only if — 3 < Xu < — 1 . The increased dimensionality appears clearly in (19.5) since the evolution of £n+i requires knowing not only £n but also £ n -i since the later fixes the feedback value during the period tn, £n+i- In other words, the knowledge of only £n does not allow to predict the future evolution of the system. To overcome this problem, several methods have been used [4] but the simplest one is to apply the correction every other period, i.e.

P2n

=

a(#n-#n-l)

P2n+1

=

0

and the evolution is fixed by the odd iterates ^

1) Oi

since the even ones are simply given by S2n —

A

us2n-1

because the system is free between times t2n-\ and t^n- So whatever the Floquet multiplier, and consequently whatever the orbit, there exists an interval of feedback parameter a for which the system is stabilized X2U-1

_

A^ + 1

u u (Xu ~ I) 2 < ^ ahuU ^< (Xu - I) 2

Such a feedback meets all the requirements given at the beginning of this section. Let us notice that as the requirement on the feedback parameter is not very stringent, convergence may be automatically obtained for several UPOs. Which orbit is eventually stabilized depends on the initial conditions, on the relative values of their Floquet multipliers and on the parameter which is used to apply the control. Pulsed correction The basic algorithm suggests to apply the correction continuously between two passages across the Poincare section plane, i.e. p(t) = pn,Vt G ]tn,tn+i[. Pulsed corrections, i.e. whose duration At is shorter than f n+ i — tn are more efficient than continuous ones because the additional freedom introduced allows to select

19.3 Feedback methods of controlling chaos

DIODE LASER

-1-( + V—-

FIBER LASER

1

1

497

DETECTION SYSTEM

I

O

GAIN WINDOW

SAMPLER HOL DERS

<e

TIMING Figure 19.1 Block diagram of the experimental set-up for controlling chaos via unstable periodic orbits in a fiber laser pumped by a laser diode.

At to optimize the correction. They are easy to implement (see next section) and consequently have been used from the beginning in the experimental approach of control of laser chaos. A complete theoretical treatment was later developed by Carr and Schwartz. Interestingly enough pulsed corrections overcome the problem linked to the fact that in the standard method, the feedback is activated while the measurements are stored. Circuit description Pulsed Control chaos also called Occasional Proportional Feedback, requires some action in a time scale At short with respect to the basic period LUR of the basic instability of the system which is of the order of a few microseconds for most lasers (see Table 1). Therefore the electronic circuit to realize that is rather easy to set up using standard electronic components. Similarly, the derivative feedback is readily achieved by an operational amplifier. A practicale control system is illustrated on Figure 19.1 for a fiber laser with control through the action on the pump parameter (i.e. No in eq.2). It is made of

498

19 Controlling laser chaos

two parts: the first one is analog computer of a first return map in the Poincare section, the second one designs the correction from these data. It First return map in the Poincare section Two sample-and-hold modules named 1 and 2 on Figure 19.1 are connected in series and triggered by the same clock. The clock signal is either periodic or triggered by a circuit that detects maxima. The clock also triggers a delay generator which allows to select the Poincare plane by adjusting the delay between the maxima and the sampling time. If a raT-periodic orbit is to be controlled, a divide-by-m circuit is inserted between the clock and the delay generator. The two sample-and-hold modules fed by the laser output detector allow to obtain simultaneously the two Poincare section values In and / n +iConditional control These two values are then compared in an operational amplifier and if their difference is smaller than some predetermined threshold e, this activates the control after some adjustable delay with respect to the original clock signal. The conditional control set up by the threshold makes sure that the control is activated only when the system explores the e-vicinity of a periodic orbit. The every other period condition is also easily introduced in the clock sequence. To summarize, five parameters are adjustable for optimal control: - the delay that selects the Poincare section plane, - the delay and duration of the control pulses, - the gain of the feedback loop, - the width e of the window for selecting periodic orbits. Example of chaos controlled laser In the laser domain, the method was originally demonstrated on a YAG laser[6], we present here results on a similar, although slower, laser. The Nd-doped optical fiber laser presents spontaneous instabilities at 10 kHz typically, leading to chaos via a period-doubling sequence. Although the mechanism of the instability has not yet been identified, it is possible to control the laser as shown on Figure 19.2. The upper picture represents the bifurcation diagram of this laser with the pump power as a parameter. In this situation, the laser undergoes a first period-doubling bifurcation at a power of 3.8 mW ansd it is chaotic at powers in excess of 4.9 mW. Using the set-up illustrated on Figure 19.1, it has been possible to stabilise e.g. the 2T periodic orbit. The lower half of figure 2 displays the bifurcation diagram of the laser with the 2T orbit controlled. The region in which the control signal is non zero is indicated by the dotted line in Figure 19.2.The control is first activated at high pump power in a regime where the laser is chaotic. When the pump power is adiabatically reduced, it is possible to track the 2T UPO. This means that the laser remains locked in this regime not only in the chaotic domain (between 6 and

19.3 Feedback methods of controlling chaos

499

4.9 mW) but down to 4.5 mW, i.e in the full domain of existence of this UPO, down to the period-doubling bifurcation where it was born.

(0

"E •

n V.

LJJ

Q HI -I Q_ <

4

5

6

PUMP POWER (mW) Figure 19.2 Example of controlling chaos in a fiber laser (a) reference diagram of the free-running fiber laser, (b) same diagram for the laser with the 2T UPO controlled. The control signal is non zero in the hatched part of the diagram.

19.3.3 Delayed feedback control of chaos Another variant of the OGY method is the delayed feedback method[29]. The principle of this method introduced in the field of nonlinear dynamics by Pyragas is similar to OGY in the sense that correction terms are determined from the divergence of the trajectory in the vicinity of the UPO on which it should be

500

19 Controlling laser chaos

stabilized. The main difference with OGY is that the divergence is continuously evaluated from the change in the trajectory with respect to its value one period of the orbit before, rather than from the discretized divergence calculated between successive passages through the Poincare section plane. To put it more formally, the correction is proportionnal to Yn(t) — Yn(t — r) instead of Y n — Yp and it is continuously applied instead of the pulsed, occasional or other time dependences in the standard method and its variants. It is clear that such a continuous delayed feedback vanishes when the trajectory follows a cycle of period r. Practically such a method is useful when the delay is easy to determine as e.g. in a driven system where the period of the UPO coincides with a multiple of the modulation period. Different methods have ben used to build the delay and the difference. They may be optical or electronic depending of the timescale of the delays. Practically delays of the order of microseconds to nanoseconds may be achieved using optical delay lines while longer delays are easy to generate by direct digital (computer) methods. In the laser domain, experiments were performed on the CO2 laser with modulated losses. The modulation frequency was chosen at about twice the relaxation oscillation frequency of the laser and the delay is of the order of microseconds. Such a delay was realized by feeding an optical fiber delay line with a signal proportional to the intensity. It allowed to drastically extend the region of T-periodic operation of the CO2 laser with modulated losses as illustrated on Figure 19.3. The upper diagram (Figure 19.3a) represents the bifurcation diagram of this laser in absence of control and shows the sequence of period-doublig bifurcations leading to chaos, crisis and switching to a next attractor. The lower one (Figure 19.3b) displays the diagram for the same laser with delayed feedback activated. It shows how the T-2T bifurcation is shifted towards large values of the modulation parameter, enlarging by a factor of 3 the T-periodic region. An interesting improvement of the delayed feedback method has been proposed by Gauthier et al. which reproduced the principle of the Fabry-Perot resonator [30]. They suggest to use a correcting function of the form Yn(t) - aYn(t - r) - a2Yn(t - 2r) - a3Yn(t - 3r) - ... Pyragas' delayed feedback is just the equivalent of two-waves interference while the Gauthier's multiple delay would correspond to the Fabry-Perot multiple interference. Just like the Fabry-Perot is more selective than the two-wave interferometer by a factor equivalent to its finesse, the multiple delayed feedback is much more effective than the standard delayed feedback in selecting a particular period. To our knowledge, it has not been applied to lasers in spite of its potential and of the possibility to implement it fully optically.

19.4 Stabilization of unstable steady states The OGY approach of control has brought into light another variant of control of chaotic systems which takes advantage of the existence of unstable steady states.

19.4 Stabilization of unstable steady states

501

15

CO

5

10

15

20

modulation amplitude (V) Figure 19.3 Stabilisation of an unstable cycle by delayed feedback (a) reference bifurcation diagram of the CO2 laser with pump modulation, (b) same laser with delayed feedback. The T-2T bifurcation is shifted towards higher modulation values in presence of delayed feedback.

Standard feedback loops used in servo control aim at correcting fluctuations of the parameters which determine the operating conditions of a system. The OGY algorithm uses very cleverly the dynamics in the vicinity of an unstable fixed point of the Poincare section. Here the idea is the same except that the unstable fixed point is in thephase space instead of being in the Poincare section. Therefore a stationary state is stabilized instead of a periodic orbit. The control algorithm consists in applying a correction to some suitable parameter such that the system falls onto the stable manifold of the unstable fixed point. This ideal situation is not easy to fulfill in a practical experiment where only approximate models and limited information are available. However it is relatively easy to implement practically since the signal and its time derivative provide independent variables and a correction signal built by combining them allows to span a 2d manifold of the phase space.

502

19 Controlling laser chaos

As for the OGY method, it is not necessary for the system to fall exactly on the stable manifold but it is sufficient if it is brought by the feedback closer and closer to the fixed point. Because the exact position of the unstable fixed point is not known a priori, the control algorithm should automatically converge towards this fixed point. Just as for the stabilization of UPOs, this property allows to track an unstable point as its position changes due to the variation of the control parameters. This may be useful to follow the position of unstable fixed points beyond the bifurcation at which they lost their stability and to check the connections between different regimes. A more interesting outcome of this method is to follow the trajectory after the control has been switched off. This gives access to the unstable manifold of this fixed point and shows how it is connected to the attractor towards which the system evolves. Figure 19.4 illustrates the efficiency of such a method in a fiber

Pump

power

(mH)

Figure 19.4 Unstable steady-state stabilization in afiberlaser. The ON state of the laser destabilizes at a pump power of 3.7 mW. The amplitudes of the emerging intabilities are given by the bold line and the stabilized steady-state by the thin line in between. laser which displays spontaneous instabilities when the pump power exceeds some threshold, here 3.7 mW, where the stable laser destabilizes via a Hopf bifurcation and then period-doubles to reach a chaotic regime. The broad traces represents the maximum and minimum amplitude of the oscillations in the free running regime and the thin trace in between the steady-state intensity obtained in the controlled state. It clearly appears that (i) this intensity is in the continuity of that obtained below the instability threshold, meaning that the observed regime results from the stabilisation of the state which destabilises (ii) the control technique tracks the unstable steady-state without requiring any resetting of the parameters and therefore

19.5 Nonfeedback control of chaos

503

(iii) the regime of continuous operation of the laser is widely extended. This method is obviously useful to suppress unwanted chaos in a laser but it should be kept in mind that for practical applications, it is often easier to slightly change the system as demonstrated by Roy et al. for the chaotic frequency doubled laser [32].

19.5 Nonfeedback control of chaos 19.5.1 Invasive vs noninvasive methods Besides the main stream of feedback control of chaos there has been many attempts to develop methods to change a chaotic system into a regular one. The simplest of these methods consists in applying to a suitable parameter an additional modulation that drives the system into some regular -often periodic- regime. It has ben known for a long time that a small periodic modulation could stabilize an otherwise unstable state and the idea is here to extend the results known for unstable steady states to time dependent regimes. There have been many variants of such methods and it was applied to several situations in particular the unstable orbit may be embedded inside a chaotic attractor or it may be just the unstable T-periodic orbit which emerges at the first period-doubling bifurcation of a cascade. Contrary to the feedback methods discussed above, the nonfeedback technique drastically alters the system under control. When such a technique is used, the feedback never vanishes, the stationary solutions are changed and not only their stability. In that sense, non feedback techniques are invasive ways to control chaos. This drawback is relevant only for basic physics applications but it does not spoil its possible practical applications. In fact nonfeedback control of chaos and unstable states is achieved via periodic and most often sinusoidal modulations, which is more convenient for both experimental implementation and analytic theory of the control. For such modulations the relevant parameters are the frequency and the amplitude of the modulation and possibly (i.e. for driven systems) the phase of this modulation.

19.5.2 Phase control The frequency of the additional modulation may be equal to that which appears in the unstable system, or to one of its subharmonics. This allows to introduce as a control parameter, the phase of the additional modulation relative to the main one. Control through phase is very attractive since it involves no additional power to drive the system and to this extent may be considered as a "soft" method compared to those which require powerful action. This method is of particular interest for driven systems in which a phase reference is easy to obtain. The control of unstable systems through the phase of a subharmonic perturbation has been applied to CO2 and doped fiber lasers [40], [41], [38], [42]. In the later case, it has also been used to tune the system from a periodic or chaotic regime to other periodic and/or

504

19 Controlling laser chaos

10 8 6 4

jnsit

CO

a

2 0 10 8 6

(b)

4

•8

2

1.76

1.78

1.80

1.82

1.84

1.86

1.88

1.90

1.92

Pump parameter Figure 19.5 Influence of the phase of subharmonic modulation on the bifurcation diagram of a laser with modulated parameters (a) reference diagram in absence of modulation, (b) with an in-phase subharmonic modulation and (c) with a dephased (<£=2TT/9) fourth subharmonic modulation.

chaotic regimes. Subharmonic modulation is extremely flexible and allows almost any kind of control to be set. This is illustrated on Figure 19.5 where bifurcation diagrams of a modulated class B laser have been reproduced. Figure 19.5a is given as a reference for the laser without additional modulation while Figures 5b and c display the bifurcation diagrams in presence of weak in- phase and dephased subharmonic modulation. Because the whole bifurcation diagram is shifted and qualitatively altered by the appearance of new attractors, it is possible to tune it by acting on the phase of the additional modulational making almost all kinds of regime accessible. Another illustration of the efficiency of this technique is given in Figure 19.6 which displays return maps for a free-running quasiperiodic laser (Figure 6a), and for the same laser with subharmonic in-phase and dephased subharmonic modulation where 8T-periodic (Figure 19.6b) and chaotic (Figure 19.6c) regimes are observed respectively.

19.6 Applications of Controlling Laser Chaos

505

(a)

(b)

(c)

In

Figure 19.6 First return maps of a fiber laser with modulated pump (a) quasiperiodic regime in absence of subharmonic modulation, (b) 8T periodic orbit with cp = 0 subharmonic modulation and (c) chaotic regime with dephased ip = TT/18 subharmonic modulation.

An analytical theory is available for the influence of second subharmonic modulation in the vicinity of the T-2T period-doubling bifurcation. It is based on a multiscale analysis using 7 as a small parameter[40]. It shows that as expected, the period-doubling bifurcation becomes unperfect because the presence of the subharmonic induces a response at this frequency even in the linear regime and a fortiori in the nonlinear one.

19.6 Applications of Controlling Laser Chaos Although the original algorithm aimed at stabilizing UPOs i.e. obtaining periodic regimes, it appears that most applications of the control fall slightly at the edge of the field except for the now developingfieldof communication and synchronization.

506

19 Controlling laser chaos

As reviewed in the following, they include unstable cw operation, measurement of the Floquet multipliers and determination of connections between stable ans unstable manifolds.

19.6.1 Enlargement of the range of cw operation The most striking use of control of chaos is the extension of the range of cw operation of lasers whose performance is spoiled by spontaneous instabilities. The intracavity frequency doubled YAG laser is the prototype of this application[31]. Near the threshold the laser emits cw radiation but its intensity exhibits spontaneous periodic and chaotic oscillations as soon as the pump exceeds times its thresholf value. In other words such a laser is stable at low output power but it becomes unstable at higher power, therefore restricting its range of possible aplications, this is the "green problem" so-called from the emission wavelength of this laser. A simple feedback loop as discussed in[6]allows to lock the laser on the stationary state which destabilizes at the first instability threshold. Note that there is an hardware alternative solution to this problem which consists in rotating the optical axis of the doubling crystal with respect those of the active medium. By changing the nature and the strength of the mode -coupling, the system looses its unstable character[24].

19.6.2 Floquet multipliers and manifold connections Floquet multipliers are readily obtained by measuring the divergence of the trajectory once the control has been switched off. When the control is suppressed, the system naturally evolves on the unstable manifold of the UPO or of the unstable steady-state on which it was locked by the feedback. After leaving the vicinity of the unstable state, it eventually approaches a stable (stationary or periodic) state through its stable manifold. Therefore by monitoring the trajectories after the control has been switched off, it is possible to determine the path which connects the unstable manifold of the controlled state to the stable one of one (possibly several) stationary states. An example of such a connection is shown on Figure 19.7. The doped fiber laser was locked on the unstable steady-state and the control is suddenly switched off. The trajectory first spirals around the unstable point revealing its unstable focus character. Then it temporarily follows a (almost) period T cycle. This cycle appears to be unstable and the trajectory progressively leaves it with alternating oscillations and it eventually converges to a stable 2T cycle. From such a transition regime, we can conclude that the unstable manifold of the fixed point is included in the stable manifold of the T-cycle, and that the unstable manifold of the latter is included in the stable one of the 2T-cycle. Note that the residence time in a particular regime (unstable fixed point or T-cycle) depends on the initial conditions and the noise in the system. It is often observed experimentally that this residence time strongly fluctuates from one experiment to the other. By measuring the divergence (resp. damping) rate of the trajectories after

19.6 Applications of Controlling Laser Chaos

507

C 3

0) Z

Ld

TIME ( 0 . 4

ms/div)

Figure 19.7 Evolution of the laser after switching-off the control. The laser was originally stabilized on the unstable steady-state. When the control is switched-off, the trajectory passes in the vicinity of the unstable T-periodic orbit before converging towards the stable 2T-periodic orbit.

LASER OFF

LAS5ER aN STABLE UNSTABLE

~ 15 •

'c/5

% 10

• •

C!

e °°

5 •

a

a

0 2

i

i"

o ~

-2 -

a

a

Q

a

2

3

4

5

6

7

8

Figure 19.8 Evolution of the amplification rate (A) and the frequency (Cl) of the transients in the vicinity of the unstable steady-state versus the pump power in a fiber laser. The laser starts emission above 3.5 mW and becomes unstable at 5.7 mW.

508

19 Controlling laser chaos

switching-off of the control (resp. switching-on a perturbation), it is possible to measure the real (A) and imaginary (ft) parts of the most unstable roots of the characteristic equation for the linear stability analysis of the steady-state solution. Results of such measurements are reported on Figure 19.8. They show that (i) the laser destabilizes via a Hopf bifurcation, i.e. a point where A changes sign while ft remains non zero and varies continuously. Note that at the laser threshold, i.e. at the bifurcation between the laser ON and the laser OFF states, Q tends to zero.

19.7 Conclusion We have reported here on several implementations of control of chaos in lasers with some emphasis on the methods rather than on the technical details specific to each implementation. This report may not be complete. The references available to the author are reported in the appendix together with some information on the control method used and the laser type. The reader needing this technical information may refer to these papers. The exploratory part of controlling chaos is now probably over and emphasis is actually put on the applications of these ideas for communication and coding with chaos. This domain is also rapidly extending and reached the field of applied physics. In that respect, lasers with their wide range of applications will probably lead to extensive new results.

Appendix : Summary of chaos controlled lasers Author

Lasertype

Roy et al. Nd*+ : YAG Reyl et al. "NMRlaser" Bielawski et al. iVddopedfiber Bielawski et al. TVddopedfiber Bielawski et al. CO2 Ciofini et al. CO2 Meucci et al. CO2 Liu&Ohtsubo diode LNP Otsuka et al. Ryan et al. diode Celet et al. iVd3+ : YAG Chizhevsky et al. CO2

Contr. Meth. Contr. par. Ref. OPF OGY OPF

deriv.FB delayedFB filteredFB modulation delayedFB modulation modulation modulation modulation

Table 19.2 Summary of controlled chaotic lasers

pump qual.fact. pump pump loss loss phase pump pump pump phase phase

[6] [8] [7] [33] [34] [37] [39] [35] [28] [41] [42]

References

509

Acknowledgements. The author wishes to thank his coworkers in this field, S. Bielawski, M. Bouazaoui, J. C. Celet, D. Dangoisse, D. Derozier, D. Dangoisse and T. Erneux. The Laboratoire de Physique des Lasers, Atomes et Molecules is a member of the Centre d'Etudes et de Recherches sur les Lasers et leurs Applications (CERLA) and is supported by the Region Nord-Pas de Calais and the FEDER.

References [1] Ya. I. Khanin, Principles of Laser Dynamics, North-Holland/Elsevier, Amsterdam (1995). [2] P. Mandel, Theoretical Problems in Cavity Nonlinear Optics, Cambridge University Press, Cambridge (1997). [3] E. Qtt, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 64, 1196-1199 (1990). [4] U. Dressier and G. Nitsche, Phys. Rev. Lett. 68, 1-4 (1992). [5] P. Glorieux, Int. J. Bif. Chaos, to be published [6] R. Roy, T. W. Murphy, Jr., T.D. Maier, Z. Gills, and E. R. Hunt, Phys. Rev. Lett. 68, 1259-1263 (1992). [7] S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 47, R2492-2495 (1993). [8] C. Reyl, L. Flepp, R. Badii and E. Brun, Phys. Rev. E 47, 267-272 (1993) [9] R. Badii, E. Brun, M. Finardi, L. Flepp, R. Holzner, J. Parisi, C. Reyl and J. Simonet, Rev. Mod. Phys.66, 1389-1416 (1994) glorieux.tex [10] S. Bielawski, D. Derozier, and P. Glorieux, Proc. SPIE 2039, Chaos in Optics, 239-249 (1993) [11] T. Tsukamoto, M. Tachikawa, T. Sugawara and T. Shimizu, Phys. Rev. A 52, 1561-1569 (1995). [12] J. R. Leite [13] D. V. Ivanov, Ya. I. Khanin, I. I. Matorin, and A. S. Pikovsky, Phys. Lett. 89A, 229-230 (1982) [14] D. J. Biswas, V. Dev, and V. K. Chaterjee, Phys. Rev. A35, R456-458 (1987) [15] F. T. Arecchi, R. Meucci, G. P. Puccioni, and J. R. Tredicce, Phys. Rev. Lett. 49, 1271-1274 (1982); J. R. Tredicce, F. T. Arecchi, G. P. Puccioni, A. Poggi, and W. Gadomski, Phys. Rev. A 34, 2073-2092 (1984).

510

References

[16] T. Midavaine, D. Dangoisse, and P. Glorieux, Phys. Rev. Lett. 55, 1989-1992 (1985); D. Dangoisse, P. Glorieux and D. Hennequin, Phys. Rev. A 36, 47754778 (1987). [17] S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 46, 2811-2822 (1992). [18] G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, Phys. Rev. Lett. 55, 339-342 (1985). [19] M. Lefranc, D. Hennequin, and P. Glorieux, Phys. Lett. A 163, 269-274 (1992). [20] M. Lefranc, P. Glorieux, F. Papoff, F. Molesti, and E. Arimondo, Phys. Rev. Lett. 73, 1364-1367 (1994). [21] F.T. Arecchi, W. Gadomski, and R. Meucci, Phys. Rev A 34, 1617-1620 (1986). F. T. Arecchi, R. Meucci, and W. Gadomski, Phys. Rev. Lett. 58, 2205-2208 ( 1987). [22] R. Meucci, M. Ciofini, and Peng-Ye Wang, Opt. Comm. 9 1 , 444-452 (1992); M. Ciofini, A. Politi, R. Meuci, Phys. Rev. A 48, 605-610 (1993). [23] T. Baer, J. Opt. Soc. Am. B 3, 1175-1180 (1986). [24] G. James, E. Harell II, and R. Roy, Phys. Rev. A 4 1 , 2778 (1990). [25] X.-G. Wu and P. Mandel, J. Opt. Soc. Am. B 4, 1970-1877 (1987). [26] R. Lang and K. Kobayashi, I. E. E. E. J. Quant. Electron., QE-16, 347-355 (1980). [27] D. Pieroux, T. Erneux [28] K. Otsuka, J.-Y. Chern and J.-S. Lih, Opt. Lett. 22, 292 (1997). [29] K. Pyragas, Phys. Lett. 170A, 421 (1992); K. Pyragas and A. Tamasevicius, Phys. Lett. 180A, 99 (1993). [30] D. J. Gauthier, D. W. Sukow, H. M. Concannon, and J. E. S. Socolar, Phys. Rev. E 50, 2343-2346 (1994). [31] Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, and I. Triandaf, Phys. Rev. Lett. 69, 3169-3172 (1992). [32] G. E. James, E. M. Harell II, C. Bracikowski, K. Wiesenfeld, and R. Roy, Opt. Lett. 15, 11441-1143 (1990) [33] S. Bielawski, M. Bouazaoui, D. Derozier, and P. Glorieux, Phys. Rev. A 47, 3276-3279 (1993). [34] S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. E 49, R971-R974 (1994); T. Erneux, S. Bielawski, D. Derozier, and P. Glorieux, Quant. Semiclass. Opt. 7, 951-963 (1995).

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[35] Y. Liu and J. Ohtsubo, Phys. Rev. A 47, 4392-4399 (1993). [36] Y. Liu and J. Ohtsubo, Opt. Lett. 19, 448-450 (1994). [37] R. Meucci, M. Ciofini, and R. Abbate, Phys. Rev. E 53, R5537-5540 (1996). [38] R. Meucci, W. Gadomski, M. Ciofini, and F. T. Arecchi, Phys. Rev. E 49, R2528-2531 (1994). [39] M. Ciofini, R. Meucci, and F. T. Arecchi, Phys. Rev. E 52, 94-97 (1995). [40] T. Newell, A. Gavrielides, V. Kovanis, T. Erneux, and D. Sukow, Phys. Rev. E (1998). [41] D. Dangoisse, J. C. Celet, and P. Glorieux, Phys. Rev. E 56, 1396-1406 (1997). [42] V. N. Chizhevsky and R. Corbalan, Phys. Rev. E 54, 4576-4579 (1996). [43] A.N. Pisarchik, V.N. Chizhevsky, R. Corbalan, and R. Vilaseca, Phys. Rev. E 55, 2455-2461 (1997).

20 Control of Chaos in Plasmas W. Klinger Institut fur Experimented- und Plasmaphysik, Universitat Kiel, Olshausenstrasse 40, 24118 Kiel, Germany

20.1 Introduction This chapter is devoted to a discussion of recent experiments and computer simulations on the control of the dynamics in ionized gases, more specifically in plasmas.1 The common spirit shared by these studies is to tame the collective dynamical behavior of the plasma by small external perturbations. Collective behavior and the notion of quasineutrality are the conceptual cornerstones of the basic definition of the plasma state [Chen, 1984]. Different from ordinary hydrodynamics in a plasma, collective behavior arises from electromagnetic forces between the charged particles, rather than friction and viscosity. This has two important implications: First, electric potentials are shielded out by space charge thus ensuring quasineutrality on length scales larger than the characteristic shielding length (Debye length). Second, waves and oscillations can develop even on long-range length scales - a most important fact in the present context. The physics of plasmas has to deal with a vast variety of waves and instabilities, their nature depending on the specific plasma conditions [Stix, 1992]. In this framework, nonlinear dynamical phenomena and turbulence are more or less intrinstic features of a plasma. Prom both the scientific and the engineering point of view, an ultimate goal remains to achieve control of nonlinear instabilities in plasmas. After introducing some basic concepts of pratical interest (Section 20.2) we discuss examples based on fluid-type and kinetic plasma instabilities. They are observed as waves or oscillations which typically develop, in their nonlinear regime, chaos and turbulence. It is important to realize that the boundaries of a plasma typically play a decisive role in the development of instabilities as well as in the energy dissipation process. Consequently, a natural approach to control the dynamics of a bounded plasma is to apply (small) external perturbations, thereby interfering in the interplay between the plasma and its boundaries. Following this idea, nonlinear oscillations of plasma diodes can be controlled by open- and closed-loop schemes (Section 20.3). Plasma instabilities become more complex when elementary atomic processes due to inelastic collisions are involved. A prominent example are ionization instabilities in glow discharges which still resist against a closed theoretical 1

An ionized gas consisting of electrons, ions, and neutral particles is not necessarily a plasma. For a comprehensive discussion of this topic we refer to the introductory textbook of Chen [Chen, 1984].

514

20 Control of chaos in plasmas

description. However, in a simplified treament one finds interesting similarities to chemical waves [Cross & Hohenberg, 1993]. The formation of ionization waves is strongly dependent on feedback effects via the external circuit. Therefore control of the dynamics can be achieved by appropriate external perturbation signals, as described in Section 20.4. Finally, in the last section, we comment on the recent developments in a demanding and technically important subject, the feedback control of turbulent magnetized plasmas (Section 20.5).

20.2 Some Basic Concepts We start the discussion by recalling some basic concepts which are of substantial importance for the experiments described later. We put emphasis on important practical application of methods of synchronization and control. For more mathematical rigor we have to refer the reader to the given references.

20.2.1 Overview over Common Chaos Control Schemes Schematic diagrams of four important control schemes are depicted in Fig. 20.1. Following the standard classification of control theory [Ogata, 1990] we first have to distinguish between open-loop and closed-loop control. The open-loop control of chaos by relatively weak, resonant harmonic perturbations is generally known as 'suppression' [Lima & Pettini, 1990] or 'taming' of chaos [Braiman & Goldhirsch, 1991]. Here, the control signal (i.e. the perturbation) is typically pre-determined rather than obtained by analysis of the dynamics of the system and is applied either to an accessible parameter or is simply superimposed. Open-loop chaos control schemes have been applied in pioneering early work [Hubler & Luscher, 1989] and there is still a high interest in chaos synchronization and suppression methods. Different closed-loop control strategies are shown in Fig. 20.1(b-d). Classical feedback attempts to achieve control by perturbing the system with a filtered, phase shifted, and amplified probing signal. The goal of classical feedback is to remove the instability, i.e., to stabilize a stationary state. In general the feedback signal makes a significant modification of the system's dynamics and there are no general rules available ensuring the stability of the control. Typically, the control scheme has to be designed specifically for the system under consideration. A somewhat different idea is followed by chaos control strategies, among which the most common ones are the Ott-Grebogi-Yorke (OGY) method [Ott et a/., 1990] and the time-delayed autosynchronization (TDAS) [Pyragas, 1992, Gauthier et a/., 1994]. Here, the goal is to stabilize particular unstable periodic orbits (UPOs).2 The main difference to classical feedback control is that the UPOs are not deformed, i.e., the controlled (regular) state is still a part of the uncontrolled dynamics. 2 It was shown previously that an infinite number of UPOs is embedded within a chaotic attractor [Grebogi et a/., 1988]. Chaotic motion may be seen as an aperiodic wandering among the infinity of UPOs.

20.2 Some Basic Concepts

515

(b)

(a)

open loop control

simple feedback

(c)

(d) OGY feedback control p

IP linear control

V X

time-delayed feedback X(t)

n

r-

map

J: Figure 20.1 Schematic diagrams of the four most important control schemes, (a) Resonant open loop control resp. taming, (b) classical closed-loop feedback control, (c) Ott-Grebogi-Yorke discrete feedback control, and (d) time-delayed autosynchronization.

The OGY scheme is basically a standard linear discrete parametric feedback control method, but acting directly in the dynamical phase space. Though based on discrete time systems, it is applicable in the continuous time case as well by considering the time discrete system obtained from the induced dynamics on a Poincar section. The OGY technique was seminal to the field and has a highly instructive geometrical interpretation (see below). The situation is more complicated in case of the TDAS scheme, where the feedback signal is continuously generated by time delay of the signal to be controlled. This makes the interpretation of the control difficult and most research on chaos control by TDAS had to rely on numerical and semiempirical results. Important recent progress in the understanding of TDAS has been achieved by applying Floquet theory [Just et al, 1997] that provides robust criteria for stability of the control. The overview of chaos control methods given here is of course far from being complete. Comprehensive reviews including extensive list of references can be found in the papers by Chen and Dong [Chen & Dong, 1993] and Shinbrot [Shinbrot, 1995].

516

20 Control of Chaos in Plasmas

20.2.2 Open-Loop Control As mentioned above, the art of open-loop control is to find the appropriate control signal for a particular stabilization problem. A major difference to feedback control is that the control signal never vanishes, even when the system is in the desired state. The subsequent discussion is focused on those open-loop control techniques already applied with some success to plasma systems. Suppression of Chaos by Resonant Perturbations Chaotic behavior can be suppressed either by adding a driving force or by modifying an accessable parameter in a pre-determined way. The dynamical system is then described by — = F(x,p, #(
(20.1)

where g(q,t) is the periodic (or stochastic) forcing term that depends on the parameter q. This introduces in the above equation an additional parameter such that the perturbed dynamical system depends on the new control parameter set (p,q). When q is varied, it is reasonable to expect structural changes of the dynamical state that belongs to a particular p-value. It was discovered by Lima and Pettini that the periodic parameteric perturbation of the Duffing oscillator significantly changes the stability of periodic orbits [Lima & Pettini, 1990]. This result has later been confirmed in experiment for the case of a magnetoelastic ribbon [Pronzoni et a/., 1991]. Consider the Duffing equation with periodic parametric perturbation d2x dt2

dx dt

o

where g(q, t) = 1 + q cos(J)£). The main effect is the occurence of resonance-like regions on the fi-axis in which the largest Lyapunov exponent vanishes meaning a stabilized periodic state. Such resonances are found close to the harmonics of the driver frequency fi ~ nu (with n G Z). While changing fi, the parametric perturbation amplitude q is held constant at a relatively weak value. A different example for taming chaotic states of a driven pendulum by applying a weak periodic external forcing was later given by Braiman and Goldhirsch [Braiman & Goldhirsch, 1991]. Stochastic Resonance Stochastic resonance (SR) is a phenomenon in which the response of a nonlinear dynamical system to weak periodic input signals is optimized by the presence of a particular level of noise [Wiesenfeld & Moss, 1995]. Though its basic principles are known since the early 80's [Benzi et a/., 1981], SR is currently undergoing a

20.2 Some Basic Concepts

517

Figure 20.2 Basic mechanism of stochastic resonance. The deformation of the double well potential determines the stability of the equilibrium points.

renaissance especially because of its high importance in biology and neuroscience. It is important to realize that SR does not achieve control in its very sense but introduces a certain coherence into a previously stochastic system. It is hence reasonable to consider it as a particular type of open-loop control scheme that does not rely on the presence of UPOs, i.e., on low-dimensional dynamics, but on multistability and the presence of noise. The basic SR mechanism can be explained in simple terms. Consider a symmetric bistable double-well system described by the (classical) potential 3 V(x) = ex - ax2 +

(20.3)

The graph of (20.3) is shown in Fig. 20.2 for (a) e = 0, (b) e > 0, and (c) e < 0. For e = 0 we obtain a symmetric double-well potential with two stable equilibrium points at d=c = ±y/2a//3. The black ball represents one of these equilibrium states and the two gray balls small deviations. If the double-well potential is perturbed by a negative force (e > 0) a deviation from the one equilibrium point +c eventually leads to a spontaneous transition to the other at - c , i.e. the equilibrium point at +c suffered a loss of stability. In contrast, a perturbation by a positive force (e < 0) increases the stability of the equilibrium point at -he. In the next step, the parameter e is replaced by A cos U[t + y/2a£(t) where £(£) represents Gaussian distributed white noise of variance o. The described changes of the stability becomes dynamic, i.e., the double-well is sinusoidally modulated and transitions between equilibria are stochastically induced by the noise. The equation of motion based on (20.3) yields the Duffing oscillator (for e — 0) and in the large-damping limit the dynamics of SR is governed by a Langevin equation at

da;

(20.4)

3 We prefer to use later the notion of 'classical potential' to avoid confusion with the plasma space charge potential that plays an important role in the physical interpretation of SR, Sec. 20.3.

518

20 Control of chaos in plasmas

Here Vo = —ax2 + fix4 is the unperturbed Duffing potential. The investigation of the stochastic differential equation (20.4) has already provided much insight into the statistical properties. The most important predictions of the adiabatic theory (ui much smaller than the characteristic rate of equilibration) are the following [McNamara & Wiesenfeld, 1989]. In the absence of modulation (A = 0) the mean first passage time between the two equilibrium points is given by the Kramers time Tk oc -e-AV^a , (20.5) a where AV is the A — a = 0 barrier height. The output power spectrum consists of odd harmonics of the (injected) driving frequency u\ superimposed on a Lorentzianlike noise: S(u) oc ( 1

J a

^°

a0

+ 7o
(20.6)

+LU

where the constants ao and 70 depend on a and 70 oc A. Finally the output signalto-noise ratio R, which is computed from the power spectrum and defined as the strength of the signal peak to the mean amplitude of the background noise is given by

(±\2

(207)

The cooperative effect between the noise and the weak signal introduces to a certain degree 'coherence' in the system. As a consequence the dynamic becomes, in the average, periodic instead of fully stachastic. This is most clearly observed in the power spectrum by a sharp peak at u\ appearing on the broad background noise, cf. Eq. (20.6).

20.2.3 Closed Loop Control Closed-loop control of chaos has the advantage of having a much less empirical character as compared to open-loop control. The output signal of a dynamical system is used to generate a control signal that ideally vanishes when the desired state is achived. This is of particular importance in many cases of practical interest where large amplitude control signals are inconvenient or costly. Ott-Grebogi-Yorke Method As already emphasized, the OGY technique is hinged on the existence of stable manifolds around periodic points (corresponding to UPOs in the time-continuous flow). The basic ideals to make small adjustments to the control parameter to shift the state close to the stable manifold of the desired periodic point. The formal description starts with a time-discrete dynamical system, described by the iV-dimensional map xn+1 =f(xn,p),

(20.8)

20.2 Some Basic Concepts

519

where x n G JRN and p G H. The control parameter p is assumed to be available for external adjustment but shall be restricted to lie in a certain (small) interval \p - Po | < 8 around a prescribed nominal value p 0 for which a chaotic attractor exists. By this condition only small changes of p are allowed and the system is controlled by tiny perturbations rather than establishing a new dyamical state. It is the objective to vary the parameter p with time in such a way that the dynamics of the system converges onto a desired periodic orbit contained in the chaotic attractor. The control strategy is the following. For the sake of simplicity we consider the case where the desired orbit has a period P = 1, i.e., we shall discuss the stabilization of unstable fixed points of the mapping (20.8). Let us denote x*(p0) the selected unstable fixed point on the attractor for the nominal control parameter value p 0 . Ergodicity of chaotic attractors guarantees that when time passes, eventually a point x n is found in the close vicinity of x*. The subsequent point x n + i is then approximately given by a truncated Taylor expansion of map (20.8). In the first order approximation, the linear map x n + 1 - x*(p0) = A[xn - x*(p0)] + B(p -po)

(20.9)

is obtained, where A = df/dx\x=x* is the NxN Jacobian matrix of the map f and B = df/dp\p=Po is an iV-dimensional column vector that describes the sensitivity of f to changes of the control parameter. For adjusting the control parameter p at any time step n, a particular linear control law is definded by (Pn ~ PO) = ~ K T • [Xn - X*(p0)] ,

(20.10)

that is, the change of the control parameter is taken to be proportional to the deviation between x n and the fixed point. The constant JV-dimensional column vector K determines the control amplitude and is therefore called the 'gain vector'. Substituting Eq. (20.10) into Eq. (20.9) we finally obtain xn,

(20.11)

where Sxn — x n — x* is the deviation from the fixed point. If K can be chosen such that the matrix (A — B • K r ) only has eigenvalues with a modulus smaller than unity the limit lim Sxn vanishes and the fixed point will be stable. The n—>oo

systematic determination of such K is a standard problem of control theory, called the 'pole placement problem' [Ogata, 1990]. Further, the condition on the size of the control parameter perturbations combined with the linear control law (20.10) yields \KT6xn\ < 6.

(20.12)

This defines a ball of radius w = 2S/\KT\. Control according to (20.10) shall be activated only if x n is found to be inside the ball. Otherwise, the control parameter is left at its nominal value po. Though in principle K can be chosen in many different ways, there is a particular choice of the gain vector K T that has been shown to be optimal with respect

520

20 Control of chaos in plasmas

*n+1 APO+SP)

nsta unstable manif

s

APO)

/

/

/

/

/

/

Ap*i

/ t

/

Figure 20.3 Geometrical description of the OGY method in the two-dimensional plane, (a) The fixed point x*(po) is crossed by its stable and unstable manifold (solid lines). It is shifted in the plane (along the dashed line) by changes of p. (b) The result of perturbing po by 8p is shown in gray. After [Shinbrot et a/., 1993].

to the time to achieve control [Romeiras et al, 1992]. This approach has been proposed in the original OGY paper [Ott et a/., 1990] and has an intuitive geometrical interpretation. For illustration, a two-dimensional mapping (20.8) is considered, where the stable and unstable manifolds of a fixed point are lines in the plane. The situation is shown in Fig. 20.3. The fixed point x* corresponds to the period-one orbit to be stabilized (more specifically, x* is a saddle periodic point. If a point x n is found to be inside the ball (20.12), the control parameter is perturbed by some Sp to be specified later. As a consequence, the next point x n + i is attracted towards x*(p + Sp) in the direction parallel to its stable manifold, and, at the same time, is repelled from x*(p + Sp) in the direction parallel to its unstable manifold. Thus, Sp may be chosen such that x n +i falls precisely on the stable manifold of x*(po)- The control parameter may then return to its nominal value po and the system will rapidly converge towards the desired fixed point x*(p0) along its stable manifold. The OGY method does not rely on the knowledge of the dynamical equations describing the system. This important feature makes it applicable and attractive to experimental situations, where appropriate model equations are often not known. We emphasize that prior to a practical application of the OGY method several practical limitations (experimental noise, degree of nonlinearity and instability, embedding reconstruction) should be considered. This is discussed for instance in the paper of Bayly and Virgin [Bayly & Virgin, 1994].

20.2 Some Basic Concepts

521

Modified OGY techniques Numerous variants along similar lines to the OGY control strategy have been developed, often motivated by experimental circumstances. In particular, the strict OGY method requires quite detailed information about the local dynamics of the system to control. Though the original OGY method has been demonstrated to be directly applicable to particular experiments, see for example [Ditto et a/., 1990], especially the computational limitations in estimating the control parameter adjustments pn in real-time are an obstacle for a widespread use of the OGY control technique. This has been the motivation for Hunt [Hunt, 1991] to introduce a technique which he called 'occasional proportional feedback' (OPF). In the OPF scheme, deviations from a preset point are periodically fed back into the system with a fixed, empirically determined gain. Rather than finding a control vector K T such that (A — BKT) has an appropriate eigenvalue spectrum (see Sec. above), the ansatz Spn =

(20.13)

is made, where X{ is one single component of the actual phase space vector x(t). The point XQ is the desired set point for that component. A block diagram of the OPF circuit is shown in Fig. 20.4. The output x(t) of the chaotic system is sam-

x(t)

ofs

V///V/, chaotic system

sync

window comparator

Figure 20.4 Block diagram of the OPF control circuit, p is an accessible control parameter of the system. Abbreviations: ofs = voltage offset, amp = amplifier, sync = external synchronizer. pled with a synchronizing frequency (often given by the external driver signal of the chaotic system). A variable offset is added to the signal to shift it into a window of adjustable width. The window comparator is activated when the signal makes a

522

20 Control of chaos in plasmas

transit through the prescribed window. When the synchronizing input is coincident with this event, the sample and hold circuit acquires the voltage value. The sampled signal is put out through the analog switch for time periods short compared to the period of the synchronizing oscillator. An amplifier with variable gain finally delivers the control signal to the chaotic system. Despite the fact that the effectiveness of this strongly simplified control scheme is not guaranteed, it has turned out to work suprisingly well in many experimental systems, see [Roy et a/., 1992] and references given in [Ott et a/., 1994, Ott & Spano, 1995]. Bielawski and coworkers proposed a slightly different approach [Bielawski et a/., 1993]. Their 'consecutive difference control' (CDC) overcomes the problem of the precise experimental determination of the map's unstable fixed points by using the differences of consecutive iterates of the map instead of the differences between each iterate and the fixed point itself. This method has the great advantage that also moving fixed point locations are directly followed and tracking (see above) becomes generic. The control law of CDC simply reads 6Pn = Ko(x«-x*_1),

(20.14)

where x% is the vector component in the unstable direction. An eigenvalue analysis of the resulting control matrix reveals the limitation of the CDC to relatively weakly unstable situations. This shortcoming disappears, however, if the alternating CDC variant is used meaning that the control is not applied every but every second cycle. In this case it is always possible to find a feedback parameter Ko which will stabilize the system.

Continuous Feedback Control Though the OGY control method is extremely elegant it has, from the practical point of view, some limitations. First, the OGY method deals with the Poincar map that is often cumbersome to determine, especially in systems without external periodic forcing. Second, the parameter adjustments appear to be rare and discrete in time. In the presence of noise, this can lead to occasional bursts into regions far from the desired periodic orbit. Third, the OGY method can stabilize only those periodic orbits whose largest Lyapunov exponent is small compared to the reciprocal of the time interval between parameter changes. Time-continuous control techniques may overcome some of these shortcomings. These considerations have stimulated considerable efforts to establish an alternative control scheme based on continuous rather than discrete feedback. The main idea of continuous feedback control is to add a particular external feedback force G to the set of first order differential equations describing the uncontrolled dynamics [Singer et a/., 1991] - F(x(*),p) + G(x(*),x*(*),*, i f ) ,

(20.15)

where x*(£) represents the trajectory to control. The purpose of the external force G is to give nudges to the orbits when they move away from the desired state. The parameter K is the adjustable feedback gain.

20.3 Plasma Diodes

523

The now most common continuous feedback control method, the TDAS scheme, was suggested by Pyragas [Pyragas, 1992]. We assume that the equations of the dynamical system are unknown, but one component X\{t) of x(t) — {xi(t),X2(t),...} can be measured as a system output. Further, the system is assumed to have an input available for external force. The TDAS scheme then specifies the feedback forcing in (20.15) as follows G1(t,K)=K[x1(t-r)-x1(t)]

(20.16)

The other components of G = {Gi, G2,...} are set to zero. The delay time r has to be chosen to coincide with the period length T of the UPO to be stabilized. For appropriately chosen feedback constant K, stabilization of the UPO is achieved by extremely small perturbations. It is important to note that external control with unlimited amplitude \G(t)\ may lead to multistability, i.e., strong initial perturbations can establish alternative stable solutions belonging to a different basin of attraction (for instance a fixed point instead of periodic orbits). Both large amplitudes of the control signal and multistability are in general undesired in most experiments. The problem is solved by restricting the perturbation to G{t)

f KA(t) if |G(t)|
if |G(t)| > Fo

.

.

(20 17)

-

where Fo > 0 is a properly chosen saturation value of the perturbation. The restriction (20.17) has two effects. First, the initial transient after activating control is in general much longer since the the system synchronizes only when the trajectory comes close to the desired UPO. Second, multistability is avoided and no other states than UPOs are stabilized. Continuous feedback control has some similarities to the open-loop control scheme discussed above. For instance, the dispersion (A^(t)) of the TDAS perturbation signal shows pronounced resonances towards small values when r approaches the period length of the UPOs [Pyragas, 1992]. Further, both TDAS and EFC can be seen as an extension of concepts of periodically perturbed dynamical systems, with the important extension that the driver signal is non-trivial and self-generated.

20.3 Plasma Diodes Among the broad class of bounded plasmas, diode-like configurations play a highly important role [Kuhn, 1994]. A plasma diode is characterized by emitter and collector electrodes where charged particles are injected and absorbed, respectively. Many of the common plasma discharge configurations are in fact diodes, especially discharges operating with direct current (dc). In this section we show in experiment and computer simulation of plasma diodes applications of some of the chaos control techniques introduced above.

524

20 Control of chaos in plasmas

electron beam ion background ne(0,t)=n0 3^.

nj (x,t)=n0

ve(0,t)=v0

x=0

Figure 20.5 Schematic representation of the Pierce-diode. Electrons are injected at the left electrode (emitter) at x = 0. After transit electrons are absorbed at the right electrode (collector) at x = L. The quasineutrality is established by a fixed ion background density.

20.3.1 The Pierce-Diode Of fundamental importance for basic studies of bounded plasmas is the Piercediode [Pierce, 1944], an idealised one-dimensional model of a collisionless plasma diode. We emphasize that the Pierce-diode has no straightforward experimental correspondent though interesting relationships between generalized Pierce-diodes and experimental thermionic diodes (see below) have been reported [Kuhn, 1984, Kolinsky et al, 1997]. A schematic diagram of the Pierce-diode is shown in Fig. 20.5. A beam of cold electrons is injected at the emitter at x = 0 and, after transit, absorbed at the collector at x — L. The two electrodes are short-circuited and grounded. The ions are assumed to be immobile and form a homogenous, neutralising background charge. The electronflowin the Pierce-diode can be described by the one-dimensional fluid equations for the density n(x,t) and velocity v(x,t) of the electron beam dn d(nv) ~dt + e 36 dv dv —"a~> at ox me ox nd Poisson's equation for the electric potential

(20.18) (20.19)

(20.20)

with boundary conditions for the potential 0(0, t) — (j)(L, t) = 0 and for the electron injection density n(0,t) = no and injection velocity v(0,t) — v$. The electric field is given by E(x,t) = —d(j)(x,t)/dx (electrostatic description). We note that the nonlinearity of the plasma is due to the convection term vdv/dx in the momentum transport equation (20.19) and the flux in the continuity equation (20.18).

20.3 Plasma Diodes

525

It is a classical result that the stability of the Pierce-diode is solely determined by the dimensionless Pierce-parameter [Pierce, 1944] (20.21) where ujpe = ^/noe2/eorae is the electron plasma frequency taken at the emitter. The Pierce-parameter represents the transit time L/v0 in units of the phase of a plasma oscillation cycle. The Pierce-model (20.18-20.20) has been studied by numerous authors and was extended in many directions. For further reading we recommend the comprehensive review by Kuhn [Kuhn, 1994]. Stability of Equilibria and Bifurcations The Pierce-diode has two different stable states: the uniform and the non-uniform equilibrium. In the uniform equilibrium the potential is constant (f){x,t) — 0, whereas the non-uniform equilibrium has a characteristic potential contour with nonzero electric fields that accelerate and decelerate the electrons. In particular non-uniform equilibria have a non-vanishing electric field E$ — E(x = 0, i) at the emitter. The bifurcation structure of the Pierce-equilibria was thoroughly investi-

a

/ /

b

A

/ -1 -

••'

I

I

]

! !

|

-2

-

I

.

.

.

i

2

Figure 20.6 Left: Bifurcation diagram of the Pierce-equilibria after Lawson [Lawson, 1989b]. The electric field EQ at the emitter is shown as a function of the Pierce-parameter a. Stable equilibrium states are indicated by full lines, unstable ones by dotted lines. Dash-dotted lines indicate transcritical bifurcations and the dashed line marks the Hopf bifurcation point. Right: Plasma potential contour (f)P(x) of the uniform (a) and a non-uniform equilibrium (b).

gated by Godfrey [Godfrey, 1987] and later extended by Lawson [Lawson, 1989b]. The discussion of the electron transit time yields the criterion for the occurence of

526

20 Control of chaos in plasmas

non-uniform equilibria ^ = 1,3,5,...

(20.22)

such that for all odd values of a/rc a transcritical bifurcation from uniform to non-uniform equilibrium takes place. The plasma potential and the electric field are made dimensionless by normalization E1 — (eL/mvfyE and (/>' = ej(mv\)(\) [Lawson, 1989b]. For the sake of simplicity we subsequently drop the prime in normalized quantities. The resulting bifurcation structure of uniform and non-uniform equilibria is shown in the diagram Fig. 20.6. After each transcritical bifurcation, the uniform equilibrium becomes unstable and is replaced by a stable non-uniform equilibrium. The linear stability theory of Pierce-equilibria is very elaborate and for details we have to refer to the literature [Kuhn, 1994]. The underlying physical background of the Pierce-instability is the known two-stream instability of an electron beam that enters at velocity v0 into a neutralizing immobile ion background. This initial situation leads to space charge waves with a dispersion [Nicholson, 1983]

(20 23)

-

Eq. (20.23) has two roots at k± = (u ± UJ^/VQ, where cupe is the injection plasma frequency, cf. Eq. (20.21). It is reasonable to consider the space charge waves as plasma oscillations, Doppler-shifted by the beam velocity vo. The Pierce-instability is the bounded-plasma counterpart of the two-stream instability and it includes the reflection of the wave at the boundaries as well as the influence of electrode surface charges [Pierce, 1944]. Basically, the Pierce-instability leads to the transition from a uniform to a non-uniform equilibrium, i.e. triggers the transcritical bifurcation. However, with regard to chaotic dynamics, the more interesting phenomenon is the presence of a small stability window just below a = Sir. As indicated in Fig. 20.6 at the stability limit at a « 2.9TT, a Hopf-bifurcation occurs and a stable limit cycle is born. It was discovered by Godfrey [Godfrey, 1987] that the establishment of periodic potential oscillations gives rise to low-dimensional chaos.

Period Doubling and Chaos The initial work on chaotic dynamics in the Pierce-diode [Godfrey, 1987, Lawson, 1989c] was based on the numerical solution of integral equations derived from the set of fluid equations (20.18-20.20). The subsequently discussed results have been obtained by a somewhat different approach, the particle-in-cell (PIC) simulation. This numerical scheme moves a large number of charged particles in the self-consistently determined electric field. PIC-simulations are made computationally efficient by forming 'super'-particles out of, say, 104 - 105 electrons and ions. The electric field is calculated at each time step by solving Poisson's equation

20.3 Plasma Diodes

527

(20.20) on a grid to which the space charge distribution is mapped. The PICsimulation scheme has the advantage of being a highly robust first-principles approach that especially allows for the appropriate treatment of plasma boundaries [Lawson, 1989a]. It is well-justified to term the PIC-simulation approach 'computer experiment' rather than 'theory'. For an introduction to PIC-simulations of bounded plasmas we refer the interested reader to the book of Birdsall and Langdon [Birdsall & Langdon, 1985].

2.85

2.855

2.86

2.865

0C/7C

Figure 20.7 Period doubling cascade to low-dimensional chaos in the Pierce-diode. The results are obtained from a PIC-simulation. The top row shows selected attractor reconstructions {4>p(L/8,t),L/8^) 5 L/2W}> where 0L/S(*) = p(L/8,t) and (j>L/2{t) = (j)p(L/2,t). The bifurcation diagram is obtained from Poincare sections of phase space attractors determined at the Pierce parameter values in the above given range. The chaos transition shown in Fig. 20.7 follows perfectly the Feigenbaumscenario and the bifurcation diagram has a striking similarity to those obtained from simple nonlinear mappings [Collet & Eckmann, 1980]. We remind the reader that the shown results are obtained from computer-simulations taking the Pierce-diode as the actual extended many-particle system. The PIC-simulation was performed

528

20 Control of chaos in plasmas

in one space dimension with Np = 8192 particles (electron super particles, the ions are taken as fixed neutralizing background charge) and Nc = 256 spatial grid cells. The chaotic attractor at a = 2.856?r shown in Fig. 20.7 resembles much Roessler's attractor [Roessler, 1976]. This indicates that similar stretch-and-fold mechanisms are involved. The dimensionality analysis of the attractor gives values very close to two and the Lyapunov spectrum has the structure (—,0, +), as required [Krahnstover et al. 1998]. During the sequence of period doubling bifurcations the orbits of periodicity P — 2n (with n e INo) are successively destabilized. It is thus appealing to attempt a stabilization of the unstable periodic orbits which are still embedded in the chaotic attractor. Discrete Feedback Control For the chaos control study a Pierce parameter value of a* — 2.856TT is chosen where a chaotic state is well developed. From a one-dimensional Poincere section we obtain 4 the first return map (/>(n + 1) = F(0(n);a*) shown in Fig. 20.8. A

-0.1

p

-0.15 -0.2

-0.25 -0.25

-0.2

-0.15

-0.1

Figure 20.8 First return map of 0 L / 2 ( ^ ) in a chaotic state at Pierce parameter value a* = 2.846TT. The inset shows the shift of the return map around the fixed point when the control parameter p = AU is varied by ±5mV. Note that on the enlarged scale the discrete points of the return map are resolved. From [Greiner et al, 1998].

careful inspection of the return map reveals a fine structure (see inset in Fig. 20.8) owing to the attractor fold in the third dimension. With regard to chaos control, the fine structure is not of great significance and may be negleted. We thus consider the return map as a unimodular, one-valued mapping. A fixed point is located at 0*(n) = <£*(n + l) = -0.21626 ±0.00006. The local Lyapunov exponent of the fixed 4

For a simplified notation we introduce the idendity (£) = L/2(t) = p(L/2,t).

20.3 Plasma Diodes

529

point given by Ai=ln

(20.24)

= 0.637 ±0.005,

is positive, clearly indicating an unstable fixed point. 5 attractor

0.2 0.3 time / ms Figure 20.9

0.4

0.5

Stabilization of the orbit of periodicity one by OGY-control [Krahnstover et a/., 1998]. The control is activated for t = 0 — 0.4 ms and then switched off. The phase space attractors for the un-controlled and the controlled state (red lines) are shown in the right.

The feedback stabilization of the fixed point * is achieved by a systematic application of the OGY-control algorithm described in Sec. 20.2.3. First of all, one has to choose an appropriate parameter p that controls the dynamics state. The straightforward choice of p — a has some numerical drawbacks that make the computations inefficient. Instead, a control parameter is chosen that globally changes the the electron velocity: we introduce a (typically small) voltage difference AC/ between the two electrodes of the Pierce diode. Asserted by the standard assumption of small parameter perturbation we keep close to the equilibrium at At/ = 0. The optimum choice of the discrete parameter corrections pn = AUn makes the distance between the <j>{n) and the fixed point 0*(n) vanish in only one step. In the one-dimensional case, setting 8xn+\ — 0 in Eq. (20.11) yields the optimum feedback parameter K = A/B = -1.17 ± 0.06 ,

(20.25)

where A B 5

=

OF.

(20.26)

= O) = -1.89 ±0.01 =

(20.27)

The numerical values given here differ slightly from those given in [Krahnstover et al., 1998], where the electron beam has a small thermal width. A thermal electron beam is known to shift systematically the bifurcation points etc. [Yuan, 1977].

530

20 Control of chaos in plasmas

To determine B simulation runs are done at small parameter variations Ap = ±0.005 V; the resulting shift of the return map showns Fig. 20.8. The estimate of the linear regime is S(f)max — 0.01. Using the control law 0(fl) e [* - <5max, * +
=»

&Pn = -K{(j){n) - *) (20.28)

with the feedback parameter (20.25) the unstable fixed point in the return map is sucessfully controlled [Krahnstover et al., 1998]. The time evolution of the stabilization process is depicted in Fig. 20.9, where a time series of the mid-potential 4>L/2(t) is shown while the OGY-control is activated. Note that in Fig. 20.9 and the subsequent one, the physical quantities are nonnormalized to give an idea of their magnitude. After an initial transient the chaotic state is established. At t « 0.26 ms the potential enters for the first time the vicinity of the unstable fixed point and the control, Eq. (20.28), is activated. As a result the single periodic orbit is immediately stabilized. The root mean square (r.m.s.) values of parameter changes, shown in the lower trace of the diagram in Fig. 20.9, are small compared to the r.m.s. values of the mid-potential, GVJG^ — 0.24%. At t = 0.4 ms the control is switched off and the unstable periodic orbit is left with exponential divergence oc exp(A^), cf. Eq. (20.24). In the right half of Fig. 20.9 the two-dimensional representation of the phase space attractor with and without control are shown. Evidently, the stabilized periodic orbit (red curve) is embedded in the chaotic attractor. 2.0 1.0 0.0 -1.0

-i-,

1

I ' I ' i I I I

?i -25 i

2.0 1.0 0.0 -1.0

0.1

0.2

-20 «

time / ms

Figure 20.10 OGY-control of period-two (a) and period-four orbit (b). The periodic states are stabilized very quickly. Two-dimensional projections of the periodic orbits are shown in the insets. The second trace in each diagram shows the control signal. From [Krahnstover et a/., 1998].

With the same approach, higher periodic orbits are equally well controlled. For a period-A7" orbit, N fixed points and control laws have to be deduced from

20.3 Plasma Diodes

531

the 2^ - th return map. For a periodicity higher than four, control was found to be difficult to achieve due to numerical noise. The stabilization of UPOs of periodicity two and four is shown in Fig. 20.10. The OGY-control is activated from the beginning of the simulation run and a controlled state is quickly established. It is important to note that for each of the TV = 2, N = 4 fixed points a control window (20.28) is activated, thus enlarging the window for 'trapping' the chaotic dynamics. Slight inaccuracies in the location of the fixed points cause a systematic tendency (towards negative values) and higher amplitudes of the control parameter corrections [Krahnstover et a/., 1998].

20.3.2 The Thermionic Diode In the previous subsection we have introduced the Pierce diode as a simple model for a bounded plasma system. The Pierce-instability gives rise to a broad variety of nonlinear dynamical phenomena including deterministic chaos. The question arises if the observations made in the particle simulation have relevance for a real plasma diode. We emphasize that the Pierce diode is a strongly simplified model and has no direct experimental counterpart. It was suggested by Kuhn [Kuhn, 1984] and discussed in some detail by Kolinsky et al. [Kolinsky et a/., 1997] that the Pierceinstability marks the onset of strongly nonlinear relaxation oscillations observed in so-called thermionic6 plasma diodes [Iizuka et a/., 1982, Greiner et al, 1993]. Since they are operated at low gas pressure (of the order of 10~4mbar), plasma diodes are collisionless any many physical effects are fully kinetic, i.e., depend on the particular form of the energy distribution functions. We note that the above described chaos which is directly based on the Pierceinst ability is by far too subtle to be observable in real experiments. Nontheless, as we shall see in the following, revealing control experiments have been performed in thermionic plasma diodes. Hysteresis and Nonlinear Oscillations The above mentioned relaxation oscillations and the related chaotic dynamics are highly nonlinear kinetic phenomena. For thermionic diodes with volume ionization the underlying physical mechanisms were investigated recently [Greiner et a/., 1995, Klinger et a/., 1995]. The most important principles are illustrated in the diagrams Fig. 20.11. The experiments are conducted in a linear magnetized thermionic discharge device. A uniform magnetic field of induction B « 10 mT is directed along the z-axis. The electrons are well confined and their dynamics is thus almost onedimensional. Primary electrons are emitted by a tungsten filament cathode and are axially accelerated by the discharge voltage between anode and cathode. A typical discharge current-voltage characteristic Id(Ud) is shown in the lower half of Fig. 20.11. There is a pronouced hysteresis in the i"d(£/d)-characteristic, accompanied by current oscillations at the transition points (see insets in Fig. 20.11). The 6

Thermionic diodes are named after their hot cathode that emits electrons by the thermionic effect. Ions are either produced by volume ionization or surface ionization (Q-machine).

532

20 Control of chaos in plasmas 20 r -e-

15

Langmuir mode

:::: UN z

t

' 10

1 10

20

aV

magnetized thermionic discharge

anode

anode glow mode

i i ::j

30

t 40

cathode

Z

50

voltage/V

Figure 20.11 Hysteresis and self-oscillations of experimental thermionic discharges. Right: Schematic diagram of the magnetized thermionic discharge device. Left: Current-volt age characteristic of the discharge The insets show diagrams of the steady-state potential contour and time series of the discharge current for self-oscillation in the Langmuir-mode and in the anode-glowmode. The dashed lines indicate the ionization potential, the dotted lines the cathode potential.

oscillation mechanisms are briefly outlined as follows. In the lower branch of the ^d(^d)-characteristic the anode-glow-mode (AGM) establishes where ionization occurs only in a small layer close to the anode where the plasma potential exceeds the ionization threshold. 7 In the AGM, due to strong electron emission at the cathode, the current flow is space charge limited. Ions stopped by charge exchange collisions with neutrals are trapped in the potential well formed by anode and cathode, thereby establishing a plasma region. Above a certain threshold value of Ud the AGM is subjected to a Pierce-instability [Kolinsky et a/., 1997] that leads to the formation of a potential humb. Ions are accelerated downhill towards the cathode where the negative space charge is partially neutralized by the positive charge of the ions. At this moment the plasma experiences a strong rise of the current flow since the space charge limited discharge current is given by — 1T —

(20.29)

where Ic is the emission current of the cathode, Tc is the cathode temperature, and — min is the potential drop at the cathode that is rosen by the postively charged ions. After the decay of the transient potential structure the reservoir of accelerated ions is exhausted and the discharge current relaxes to the initial value. This relaxation oscillation cycle repeats periodically and a steady current oscillation in a frequency range 100 — 1000 Hz is established. A detailed description of the oscillation mechanism is found in the papers of Greiner et al. [Greiner et al, 1993, Greiner et aL, 1995]. The oscillations observed in the upper 7 Electrons emitted by the cathode gain their kinetic energy by acceleration due to the electric field in sheath region of the discharge.

20.3 Plasma Diodes

533

branch of the I&(Ud)-characteristic have been experimentally studied by Ding and coworkers [Ding et a/., 1996]. The instability develops here in the Langmuir-mode (LM) where again a potential drop at the cathode reduces the discharge current according to relation (20.29), however to a much smaller magnitude. The current oscillation cycle is here due to the formation of an ionization front located where the plasma potential exceeds the ionization threshold. Slight fluctuations of the plasma potential make the front moving forth and back thereby producing ion bunches that travel at low speed into the negative space charge region. As for the AGM-oscillations, these ion bunches neutralize the space charge and cause the current oscillations. For completeness we mention the third discharge state, the temperature-limited-mode (TLM). In Fig. 20.11 It establishes at a discharge voltage Ud > 33 V. As suggested by the name, the discharge current is not anymore limited by space charge but by cathode temperature. The TLM seems to be stable typically. We note, however, that low-frequency oscilations in the TLM have been reported [Arnas Capeau et a/., 1995]. It was speculated that the oscillation mechanism is hinged on the three-dimensional nature of the experiment. The above described self-oscillations are the birth place of the chaotic dynamics of thermionic discharges. For AGM-oscillations, strong periodic forcing of the discharge gives rise to period doubling bifurcations and chaos in the discharge current and plasma potential. This pioneering observation of Cheung and Wong [Cheung & Wong, 1987] ecouraged further investigations and a descriptive model of the related plasma kinetics was later developed [Greiner et a/., 1993]. However, similar to the dynamics of the driven van der Pol oscillator, the chaos is relatively weak, i.e., typically much dominated by the periodicity of the external driver [Parlitz & Lauterborn, 1987]. This feature makes it being a rather poor candidate for chaos control experiments. Chaotic oscillations that develop out of LM-oscillations without any external forcing have been described in several papers [Qin et a/., 1989, Fan et a/., 1992, Ding et a/., 1993]. However, the LM-oscillations have the hadicap of being intrinsically noisy, and noise is troublesome for most chaos control techniques (see Sec. 20.2). Nevertheless, open-loop techniques proved to be applicable and shall be described subsequently.

Suppression of Chaos Since chaos control in thermionic discharges is distincly difficult the most robust control scheme, open loop control by parametric perturbation, is most promising. It was shown by Ding and coworkers [Ding et al, 1994] that chaotic LM-oscillations with a low attractor dimension 2 < D < 3 are controlled by a relatively moderate modulation of the discharge voltage Ud(t) = U0 + U{ COS(2TTfit).

(20.30)

The perturbation is resonant since the modulation frequency f\ = 1.8 kHz was chosen to coincide roughly with the characteristic LM-oscillation frequency of the discharge. The result of the control experiment is shown in Fig. 20.12. Without any modulation U\ = 0 V the power spectrum of the discharge current shows a

534

20 Control of chaos in plasmas

0.0

8.0

2.0

4.0

6.0

8.0

Figure 20.12 Suppression of chaos in an experimental thermionic discharge plasma. Shown are the power spectra of the discharge current. The discharge voltage modulation amplitudes are (a) U\ — 0 V, (b) U\ = 100 V, (c) Ui = 200 V, (d) Ui = 300 V. After [Ding et al, 1994].

broad distribution with a local maximum at /o « 1.8 kHz. There is no significant change of the power spectrum for modulation amplitudes U\ < 60 mV. From the power spectra shown in Fig. 20.12(b-d), it can be seen that as U\ increases above this threshold value the level of the broadband noise decreases proportionally. Furthermore, sharp spectral peaks at integer multiples of f\ become more and more visible, thus indicating periodic oscillations. This finding is in nice agreement with the chaos suppression concepts discussed in Sec. 20.2. In their paper, Ding et al. [Ding et a/., 1994] have sucessfully demonstrated the resonance-like behaviour when varying the modulation frequency, i.e., the Lyapunov exponent steeply falls when f\ is close to integer multiples of / 0 , as observed by Lima and Pettini [Lima & Pettini, 1990] in the parametrically forced Duffing oscillator (20.2). Stochastic R e s o n a n c e Thermionic discharges have exacly the necessary features for applying the concepts of stochastic resonance (SR) to suppress irregular behaviour: Bistability and intrinsic stochastic noise. As already mentioned in Sec. 20.2 SR is not chaos control in the sense of stabilizing unstable periodic orbits. SR is a bona fide resonance and may be used to reduce stochasticity in the dynamics of thermionic discharges. We note here that SR was studied experimentally in a magnetoplasma [I & Liu, 1995].

20.3 Plasma Diodes

535

In their paper, the authors put the emphasis on the SR-effect rather than on the control idea, this is why we shall not go into further detail. In a recent SR-experiment conducted in a thermionic discharge it was shown that small periodic perturbations significantly reduce the low-frequency noise level which is due to stochastically induced jumps between two oscillatory states [klinger & Piel, 1998]. The initial situation is depicted in Fig. 20.13. Shown are time series

0.1

1 f (kHz)

10

0.1

1 f (kHz)

10

0.1

1 f (kHz)

Figure 20.13 Stochastic behaviour of a thermionic discharge in the close vicinity of the transition points of the hysteresis loop shown in Fig. 20.11. Time series of the discharge current for gradually increasing discharge voltages Ud of (a) 19.7 V (pure AGM-oscillations), (b) 20.3 V, (c) 21.0 V (pronounced stochastic state), (d) 21.2 V, (e) 21.3V (pure LM-oscillations). The power spectra (f-h) correspond to time series (a), (c), and (e), respectively. After [Klinger k. Piel, 1998].

of the discharge current I&{t) and their power spectra for three different states. At a discharge voltage of U<\ — 19.7 V stationary, coherent AGM-oscillations are established and the power spectrum is peaked at the oscillation frequency / 0 and

536

20 Control of chaos in plasmas

Figure 20.14 Equilibrium curve of the classical potential (20.31) at constant a = —120. At two different it-values the classical potential is shown and the dark balls indicate the stability change. The vertical dashed lines mark the instability points where a transition to the other stable state occurs.

its higher harmonics, Fig. 20.13(a) and (f). When the discharge voltage is increased both the AGM-oscillation amplitude and frequency rise [Klinger et a/., 1995]. Above a level of U& = 20.3 V the AGM-oscillations are occasionally interrupted by small 'bursts' as shown in Fig. 20.13(b). The bursts turn out to be few cycles of LMoscillations. For increasing discharge voltage the occurence of these bursts becomes more frequent and their length, in the average, longer. The time series and the corresponding power spectrum for U<\ = 21.0 V are shown in Fig. 20.13(c) and (g), respectively. The time series shows apparently a stochastic sequence of AGMand LM-oscillation periods. The power spectrum has a much more complicated structure when compared to the previous one. It shows spectral peaks that correspond to coherent AGM-oscillations (/o = 1.8 kHz and higher harmonics) and LM-oscillations (/i = 4.2kHz and higher harmonics). The stochastic transitions between the^two oscillations lead to the increased low-frequency broadband noise (below 200Hz). It is the goal of SR-control to reduce systematically this noise. Further increase of the discharge voltage increases the length of the bursts until only LM-oscillations remain, Fig. 20.13(d) and (e). The power spectrum of the pure LM-oscillations shown in Fig. 20.13(h) is peaked at the oscillation frequency / i = 4 . 5 kHz with some low-frequency noise owing to intrinsic incoherence. In comparison to the power spectrum Fig. 20.13(g) the low-frequency noise level is however much lower (by roughly - 1 0 db). The bistability of thermionic plasma diodes becomes evident by the hysteresis in the /d^d^characteristic (see above). Hysteresis and sudden jumps are often observed in discharge plasmas, see for instance [Bosch & Merlino, 1986a, Bosch & Merlino, 1986b], and Knorr [Knorr, 1984] suggested an phenomenological description based on bifurcation theory. It is introduced a classical potential V(i) with an order parameter i, the appropriately normalized and rescaled discharge current. The expression for the classical potential

537

20.3 Plasma Diodes

-30

g-40 -60

§-40

i- 50 -60

^K

^-30

§-40 w"50 -60 10"1

10" f(KHz)

10'

Figure 20.15 Time series (left) and power spectra (right) of (a) the unperturbed discharge, (b) modulation amplitude U\ = 80 mVss and frequency f\ = 75 Hz, (c) modulation amplitude U\ = 80mVss and frequency f\ = 150 Hz. The top trace in the time series is the modulation signal. The insets in the power spectra show enlarged the frequency range 0 — 100 Hz. After [Klinger & Piel, 1998].

reads

= i4 +ai2

(20.31)

where u is the (normalized) discharge voltage and a is a parameter. Eq. (20.31) is the canonical form of a cusp catastrophe [Jackson, 1991]. The set of equilibrium points is given by dV/di = 0 which is shown in Fig. 20.14. Starting at low uvalues the initially stable equilibrium state becomes unstable at the transition point with a vertical tangent in the equilibrium curve. It occurs a sudden jump and the other equilibrium state is established. The same happens in the reversed direction when u is decreased while sitting on the upper branch. This is considered as a heuristic model of the hysteresis in the establishment of the AGM and the LM/TLM in thermionic discharges. It is important to note, however, that the two equilibrium states are non-stationary, i.e., oscillating states. The classical potential (20.31) has the same form as the generic double-humb potential Eq. (20.3) in the general description of SR in Sec. 20.2. The stochastic behaviour shown in Fig. 20.13 is understood as follows. The hysteresis curve of the magnetized thermionic discharge is very narrow, only a few Volts. Close to the transition points the above described strong nonlinear oscillations occur. Stochastic fluctuations in the oscillation amplitude (incoherence) occasionally induce transitions AGM <-» LM by increased and decreased ion production, respectively. The discharge voltage determines the closeness to either the AGM- or the LM-oscillation mode and thus controls the statistics of the sudden jumps, as evident from inspecting Fig. 20.13. To investigate the influence of a small perturbation signal, the dc discharge

538

20 Control of chaos in plasmas

30 25

(a)

o Q O

20

-

:15 -

10 Q

5 O

0

50 driver voltage (mV)

100 0

100 200 300 driver frequency (Hz)

Figure 20.16 Signal-to-noise ratio (SNR) dependency of (a) driver amplitude at fixed driver frequency f\ = 35 Hz and (b) of driver frequency at fixed driver amplitude U\ = 100mV. After [Klinger & Piel, 1998].

voltage UQ = 21.1V is modulated by a weak signal of a few tenths of millivolts, cf. (20.30). The modulation frequency is chosen to be in the order of the Kramers time (20.5) which is estimated from the statistics of the unmodulated discharge. The result of the experiment is shown in Fig. 20.15. Without any modulation a stochastic sequence of AGM- and LM-oscillations is found [Fig. 20.15(a)]. Modulation with a small signal of amplitude U\ — 80 mVss significantly changes the transition behaviour [Fig. 20.15(b)]: At each minimum of the modulation signal there is increased pobability for an LM -» AGM-transition and at each maximum the same happens for the AGM -* LM-transition. As a result a sharp peak arises at f\ in the power spectrum to the expense of the noise underground. The low-frequency part of the power spectrum has the form (20.6) predicted by the adiabatic theory of SR. If the modulation frequency is chosen almost twice the inverse Kramers time, the SR effect is still visible but less pronounced [Fig. 20.15(c)]. Further evidence for the resonance-like nature of the modulated plasma diode is obtained by systematic evaluation of the power spectra. The results are shown in Fig. 20.16. Modulation of the discharge at fixed driver frequency f\ = 35 Hz leads to rapidly increasing SNR, Fig. 20.16(a). In the weak amplitude range U\ < 50mVss the SNR increases roughly quadratically with the driver amplitude, as predicted by adiabatic theory, Eq. (20.7). For U\ > 50mVss the SNR increases linear with the driver amplitude. As outlined in Sec. 20.2.2 SR is a bona fide resonance and consequently a resonance peak is not observed. The diagram Fig. 20.16(b) shows the SNR dependence on the driver frequency f\. The SNR monotonously decreases for increasing driver frequency above the inverse Kramers time T^1 « 30 Hz until a saturation level is reached at f\ « 180 Hz. The above presented findings support the concept of SR as a bona fide resonance. The stochastic behaviour of the thermionic diode in its hysteresis regime is

204 Ionization Waves

539

very efficiently suppressed by weak modulation of the discharge voltage. This becomes most evident in the dramatic change of the SNR, Fig. 20.16. We emphasize that SR is a control of stochasticity and not of chaos, i.e., the coherence introduced by SR is not the result of stabilization in phase space.

20.4 Ionization Waves Ionization waves propagate in the positive column of simple cold-cathode glow discharges. Compared to the plasma diodes introduced above, the gas pressure of a glow discharge is typically three orders of magnitude higher. This has two important implications. First, the particle motion is mainly diffusive due to frictional forces. Second, inelastic and superleastic collisions transfer energy from electrons to neutral gas atoms and vice versa. By use of nobel gases like neon, the most important case in the present context, stepwise ionization processes are introduced owing to metastable levels. The interplay of energy transfer between neutral gas atoms and electrons leads to low-frequency instabilities of the ionization degree of the plasma.8 This instability forms in the positive column a chain of striations that propagate axially, the ionization waves [Oleson & Cooper, 1968]. Ionization waves are essentially one-dimensional convective waves that incorporate electron temperature fluctuations to lead to a local change of ionization and excitation of neutrals.

20.4.1 Basic Theory The modelling of ionization waves is a difficult task due to the rather complex atomic physics ('chemistry') involved. It is nevertheless revealing to discuss the basic principles and the main results of the linear stability theory. Glow discharges are typically operated in long cylindrical glass tubes and the plasma density is given by the balance between particle production (ionization) and radial losses due to ambipolar diffusion of electrons and ions to the glass walls. This situation can be described in terms of a one-dimensional fluid model that consists of particle balance equations for the ions and the metastable atoms. The ion particle balance equation reads [Pekarek & Krasa, 1974] W-D'TT + Z-^, (20.32) at ox2 7i where n\ is the ion density in the center of the plasma column, DA is the ambipolar diffusion coefficient [Chen, 1984], and TI = A 2 /D a is the characteristic particle loss time by radial diffusion [Franklin, 1976]. With A = r o /2.4 we denote the effective transverse discharge dimension of a tube of radius ro. The source term Si describes particle production and is given by nenmzmoo 8

+ n2mzmm ,

(20.33)

Note that, as in thermionic diodes, glow discharges have an ionization degree far below 1%. Rougly speaking, the plasma is always embedded in a 'sea' of neutral gas.

540

20 Control of chaos in plasmas

where n e is the electron density. The source function (20.33) consists of three terms, the ionization from the atomic ground level (density n a , rate £ooo)5 ionization from metastable levels (density n m , rate z m o o ), and ionization by pair collisions between metastables (rate zmm). A balance equation similar to (20.32) governs the metastable density ^

^

+ »e"a*bm - - ^ .

(20.34)

The metastable production is due to excitation from the atomic gound state and the loss rate ^m

=

~T~2 ^~ ^a^mO "I" ^e^moo H" ^ m ^ m m

(ZU.OOJ

is determined by radial diffusion (with diffusion coefficient Dm), deexcitation by collisions with atoms, deexcitation and ionization by electrons, and ionization by pair collisions between metastables. The two equations (20.32) and (20.34) are coupled by their source and loss terms on the r.h.s. The rate coefficients zap for the various atomic processes introduced above are of course functions of the electron energy. The electron energy distribution stongly deviates from a Maxwellian. In particular its high-energy tail is depleted by inelastic collisions. In a simplified view the mean electron energy Ue (rather than the temperature Te) is determined by the balance equation [Pekarek & Krasa, 1974, Franklin, 1976] \

^

= -W%EX

- neH(Ue),

(20.36)

where the axial heat convection is balanced by Joule heating jeEx in the axial electric field and loss due to collisions with neutral atoms. The loss function H depends on the mean electron energy only. Balance equation (20.36) closes the set of model equations for the positive column. There are two most important types of ionization waves, the r- and the pwaves. r-waves are ion-guided ionization waves and influence of metastables can be neglected. For p-waves this assumption is not anymore justified and since the metastable density itself is modulated by the passage of the waves, the full set of balance equations for n\ and nm has to be considered simultanously. For this reason, p-waves are called metastable-guided ionization waves. The stability analysis was carried out by Pekarek [Pekarek, 1971] and a typical dispersion relation is shown in Fig. 20.17(a). Undamped p-waves are found in the regime dfi,re/dK < 0. Since Clre/K > 0 ionization waves are backward waves, i.e., the group velocity is directed towards the anode wheras the wave fronts propagate from anode to cathode. Note that the wave frequency is normalized to the lifetime of the metastables, meaning that p-waves are low-frequency. We briefly comment on the physical mechanism of wave propagation. Formally, the dispersion diagram in Fig. 20.17(a) bears the complete information about the propagation of p-waves. Nevertheless it is worthwhile to consider the basic physical

20.4 Ionization Waves

541

Figure 20.17 (a) Dispersion and damping of metastable-guided ionization waves. Real part (upper curve) and imaginary part (lower curve) of the wave frequency 0 are normalized to the lifetime of metastables r m . The wavenumber is normalized by the energy relaxation length K = kL. After [Pekarek, 1971]. (b) Schematic diagram of ionization wave propagation. The solid line in the upper graph is the ion density, the dashed line the electron density. Dotted lines indicate the equilibrium state.

processes that lead to propagation of density perturbations. Initial short wavelength density perturbations Sn evolve by ambipolar diffusion thus producing a space charge electric field SE. The ambipolar field superimposes on the static electric field Ex which is increased on the cathode-side and decreased on the anode-side slope of the perturbation [Fig. 20.17(b)]. Clearly the perturbation of the electric field is TT/4 out of phase with the density perturbation. Since the current density j e is constant the Joule heating jeEx has a maximum shifted towards the cathode. The accelerated electrons increase, the decelerated electrons decrease ionization and the density perturbation (and therfore the wave front) propagates from anode to cathode, as observed in experiment. It has to be beared in mind that this strongly simplified physical picture gives only the essentials of the proagation mechanism and for quantitative comparisons, numerous additional effects have to be taken into account [Pekarek & Krasa, 1974, Franklin, 1976, Raizer, 1991]. After discussing the governing equations, the dispersion properties, and the propagation mechanism we raise the question why ionization waves are self-excited. It was discovered by Achterberg and Michel [Achterberg & Michel, 1959] that each time when an ionization wave packet reaches the anode, the discharge voltage changes accordingly. The current in the glow discharge is limited by a series resistor and any current change is immediately converted into a voltage fluctuation which launches a new, secondary wave packet at the cathode. If the perturbation is sufficiently amplified by the positive column, the process can attain a steady oscillatory state.

542

20 Control of chaos in plasmas

cathode

Figure 20.18 Schematic diagram of experimental device for ionization wave studies. The local light flux is picked up by an array of photo transistors.

20.4.2 Experiment and Transition to Chaos Among the early reports on probably low-dimensional chaos in plasmas are ionization wave experiments. We have to distinguish between autonomously operated discharges and non-autonomous ones, the latter periodically perturbed by a weak signal superimposed on the dc-discharge voltage. Fig. 20.18 shows a schematic diagram of a typical experimental arrangement for ionization wave studies. Most investigations are performed in simple glass discharge tubes of 400 — 800 mm length and 20 — 30 mm inner diameter. The neon gas pressure is between 0.5 — 5mbar. The discharge current is limited by a series resistor in the external circuit and the discharge current may be varied between 5 - 2 0 mA. Fluctuations of the discharge current, the local electric field, and the integral light emission flux are recorded using digitizers. Since the light emission is a local but non-intrusive diagnostic (in contrast to electrical probes) large arrays of photo-transistors or charge-coupleddevices (CCD) may be used without perturbing the plasma dynamics. This allows one to abserve the spatio-temporal evolution of the essentially one-dimensional ionization wave dynamics. We first discuss briefly the chaotic dynamics of autonomous discharges. Braun et al reported a period doubling route to chaos in the discharge current of a helium capillary discharge [Braun et al, 1987]. Similar observations were made in different discharge devices and it was realized that nonlinear ionization waves are the origin of the observed chaotic behavior [Ohe h Tanaka, 1988]. We note that period doubling is not the only transition szenario to ionization wave chaos and turbulence and a variety of different observations were made [Ohe, 1989]. In nonautonomous helium discharges period doubling was found, too, but the transition to chaos follows a more complicated scenario [Wilke et al, 1989]. Detailed investigations were conducted in non-autonomous neon discharges [Albrecht et al, 1993, Weltmann et al, 1993, Klinger et al, 1993]. Here, the nonlinear interaction between the self-excited p-waves and the external forcing leads to quasi-periodic, mode-locked, or chaotic states. The transition szenario to chaotic behaviour is

20.4 Ionization Waves

543

typically the Ruelle-Takens-Newhouse szenario were the two- resp. three-torus phase space attractor (quasiperiodic states) is destabilised and suffers a direct transition to chaos [Albrecht et al, 1993]. The chaotic phase space attractor was shown to be low-dimensional [Weltmann et ah, 1993] and many detailed phenomena strongly resemble the dynamics of the periodically driven van der Pol oscillator [Klinger et a/., 1993].

20.4.3 Control of Ionization Wave Chaos Ionization wave chaos of neon glow discharges is considered as being relatively well understood and up to now all experimental work on ionization wave chaos control has been done in neon glow plasmas. In the non-autonomous discharge, the application of OGY discrete feedback control is appealing since we obtain the Poincar section without any effort by stroboscopic sampling. Continuous feedback control does not rely on the construction of the Poincar section and is thus more advantageous to use in the autonomous discharges. Discrete Feedback Control A neon glow discharge tube is operated at a gas pressure of p = 1.3mbar and a discharge current of 1^ = 10 — 20 mA. In this regime, monchromatic (regular) pwaves propagate. By modulating the discharge current with the sinusoidal output signal of a function generator, a quasiperiodic state is established and above a certain threshold value of the modulation degree m = /d//d> a low-dimensional chaotic attractor establishes via break-up of the two-torus. The modulation degree (m is typically in the realm of only a few percent) is taken as the control parameter of the dynamics. To observe experimentally the dynamical state of the positive column, Ex and $ have proven to be the best choice, mainly because they are easy to access and relatively unloaded with noise. The stroboscopic sampling of the observable yields the topological equivalent of the Poincar section. It is then rather straightforward to apply the OPF-variant (see Sec. 20.2) of the OGY control strategy [Weltmann et ai, 1995]. Fig. 20.19 shows the initial chaotic state of the glow discharge. The discharge current is /a = 13.9 mA and the modulation degree is m = 5%. The observed fluctuating quantity is the axial electric field strength Ex(t) measured by a pair of Langmuir probes located at x — Ax and x 4- Ax with Ax = 2.5 cm. The time series of the electric field fluctuations shows a quite irregular amplitude distribution [Fig. 20.19(a)]. There is, however, a pronounced pseudo-periodicity as typical for non-autonomous systems. In Fig. 20.19(b) the corresponding Fourier spectrum is depicted. The sharp spectral peak is just the external driver signal (frequency /i = 4450 Hz) and below f-x the spectral distribution is broad with a maximum at roughly /i/2. The scheme and the electric circuit of OPF control were introduced in Sec. 20.2, cf. Fig. 20.4. We are now able to apply OPF control to stabilize different UPOs and a typical result is shown in Fig. 20.20. Starting with the chaotic state, it is desired

544

20 Control of chaos in plasmas

§.-80

0

1 2 3 4 frequency/kHz

2

5

4 6 time/ms

8

Figure 20.19 Initial chaotic state of the periodically forced glow discharge experiment: (a) Fluctuations of the integral light emission flux, (b) corresponding Fourier spectrum. From [Weltmann et a/., 1995].

(a)

control on

X

contro information 2000 time/T

if

lr

; 3000

>

" 4000

Figure 20.20 Experimental control of ionization wave chaos. (a) Stroboscopically recorded time series of the local electric field strength over a time period during which the control is activated, (b) The first return map of the dynamics shows both the Poincar section of the chaotic attractor (gray dots) and the fixed points of the stabilized orbit (black dots), (c) Third return map of the dynamics. Note that the fixed points lie along the Xn = X n + 3line. From [Weltmann et a/., 1995].

20.4 Ionization Waves

545

to stabilize the orbit of the arbitrarily chosen periodicity P = 3. Fig. 20.20(a) shows the stroboscopically recorded time series of electric field fluctuations, further denoted by Xn with n index of full driver signal periods. The second trace in the diagram is the difference signal 8X\ } = Xn+3 - Xn as the feedback control is activated. During a short time interval, the modulation degree m (control parameter) is adjusted by the correction Srrik = OLX^ with the goal to achieve the desired period three state. Indeed, the three fixed points of the P = 3-orbit are exponentially approached and after roughly 30 driver periods (« 7 ms) the periodic state is fully established. This means that an UPO of periodicity P = 3 embedded in the chaotic attractor has been successfully stabilized. Further evidence for UPO stabilization is added by inspecting the first and the third return map of the stroboscopically recorded data [Figs. 20.20(b) and (c)]. The fixed points of the UPO are found to be located in dense regions of the Poincar section of the chaotic attractor. In the third return map, the fixed points are ordered along a straight line defined by Xn = X n + 3 , as required. This indicates that the stabilized periodic orbit was actually embedded in the chaotic attractor.

1

2

3

4

frequency/kHz

Figure 20.21 Stabilization of a high-periodic orbit with periodicity P = 16. (a) Power spectrum of the local electric field fluctuations before (gray) and after (black) the activation of control. In the controlled state, spectral components are found at integer multiples of /i/16 only, (b) The first return map of the controlled state (black dots) shows the 16fixedpoints of the stabilized orbit. After [Weltmann et a/., 1995]. In conclusion, the OPF control circuit proved to perform stabilization of the actual dynamics of the system rather than establishing a new system with a new dynamical behaviour. Basically control can be achieved for periodic orbits of any desired periodicity P. It depends, however, on the stability properties of the desired UPO if successful control is possible at all. For example, if the linear neighbourhood of fixed point is left after less than one driver period (meaning a strongly unstable periodic orbit), control is barely possible. Experimentally it was demonstrated here the control of periodic orbits up to P = 32. In Fig. 20.21 it is shown such a stabilized high-periodic orbit, here of P — 16. Finally, we note that the open loop

546

20 Control of chaos in plasmas

control of the chaotic state is possible but by far less efficient. If the driver signal is just amplitude-modulated by a resonant periodic signal, much higher changes of the control parameter m are required to suppress the chaos. Continuous Feedback Control The above described OGY-type control techniques generally require a detailed knowledge of the n-th return map of the dynamics. This is no serious obstacle in periodically driven systems, where the driver period can be taken as clock to record the fluctuations stroboscopically. However, frequently chaos occurs in atonomous systemes without any external periodic driving force and the technical requirements on the on-line construction of the return map become technically demanding, especially if the characteristic time scale of the fluctuations is below several tenth of milliseconds. In such cases, continuous feedback control is much more favourable since it does not rely on the knowledge of the Poincar section. We describe subsequently experimental studies where autonomous ionization wave chaos is controlled by continuous feedback based on the TDAS scheme proposed by Pyragas (see Sec. 20.2).

100

130

time (ms)

Figure 20.22 Continuous feedback control of autonomous ionization wave chaos. Time series of the integral lightfluxfluctuations.Feedback control is activated at t = 100 ms. After an initial transient, an UPO of periodicity P = 1 is stabilized. From [Pierre et a/., 1996]. The initial situation is quite similar to that described in the previous subsection. The main difference is, that no external periodic driving force is applied and ionization wave chaos occurs autonomously. The dynamical behaviour of the positive column is monitored by picking up the integral light emission flux $(t) = X(i) using optical fibers. In the TDAS scheme the control signal is obtained by timedelay of the observable GT(t) = K[X(t - r) - X(t)], where K is the feedback constant [cf. Eq. (20.16)]. If r is chosen to be equal to the period duration T of the desired periodic orbit, control should be achieved provided that the stability conditions for TDAS are met [Just et al, 1997]. We consider first the practical issue

20.4 Ionization Waves

547

how to obtain the time-delayed signal X(t — r). This task may be accomplished either by delay-lines [fast systems [Gauthier et al, 1994]] or digitally stored data (slow systems). Since the p-waves discussed here have a relatively low frequency / < 5 kHz, a time-delay may be obtained by storing the intermediatley digitized data in a first-in-first-out (FIFO) memory. The application of the TDAS scheme is then straightforward and the experimental observation is shown in Fig. 20.22 [Pierre et al., 1996]. The amplified feedback signal is added to the discharge current. The feedback parameter K as well as the time delay r are chosen empirically, where the choice of r is guided by the fluctuation spectrum of the uncontrolled system (note that the UPO has to be a significant Fourier component in the spectrum). As shown in Fig. 20.22, the goal of the stabilization of a P = 1 UPO is sucessfully achieved. After an initial transient, the discharge is stabilized on a periodic orbit of periodicity P — 1. There is, though small, still a nonvanishing periodic control signal even in the controlled state. This hints that the control is not optimal, blaming additional effects like transit time phenomena and dispersion [Pierre et al., 1996]. In particular, it was found that the success of TDAS control depends much on the specific choice of the axial position where the light flux is detected. The problem of the influence of transit time pheomena, i.e, when the physical system reacts with some additional delay to changes of the control parameter of system variables is a subject of current research. The technical challenge to provide an online time-delay is elegantly circumvented by taking benefit of the wave character of the observed temporal fluctuations [Mausbach et al, 1997]. The time delay r is replaced by a spatial displacement £ such that the condition u/k = v^ = £/r holds, where r is fixed by the period of the orbit to be stabilized and the wavelength of the corresponding mode, i.e., r = nX/vfi. The control law then reads Gz(t) = K[X(x - $,t) - X(x,t)}

(20.37)

The feedback signal (20.37) is easily obtained using a differential amplifier and two spatially displaced detectors. Fig. 20.23(a) shows the time series of the integral light fluctuations when the feedback control is activated. Starting in a chaotic state, after a transient time of roughly 300 ms an orbit of periodicity P = 1 is stabilized by feedback control. This finding is supported by estimating the dimensionality and the Lyapunov-spectrum before and after control is applied [Mausbach et al., 1997]. The phase space attractor of the dynamics is obtained by delay-time embedding [Packard et al., 1980]. The (uncontrolled) finite-dimensional chaotic attractor shown in Fig. 20.23(b) has a positive Lyapunov exponent, a vanashing, and negative ones. In the controlled state, the phase space attractor Fig. 20.23(c) is one-dimensional and the largest Lyapunov exponent is close to zero, while the other exponents have negative values. We note that the transient until full control is achieved is considerably longer than in the TDAS experiment described above. This is believed to be due to dispersion effects and the amplification property of the positive column. Such effects also imply a small but nonzero difference signal, even in the controlled state.

548

20 Control of chaos in plasmas

wmmmmmm iv control signal

\

control on

360

380

Figure 20.23 Control of autonomous ionization wave chaos by spatially derived feedback, (a) Time series of the integral light flux fluctuations during which the feedback control is activated (at t = 58 ms. After an initial transient, an UPO of periodicity P = 1 is stabilized. Takens-reconstruction of the phase space attractor in (b) the uncontrolled and (c) the controlled state. From [Mausbach et a>l, 1997].

Spatiotemporal Dynamics of Control The propagation of ionization waves occurs essentially in the axial direction due to diffusion in the static electric field (see above). The spatiotemporal dynamics of ionization waves is thus two-dimensional in (x, t)-space. It is revealing to investigate the spatiotemporal evolution of ionization waves when chaos control is achieved [Mausbach et al, 1998]. It was chosen the control method described above, where the feedback signal is obtained from spatially derived detectors. The spatiotemoral dynamics of light fluctuations is observed by a linear array consisting of 64 photodetectors. Fig. 20.24 shows the experimental results. The space-time diagram of the initial chaotic state is shown in Fig. 20.24. The dynamical features of ionization wave chaos resembles chemical turbulence in many details. The wavefronts are strongly phase-modulated and occasionally phase de-

20.4 Ionization Waves

549

' " ' '

0.5

gO.5| CC CL CO

Jn 5 time (ms)

10 10

5 time (ms) Figure 20.24 Spatiotemporal evolution of chaos control of ionization wave chaos, (a) Chaotic state, (b) Control is activated. The control leads to a spatiotemporal perturbation front that propagates with group velocity from cathode to anode. The space axis is normalized to the length L of the positive column, the amplitudes of the light fluctuations are normalized to their maximum.

550

20 Control of chaos in plasmas

fects occur. This is taken as motivation to consider ionization wave turbulence as being essentially weak phase turbulence [Niedner et al, 1997]. We remind the reader that the ionization wave dynamics is based upon reaction-diffusion type equations, cf. Eq. (20.32) and (20.34). The basic description of ionization waves is thus very similar to that of chemical waves, where a vast amount of research efforts have recently been devoted to [Cross & Hohenberg, 1993]. Reaction in chemical waves is given by the elementary processes among different constituents, wheras reaction in ionization waves is based on the elementary processes in single-type gas atoms. In both cases diffusion plays a major role, but in case of ionization waves the static axial electric field and ambipolar diffusion are decisive for the formation and propagation. Nevertheless, the phase turbulence nature seems to be robust enough to be common to both physical situations. The space-time evolution of the controlled ionization wave dynamics is depicted in Fig. 20.24(b). The control is activated at t = 0 ms and a fully controlled regime is established at approximately 8 ms. The time to achieve control is much shorter compared to Fig. 20.23, owing to improved adjustment of the photo detectors. The establishment of the controlled state has an interesting spatiotemporal history. Stating from the cathode side x = 1, where the feedback signal is applied, a front propagates towards the anode side. The front separates both spatially and temporally the chaotic from the controlled regime. The latter shows a regular pattern of monochromatic travelling waves with a well-defined phase velocity, in contrast to the spatiotemporal chaos Fig. 20.24(a). The perturbation front propagates at a speed of vt « 7 • 104 m/s which is in agreement with the group velocity of p-waves in the present parameter regime [Franklin, 1976]. As discussed above, the ionization instability is forming backward waves where group velocity and phase velocity have the opposite direction. In fact the phase fronts in Fig. 20.24(a,b) always propagate from anode to cathode, wheras the perturbation front takes exactly the opposite direction. We emphasize that the control of ionization wave chaos is still far away from being an experimental example for the control of spatiotemporal chaos. Though the control process itself has an interesting spatiotemporal history, the dynamics of ionization waves is dominated by the boundaries and especially the external circuit. The interaction of the ionization waves with the current flow in the external circuit and the plasma conditions at the electrodes is far more important than wave-wave interaction which is a hallmark of trubulence. We therefore tend to consider the ionization wave experiments as being just half-way between chaotic (global) plasma oscillations and turbulence. This supports the widely accepted conjecture that the control of fully spatiotemporal systems requires more than only one parameter and demands possibly nonlinear feedback laws.

20.5 Taming turbulence A plasma is certainly an extended dynamics sytstem and is intrinsically nonlinear, for example given in the fluid description by the convection term (v • V)v in the

20.5 Taming turbulence

551

momentum balance equation. As a consequence, many plasma instabilities can develop turbulent states and it is appealing to use feedback control schemes for taming the irregular dynamics. However, the research on feedback control techniques to deal with spatiotemporal systems is still in its infancy since most basic concepts rely on low-dimensionality of the phase space, a condition that is typically violated in turbulence. There are nevertheless efforts in plasma physics to challenge turbulence by active feedback and we shall give a few examples of recent developments. Drift Wave Chaos and Turbulence Drift wave turbulence is of particular interest for magnetic confinement of plasmas.9 This is because in the presence of irregular electric field fluctuations E(t) it is possible to have strong E x B-transport perpendicular to the magnetic field. This plasma transport channel in magnetically confined (esp. fusion) plasmas is often called 'anomalous' [Wagner & Stroth, 1993]. It was shown recently in a linear magnetized plasma device [Hansen et a/., 1994] that the route to chaos and turbulence strictly follows the Ruelle-Takens-Newhouse scenario [Klinger et al, 1997b]. The control parameter is the E x ^-rotation frequency of the plasma column that is determined by the bias of a grid separating the plasma source from the magnetized plasma [Klinger et a/., 1997a]. Fig. 20.25 shows the temporal dynamics of the turbulence transition. The first Hopf bifurcation occurs at the linear instability onset of a drift mode, Fig. 20.25(a). The phase space attractor is a limit cycle and the frequency power spectrum is sharply speaked at / 3 = 13.5 kHz and its higher harmonics. The next Hopf bifurcation, Fig. 20.25(b), introduces a second drift mode with frequency /i = 6.9 kHz. Parametric interaction of the two modes results in a multipeaked spectrum. The significant components appear at sums and differences of integer multiples of the two mode frequencies fa = ij\ ± jfy. The phase space attractor of such a quasiperiodic state is a two-torus and the time series has no periodicity. If the control parameter is further increased, mode-locking between the two drift waves occurs, Fig. 20.25(c). In comparison to the quasiperiodic state, the frequency power spectrum is greatly simplified. There are only two strong peaks (and higher harmonics) left at /i = 7.9 kHz and fa — 2/i = 15.8 kHz, where the higher frequency peak is the stronger one. Such a subharmonic spectrum is a hallmark of higher periodic behavior, as apparent from the double loop phase space attractor and the time series with periodicity two. A further increase of the control parameter leads to the gradual dissolution of the mode-locked state, Fig. 20.25(d). Inspecting the time series it is found that the periodicity experiences a slow evolution until it is occasionally interrupted. This means that the periodic orbit has become unstable and the excursions to irregular behavior broaden the spectrum, increase the noise level, and fill up previously empty phase space regions. The further increase of the control parameter leads to strongly irregular behavior, Fig. 20.25(e), 9 For an introduction to drift wave physics we refer the interested reader to the textbooks of Chen [Chen, 1984] and Nicholson [Nicholson, 1983].

552

20 Control of chaos in plasmas

10

20 30 40 frequency (kHz)

10

20 30 40 frequency (kHz)

10

20 30 40 frequency (kHz)

Figure 20.25 Temporal dynamics of the transition to drift wave turbulence. In each subfigure, the time series of densityfluctuations,its power spectrum (the dashed line indicates the ion cyclotron frequency omegaci = eB/rm), and the phase portrait are shown. Prom (a) to (f) the grid bias and thereby the E x i?-rotation freqency is increased. From [Klinger et a/., 1997b].

and finally to developed turbulence with a broad, noiselike spectrum, intermittent density flutuations, and a scattered phase space distribution, Fig. 20.25(f). As reasoned above in a narrow regime of the control parameter, when the previous periodic state is destabilized, a low-dimensional phase space attractor is established [Klinger et al, 1997b]. By a standard procedure [Lanthrop & Kostelich, 1989] unstable periodic orbits (UPOs) were extracted out of the chaotic phase space attractor and examples are shown in the diagrams Fig. 20.26. Time series of plasma density fluctuations are recorded by electric probes. The phase space is again reconstructed by delay-time embedding (see also Sec. 20.4). The UPOs are located by observing the recurrence time of trajectories into a small phase space volume. In the weakly developed turbulent state, UPOs of periodicity P = 1 and P = 2 are found, correponding to two interacting drift wave modes. The instability of the P — 2-orbit is demonstrated in Fig. 20.26 by plotting longer time series and phase space trajectory. The periodic orbit shows a development on a relatively slow time scale until the periodicty is suddenly lost and the behaviour becomes irregular instead. The existence of UPOs in weakly developed drift wave turbulence encourages experiments on chaos control. It turned out, however, that single-parameter control (TDAS with perturbation of the grid bias) has a rather poor performance

20.5 Taming turbulence

UPO P=1

I/ V V time (ms)

UPO P=2

TRANSITION

time (ms)

time (ms)

553

0.3

Figure 20.26 Unstable periodic orbits (UPOs) in weakly developed drift wave turbulence. The top row shows (from left to right) phase space reconstructions of UPOs of periodicity one and two and the instability of the period-two UPO. The bottom row shows the corresponding time series.

and is extremely difficult to adjust. There are two reasons that make control of drift wave turbulence difficult. First, the control parameter (the rotation frequency of the whole plasma column) acts rather globally on the plasma equilibrium and has a dynamical response below the drift wave time scale. Second, the regime with low-dimensional chaos is narrow and must be seen as a precursor to turbulence. Changes in the control parameter are thus likely to bring the plasma into the regime of drift wave turbulence where the concepts of standard chaos control fail. Current work on taming drift wave turbulence follows the idea to try open-loop control by azimuthally distributed local perturbation. Though first results are encouraging [Latten, 1997] the performance of the control method is not always good and further investigations are required. Additionally, future experiments are suggested to attack the control of drift wave turbulence by nonlinear multi-parameter control. Feedback Stabilization of Turbulence Much interesting progress was made in the field of feedback stabilization of the actual plasma instability [Tham et aL, 1991, Tham & Sen, 1992a, Tham & Sen, 1992b ]. The basic mechanism of the suppression is the cancellation of charge separation (thereby of electric fields) inherent in an instability by a modulated ion

554

20 Control of chaos in plasmas

100 120 Frequency (kHz)

140

Figure 20.27 Feedback suppression of an ion-temperature-gradient driven mode for the two feedback gains of 0.005 and 0.009. From [Tham & Sen, 1992b].

beam. The injected ion beam is typically nonperturbing to the plasma, constituting only 1% of the plasma background density. The classical feedback scheme (cf. Fig. 20.1(b) in Sec. 20.2) is used where the output signal of an electric probe in the plasma is first passed through a preamplifier, a bandpass filter, and a phase shifter and then passed through a second amplifier. The output signal of the amplifier is used to modulate the ion beam energy. An experimental result of the suppression of an ion-temperture-gradient (ITG) driven mode [Greaves et a/., 1992] is shown in Fig. 20.27. By increase of the feedback gain (ratio of electron beam density to equilibrium plasma density) the ITG mode was completely suppressed at 0.009. The feedback gain required for suppression of the instability depends on the particular discharge configuration, but the feedback scheme seems to be sufficiently robust to achieve control in a relatively broad parameter range [Tham & Sen, 1992b]. The control of instabilities by classical feedback is definitely a successful approach and finds applications in many different fields of plasma physics, for instance in fusion physics [Zhai et a/., 1997]. For each type of plasma instability, however, an individual feedback scheme has to be developed since there is no general theory available. From a basic point of view, this feedback technique attempts to stabilize a fixed point, starting either from a limit cycle or even from a trubulent state. It is important to note that the actual concepts of chaos control follow a different idea, the stabilization of unstable but periodic (or even stationary) states that are embedded in the uncontrolled dynamics.

20.6 Summary and outlook In the present contribution an overview was given over recent developments in experimental chaos control of plasmas. The established control techniques, both

References

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open-loop and closed-loop, have proven to find applications in very different types of plasma experiments. The OGY control scheme was systematically applied to the Pierce-diode, a simple model system of a bounded plasma diode. Its simplified variant, OPF control, is successful in the control of ionization wave chaos. TDAS control has proven to be a robust technique in ionization wave chaos, too, and the required time delay may be directly obtained by spatially displaced detectors. Open-loop control is an appealing simple technique and were chosen in first work on chaos control in experimental plasma diodes. Stochastic resonance received much recent attention as an open-loop control technique since it introduces a certain ceherence in stochstic rather than chaotic behaviour. This effect was shown to be highly significant in oscillating thermionic plasma diodes. Finally, we gave a few comments on the prospects of taming turbulence in plasmas. This is for sure the most demanding subject of current research and it is expected that future developments in the control of dynamical systems will contribute to progress in this important field.

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[Lawson, 1989a] Lawson, W. S. 1989a. Particle Simulation of Bounded ID Plasma Systems. J. Corn-put. Phys., 80(2), 253-276. [Lawson, 1989b] Lawson, W. S. 1989b. The Pierce Diode with External Circuit. I. Oscillations About Nonuniform Equilibria. Phys. Fluids, Bl(7), 1483-1492. [Lawson, 1989c] Lawson, W. S. 1989c. The Pierce Diode with External Circuit. II. Chaotic Behavior. Phys. Fluids, Bl(7), 1493-1501. [Lima & Pettini, 1990] Lima, R., & Pettini, M. 1990. Suppression of Chaos by Resonant Parametric Pertubations. Phys. Rev. A, 41(2), 726-733. [Matsumoto et al, 1996] Matsumoto, EL, Yokoyama, H., & Summers, D. 1996. Computer Simulation of the CHaotic Dynamics of the Pierce Beam-Plasma system. Phys. Plasmas, 3(1), 117-191. [Mausbach et al, 1997] Mausbach, T., Klinger, T., Piel, A., Antipo, A., Pierre, T., & Bonhomme, G. 1997. Continuous Control of Ionization Wave Chaos by Spatially Derived Feedback Signals. Phys. Lett. A, 228, 373-377. [Mausbach et al, 1998] Mausbach, T., Klinger, T., & Piel, A. 1998. Spatiotemporal evolution of controlled chaos in a discharge plasma. Phys. Rev. E. in preparation. [McNamara & Wiesenfeld, 1989] McNamara, B., & Wiesenfeld, K. 1989. Theory of Stochastic Resonance. Phys. Rev. A, 39(9), 4854-4869. [Nicholson, 1983] Nicholson, D. R. 1983. Introduction to Plasma Theory. New York: Wiley. [Niedner et al, 1997] Niedner, S., Schuster, H.-G., Klinger, T., & Bonhomme, G. 1997. Biorthogonal decomposition analysis of ionization wave turbulence. Phys. Rev. E. submitted. [Ogata, 1990] Ogata, K. 1990. Modern control enineering. 2 edn. New Jersey: Englewood Cliffs. [Ohe, 1989] Ohe, K. 1989. Nonlinear Phenomena of Ionization Waves. Page 150 of: Zigman, V. J. (ed), Proc. of XlXth ICPIG, Invited Papers. [Ohe & Tanaka, 1988] Ohe, K., & Tanaka, H. 1988. A Strange Attractor and its Dimension of Ionisation Instability. J. Phys. D: Appl. Phys., 21, 1391-1395. [Oleson & Cooper, 1968] Oleson, N. L., & Cooper, A. W. 1968. Moving Striations. Adv. Electron. Electron Phys., 24, 155-278. [Ott & Spano, 1995] Ott, E., & Spano, M. 1995. Controlling chaos. Physics today, 5, 34-40.

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[Ott et al, 1990] Ott, E., Grebogi, C., & Yorke, J. A. 1990. Controlling Chaos. Phys. Rev. Lett, 64(11), 1196-1199. [Ott et al, 1994] Ott, E., Alexander, J. C, Kan, I., Sommerer, J. C, & Yorke, J. A. 1994. The transition to chaotic attractors with riddled basins. Physica D, 76, 384-410. [Packard et al, 1980] Packard, N. H., Crutchfield, J. P., Farmer, J. D., & Shaw, R. S. 1980. Geometry from a time series. Phys. Rev. Lett, 45, 712-716. [Parlitz & Lauterborn, 1987] Parlitz, U., & Lauterborn, W. 1987. Period-doubling Cascades and Devil's Staircases of the Driven van der Pol Oscillator. Phys. Rev. A, 36(3), 1428-1434. [Pekarek, 1971] Pekarek, L. 1971. Ion Waves and Ionization Waves. Page 365 of: Davenport, P. A. (ed), Proc. of Xth ICPIG, Invited Papers. [Pekarek & Krasa, 1974] Pekarek, L., & Krasa, J. 1974. Ionization Waves in Plasmas. Pages 915-957 of: Vujnovic, V. (ed), Proc. of. 7th Yugoslav Symposium and Summer School on the Physics of Ionized Gases, Invited Lectures. [Pierce, 1944] Pierce, J. R. 1944. Limiting Stable Current in Electron Beams in the Presence of Ions. J. Appl. Phys., 15, 721-726. [Pierre et al, 1996] Pierre, T., Bonhomme, G., & Atipo, A. 1996. Controlling the chaotic regime of nonlinear ionization waves using the time-delay autosynchronization method. Phys. Rev. Lett, 76, 2290-2293. [Pyragas, 1992] Pyragas, K. 1992. Continuous Control of Chaos by Self-Controlling Feedback. Phys. Lett. A, 170, 421-428. [Qin et al, 1989] Qin, J., Wang, L., Yuan, D. P., Gao, P., & Zhang, B. Z. 1989. Chaos and Bifurcations in Periodic Windows Observed in Plasmas. Phys. Rev. Lett, 63(2), 163-166. [Raizer, 1991] Raizer, Y. P. 1991. Gas Discharge Physics. Berlin: Springer. [Roessler, 1976] Roessler, O. E. 1976. An equation for continuous chaos. Phys. Lett A, 57, 397-398. [Romeiras et al., 1992] Romeiras, F. J., Grebogi, C, Ott, E., & Dayawansa, W. P. 1992. Controlling chaotic dynamical systems. Physica D, 58, 165-192. [Roy et al, 1992] Roy, R., Murphy, T. W., Maier, T. D., Gills, Z., & Hunt, E. R. 1992. Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system. Phys. Rev. Lett, 68, 1259-1262. [Shinbrot, 1995] Shinbrot, T. 1995. Progress in the control of chaos. Adv. Physics, 44(2), 73-111.

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21 Chaos Control in Spin Systems H. Benner and E. Reibold Institut fur Festkorperphysik, Technische Universitat Darmstadt, D-64289 Darmstadt, Germany

21.1 Introduction Spin systems are considered to be intriguing paradigms for studying nonlinear dynamics in real experimental systems. The reason for this is related to intrinsically nonlinear interactions already present in their basic equation of motion, M(r, t) = M(r, t) x H c //(r, t) + dissipation.

(21.1)

The evolution of the magnetization M(r, t) in time and space is determined by the torque of an effective field

(21.2) For ferromagnetic insulators considered here, this effective field is composed of external dc and ac magnetic fields as well as of the internal demagnetizing and single-site anisotropy fields, the dominating exchange field, and the dipolar field. Since the internal fields themselves depend on the magnetization, they all give rise to nonlinearities. Usually, in textbooks this equation is only discussed in a linearized form restricting its application to weak excitation (h < H, \A-M.\, etc.) and small deviations of M(r, £) from thermal equilibrium M o . However, even when considering only the uniform part of magnetization, the exact solution of (21.1) shows a bistability ("foldover effect") at moderate excitation amplitudes [1]. Moreover, the additional effect of the non-local exchange and dipolar fields may result in more complicated threshold phenomena indicating self-induced formation of spatio-temporal structures. Part of these phenomena have been well known for decades [2, 3, 4] from highpower ferromagnetic resonance (FMR) experiments, and have theoretically been explained to result from the parametric excitation of spin waves through transverse or parallel pumping on the uniform magnetization. Suhl's famous first- and secondorder instabilities [5] as well as the parallel pumping instability [6] are related to the details of this parametric process, where pairs of spin waves are excited at half the pumping frequency, Uk == ^ / 2 , or directly at ujk — UJ. Parametrically excited spin waves show a variety of nonlinear phenomena, low-frequency auto-oscillations, and a very complex multistability accompanied by sequences of bifurcations. Period doubling routes, quasiperiodicity and mode-locking, various types of intermittency

564

21 Chaos Control in Spin Systems

and chaos have been widely studied and reviewed in the literature [7, 8, 9]. In spite of the efforts to observe, to analyse and to understand the chaotic behaviour of nonlinear systems, irregular and unpredictable motion is generally not desired in practical applications. Thus, the suppression or control of chaos has become the subject of intensified research recently. Different strategies have been proposed of how to change the irregular into regular motion (for a review see e.g. [10]). Simple non-feedback methods are generally based on the indirect change of some system parameter and require higher perturbation power. More sophisticated feedback methods, as described in the preceding chapters, aim at the stabilisation of existing unstable periodic orbits. Since the latter techniques make use of the intrinsic properties of the underlying chaotic attractor, they generally can be run with very small controlling power, but require detailed information on the system and are rather intricate to perform. The specific challenge of analyzing and controlling real spin systems is mainly related to their complex dynamic behaviour and to the fast time scale of regular and irregular auto-oscillations, which is of the order of a microsecond or even less. Therefore, sophisticated concepts which are based on real-time numerical analyses are generally too slow and can only be applied in a modified way. Here, in an exemplary way, we focus on recent experimental results obtained on the low-dissipative ferromagnet yttrium iron garnet (YIG) at the first-order Suhl instability (Section 21.2). Since the complexity of this system can be varied within a wide range by the proper choice of external parameters (e.g. pumping frequency, input power, magnetic field, orientation, and temperature), it could be used as a (nearly) ideal model system for probing general concepts which generally work quite well in computer simulations or in tailored mechanical or electrical devices, but often fail in real systems, especially in solid state physics. In Section 21.3 we demonstrate the suppression of chaos by fast parametric modulation [11]. It is shown that the increase of modulation amplitude actually results in an effective variation of the modulated control parameter, and the obtained suppression of chaos is essentially related to a global change of the system. A critical review of the literature shows that most of the present attempts to "control" spin-wave chaos, even when sold under the label of popular feedback techniques, are basically related to this mechanism. In contrast, the method of Ott, Grebogy, and Yorke [12] is based on the precise knowledge of existing unstable periodic orbits, eigenvalues, and eigendirections, and allows the suppression of chaos with very slight external perturbations by making use of the intrinsic system dynamics in a rather sophisticated way (Section 21.4). Application to spin-wave chaos is only possible in terms of modified schemes, e.g. "occasional proportional feedback" [13], which have been implemented by means of a fast analog feedback device. The time-delayed feedback method of Pyragas [14] may be considered as an efficient compromise combining simplicity of application with low controlling power (Section 21.5). Though this method has been successfully applied to many real experiments for some years, a theoretical understanding was supplied only very recently [15] and its efficiency can now be optimized.

21.2 Ferromagnetic Resonance at Spin-Wave Instabilities

565

21.2 Ferromagnetic Resonance at Spin-Wave Instabilities Existing nonlinear theories describing the dynamics of spin systems are mainly based on a perturbative approach, i.e. on the idea of a set of weakly coupled eigenmodes with nonlinear couplings between them becoming efficient only at high amplitudes. Appropriate eigenmodes of eq. (21.1) are the uniform mode m o (t) = bo exp(zu;o£), which is driven by the external microwave field, and non-uniform spin waves and magnetostatic modes m^(r,t) — b^.(r) exp(—iu)^t\ which are not directly affected by the microwaves. Expanding the magnetization M(r,£) into eigenmodes, eq. (21.1) yields a set of coupled amplitude equations with nonlinear couplings determined by the terms of higher-order ("multi-magnon processes"). Suhl's first-order spin-wave instability [5], to which our experiments refer, is based on the parametric excitation of spin waves through transverse pumping on the uniform mode. The parametric process is characterized by the decay of the pumped uniform mode into two spin waves of half the pumping frequency u^ = UJP/2 and opposite wave vectors (k, -k), according to the conservation of energy and quasimomentum. This instability can either be observed off resonance (i.e. with the microwave pumping frequency far away from the usual ferromagnetic resonance, up / LOQ) as a subsidiary absorption, or directly on the FMR line (LJP = LJQ) within the coincidence regime. Note that in ferromagnetic spheres u0 = 7 H is proportional to the magnetic field (7 is the gyromagnetic ratio), while u^ depends in a more complicated way on H as well as on the magnetization and on the magnitude and direction of wave vector k.

21.2.1 Experimental Set-Up A prototype ferromagnet used most often for the investigation of nonlinear spin dynamics is yttrium iron garnet (YIG) Y^Fe^On. To be exact, YIG represents a ferrimagnet of cubic symmetry. Its magnetic Fe 3 + ions (6S$/2) are located on tetrahedral sites (3 per unit cell) and octahedral sites (2 per unit cell). Magnetic order occurs below 550 K. The magnetic moments on either site form ferromagnetic sublattices of antiparallel orientation with weak preference of the (111) direction. The excitation of " antiferromagnetic" modes occurs at much higher frequencies than applied in the present FMR investigations, so one merely probes the "ferromagnetic" properties of the net magnetization, which is still rather strong at room temperature. The most important property of YIG, however, is its low dissipation. The resonance line of high-quality samples shows a width of less than 0.5 Oe, resulting in extremely low thresholds for spin-wave instabilities (typically some 10 fiW pumping power in the coincidence regime). In view of such low thresholds high-power FMR experiments can be performed with a conventional ESR spectrometer. We have studied the subsidiary absorption at about 9.3 GHz. Instead of a standard reflection-type cavity we used a bimodal transmission-type cavity of quality factor 3000, see Fig. 21.1, which allows a nearly complete separation of the

566

21 Chaos Control in Spin Systems

Figure 21.1 Experimental set-up. The cross-section through the bimodal cavity (r.h.s.) shows the magnetic component of the microwave mode used for excitation of the YIG sample (full) and of the degenerate orthogonal mode selecting its response (dashed). The transmitted signal is amplified, rectified, and digitally recorded.

strong microwave input power from the weak time-dependent output signal. This way the signal-to-noise ratio was improved by almost 20dB. The squared amplitude of the driving field h at sample position is proportional to the input power Pin, which was supplied by a microwave generator, and the transmitted signal Ptr is proportional to the squared amplitude | bo | 2 of the uniform mode. By means of a digital oscilloscope and an integrating voltmeter we recorded both the time dependence of Ptr (t) and its time average Ptr on variation of input power P{n and magnetic field H. The data presented below were obtained at room temperature on a highly polished sphere of pure YIG, 0.71 mm in diameter, and the magnetic field was applied either in (100) or in (111) direction.

21.2.2 Observed Phenomena The subsidiary absorption is manifested as an additional absorption structure at lower field, which is well separated from the FMR main resonance and shows a drastic broadening with increasing microwave power, accompanied by auto-oscillations and sequences of bifurcations. We have systematically analysed [16] the dynamic behaviour of the subsidiary absorption signal at fixed pumping frequency, as presented in Fig. 21.2. The lower line shows the dependence of the Suhl threshold on H (the so-called butterfly curve). (Here and in the corresponding figures below Pin was normalized to the minimum threshold.) The next line indicates a Hopf bifurcation and corresponds to the onset of auto-oscillations. Further bifurcation lines above separate regimes of different time behaviour, e.g. period doublings, quasiperiodicity, intermittency, or chaos. The steep increase of the threshold at 2.2 kOe

21.2 Ferromagnetic Resonance at Spin-Wave Instabilities

567

16 H

12 -

I

8 -

o o

E 4H 1 st order Suhl threshold 0 1600

1800

2000

2200

magnetic field (Oe)

Figure 21.2 Dynamics in the subsidiary absorption regime (y — 9.26GHz) with respect to magnetic field H and input microwave power Pin. The lowest line indicates the Suhl threshold, the lines above separate regimes of different time behaviour, e.g. period doublings (P2, P4), quasiperiodicity (QP), or chaos. Intermittency is observed in several parameter regimes, e.g. type II at 1650Oe/12 - 15dB; type III at 1900Oe/ll - 15dB; crisis-induced intermittency at 1900Oe/16dB.

indicates that the bottom of the spin-wave band becomes larger than o;p/2, and the parametric excitation of spin waves is no longer efficient.

21.2.3 Routes to Chaos As a general result, we found that a global correspondence to one of the wellknown scenarios of Feigenbaum, Ruelle-Takens-Newhouse or Pomeau-Manneville [17] does not occur, but instead a variety of parts from all of them. This obviously corresponds to the fact that the nonlinearities of a real system are more complicated and based on a larger number of internal degrees of freedom than those of the simple models from which these standard routes have been derived. The physical meaning of the degrees of freedom is probably that of specific eigenmodes or a collective motion of several of them. Period-doubling routes, as reported previously from both transverse and parallel pumping experiments [18, 19], were observed up to period 8, but occurred rather seldom. Even more rarely we actually observed a sequence of period triplings (not to be confused with a period-3 window!) up to

568

21 Chaos Control in Spin Systems

0.20

t [ms]

0.32

t [ms]

Figure 21.3 Different types of intermittency in spin-wave instabilities, a) PomeauManneville type I, b) Pomeau-Manneville type III, c) chaos-chaos intermittency due to a homoclinic crises, d) on-off intermittency, e,f) extended time scales of c,d.

period 9. More often, however, only a single period doubling was found, which remained stable for a rather extended parameter range and then changed directly over to chaos. Quasiperiodicity with up to three fundamental frequencies was observed both in subsidiary absorption (see Fig. 21.2) and in the coincidence regime. A few dB above the threshold, the FMR signal starts to auto-oscillate with typical frequencies ranging from 100 to 400kHz. Slightly above, a second fundamental frequency occurs - corresponding to a second Hopf bifurcation - together with several mixing frequencies and harmonics, which indicate that the attractor is a 2-torus. Very rarely, we also found a third fundamental frequency occurring within an extended parameter range. More often, instead of a third Hopf bifurcation and a collapse of the resulting 3-torus to chaos [20], we observed the spin system switch over to a coexisting stable attr act or. The changeover to chaos was generally accompanied by a jump of Ptr, but did not arise from a 2- or 3-torus. Hence it could not be related to a Ruelle-Takens-Newhouse scenario. Instead we suppose that the chaotic behaviour results from a sudden increase of the number of coupled modes, which is related to a global symmetry-breaking bifurcation [16] and does not follow one of the standard routes. We also observed various kinds of intermittency starting from a fixed point, a limit cycle, a 2-torus, or even alternating between different chaotic states (see Fig. 21.3). Often specific types could already be distinguished from their character-

21.3 Nonresonant Parametric Modulation

569

istic time behaviour, but also from a reconstruction of the generating map and from the distribution and scaling behaviour of the laminar lengths [17]. The observed signals could clearly be attributed to each of the Pomeau-Manneville types I, II, or III or to crises [16, 21]. It is interesting to note that 'classical' intermittency changing between regular and irregular behaviour was generally observed in parameter regimes where the system remains low-dimensional, whereas chaos-chaos intermittency occurred at higher dimensions, especially in the coincidence regime. Here we found that in most cases the observed chaos-chaos intermittency shows a scaling like Pomeau-Manneville type III, which is not consistent with the common interpretation of arising from a crisis [22]. Instead, we suggested an interpretation in terms of 'on-off intermittency' based on a global symmetry-breaking bifurcation [23]. The underlying physical mechanism is probably that of the transitory excitation of an additional spin-wave mode through a 3-magnon process [16, 21, 24]. The analysis of the observed chaotic behaviour shows a significant difference between the conditions of resonant and nonresonant pumping: In the coincidence regime we generally observe higher-dimensional chaos ("hyperchaos") with correlation dimensions JD2 = 7... 15, whereas in subsidiary absorption low-dimensional ("marginal") chaos of dimension D2 = 2...3 prevails [16]. Accordingly, our attempts to suppress or control the irregular time behaviour were limited to the latter case.

21.3 Nonresonant Parametric Modulation We first focus on the method of nonresonant parametric modulation, which actually requires no knowledge of the system and can be applied to a wide class of systems. We demonstrate its simple applicability to spin-wave chaos observed at higher pumping power in subsidiary absorption. The large universality, however, can only be obtained by lifting the condition of low control amplitudes. Thus, parametric modulation actually results in an effective change of the modulated control parameter, and the obtained suppression of chaos is essentially related to a global change of the system.

21.3.1 Analytical and Numerical Approach Theoretical understanding of this method can be achieved by considering the arbitrarily chosen example of a damped and driven Duffing oscillator x - ax + fix3 = -jx + Fcos((jt),

(21.3)

with the control parameter a being modulated with a high frequency Q !^> u> : a(t) =a[l + ecos(Slt)] .

(21.4)

By means of an analytical technique based on the separation of slow and fast time scales of x(t) and by comparing independent harmonics of the rapidly oscillating terms [11], we found that the dynamics of the slowly varying part x(t) (averaged over an oscillation period 2TT/O ) approximately follows the same equation of motion

570

21 Chaos Control in Spin Systems

(21.3) as the original Duffing oscillator without modulation. However, this equation has renormalized coefficients a, J3, and 7, which up to the order of a2/ft4 are given by

[ f (hY] ^ = /?' ^ = J'

(2L5)

Thus, for the slow timescale, the continuous increase of the modulation amplitude e acts in the same way as a reduction of the effective parameter a. This means that fast parametric modulation should result in the same scenario as obtained when directly decreasing the respective control parameter. By means of the well-known Melnikov criterion [25], we were able to calculate the values of a (or e) where the chaotic behaviour is suppressed and the signal becomes regular. For the case of small modulation amplitude, the resulting condition for the suppression of chaos then reads 6 > yfljOL n .

(21.6)

We checked these analytical results by numerical simulations of eqs. (21.3),(21.4), calculating the time-dependent signal, the power spectrum, and the relevant Lyapunov exponent. For certain initial conditions (ao = / 3 = CJ = 1 , F = 0.35 and 7 = 0.4) we found a symmetric chaotic attractor. Then, switching on the fast (Q, = 10) parametric modulation and increasing its amplitude e from 0 to 20, the system actually steps back through a scenario of different bifurcations, periodic windows, and finally ends up in periodic oscillations [11, 16]. For further illustration of the scenario, we extracted the characteristic Lyapunov exponent Ai from the simulated time series with respect to the modulation amplitude e (Fig. 21.4, top). Note that there is only one relevant exponent Ai for the Duffing oscillator, while the second one is determined by dissipation: A2 = — 7 - Ai. l Two important results were obtained [11]: (i) The exponent Ai becomes negative for large values of e , and a regular motion is actually recovered, (ii) Periodic windows with a negative exponent Ai are observed, and bifurcations occur as indicated by the vanishing of Ai . The equivalence between the continuous increase of the modulation amplitude and the continuous decrease of the effective control parameter a in the unmodulated system was shown by calculating the Lyapunov exponent of the unmodulated system in dependence on a (Fig. 21.4, bottom). The equivalence was emphasized by rescaling the x-axis according to eq. (21.5), i.e. by using the nonlinear scale /(a) = y/2(ao — a) fl/ao. This direct comparison of numerical simulations yields striking evidence of the fact that the direct variation of an external parameter gives rise to the very same scenario as the high frequency parametric modulation, and that the basic mechanism of chaos suppression by fast parametric modulation can be well understood to arise just from a renormalization of external control parameters. The analytical approach formulated above can, in principle, be applied to any system of differential equations. The effect of modulation on one of the control lr

The formal inclusion of a third exponent related to the external periodic forcing does not change this balance since A3 = 0 .

21.3 Nonresonant Parametric Modulation

571

0.21

10 ,

15

f(oc) Figure 21.4 Numerical simulation of chaos suppression by fast parametric modulation. Top: Decrease of the largest Lyapunov exponent with increasing modulation amplitude e for constant parameter a = ao- Bottom: For comparison: Scenario without modulation, but on direct variation of a, using a nonlinear scale related to eq. (21.5), f(a) = y/2(ao — a) O/ao-

parameters, as depicted by eq. (21.5) for the case of the Duffing oscillator, however, specifically depends on the system under consideration. For example the modulation-induced change of an effective parameter could be of opposite sign, which means that chaos suppression is not achieved in every attempt. In all cases of resonantly driven oscillators, however, the sign is not relevant, since any shift of "eigenfrequencies" will move the system out of resonance and, thus, increase the threshold of chaotic excitation.

21.3.2 Experimental Suppression of Spin-Wave Chaos The generality of the fast modulation approach could be demonstrated by applying it to the real experimental situation. We chose the chaotic regime around H = 1900Oe and Pin — 11.5dB in Fig. 21.2. The chaotic behaviour was analysed [16] to be of marginal type, i.e. characterized by only one positive Lyapunov exponent and a correlation dimension between 2 and 3. This chaotic regime is entered from the high field side through the following scenario (reverse to the dashed arrow in Fig. 21.2): Starting at 1950Oe/11.5dB inside a period-1 regime (PI) and decreasing H, one passes through a period-2 regime (P2), and finally through classical type III-intermittency to a chaotic state at 1850Oe. Because of the resonance behaviour of spin-wave excitation, the decrease of field at fixed frequency may be considered equivalent to an increase of frequency at fixed field, which is hard to check in our experiment for technical reasons. So, the scenario expected for increasing pumping frequency should read period-1 oscillation -> period-2 oscillation -» intermittency type III -» chaos. In our modulation experiment [26], we fixed the control parameters in the chaotic regime at H - 1900Oe and Pin = 11.5 dB. (With respect to Fig. 21.2 the bifurcation lines were slightly shifted to higher field owing to a different sample orien-

572

21 Chaos Control in Spin Systems

tation.) A time series of the unperturbed chaotic attractor is shown in Fig. 21.5. We then switched on a frequency modulation of the microwave field: UJP

= UJPJ0 + ALJ sin(ftt) .

(21.7)

The modulation frequency was fixed at ft/27r = lA90MHz, and the modulation depth Au was continuously increased, which should correspond to a decreasing effective pumping frequency. The time series and corresponding power spectra observed for increasing modulation depth are presented in Fig. 21.5. The uppermost diagrams (ALJ = 0) are related to the unmodulated chaotic system. The spectrum is broadband around some typical system frequency f8. After switching on the modulation, its frequency appeared as the strongest peak in the power spectrum. Increasing ALJ/2TT to 2.0MHz we observed intermittency with the laminar phases showing a period-2 oscillation of diverging amplitudes. This is typical for intermittency type III. Simultaneously, the broad spectrum decreased in amplitude and a sharper peak at a system frequency fs = 750kHz developes from the noisy background, accompanied by a weak subharmonic at / s / 2 . For ALJ/2TT — 3.6MHz the broad bump had moved down to the noise level, indicating the successful suppression of chaos. Simultaneously, the subharmonic at fs/2 showed up very clearly, but vanished again at higher modulation depths. Thus, by increasing the modulation depth we observed the scenario: chaos —> intermittency type III -> period-2 oscillation —> period-1 oscillation. This is just the reverse of what we observed on direct variation of the magnetic field H and confirms that high-frequency parametric modulation results in a " route out of chaos" as predicted by our analytical and numerical calculations. The scheme of parametric modulation seems to be applicable to almost every system. In principle, it should be capable of suppressing even more than only one positive Lyapunov exponent and, therefore, is basically not restricted to marginal chaos. Moreover, parametric modulation is very easy to apply, since it requires no knowledge about the system under consideration, and even the sign of its influence is of no importance in many cases. In real experiments it allows a simple access to fast systems, but it has to be classified as a brute force method which drastically changes the system. The indirect variation of a system parameter results in new regular orbits of different topology and larger size, which became evident in our experiment when comparing the reconstructed attractors of the unperturbed and of the modulated system. In spite of the very small change of pumping frequency by less than 10~3 the resulting regular orbit was far outside the original attractor and of different shape. Previous control experiments on spin-wave turbulence by Azevedo [27] and by Ye [28] ("open-loop control') were based on exactly this mechanism. In both papers parametric perturbations were made on the magnetic field. Profiting from the very sharp resonance condition of YIG (AH/Hres ~ 10~4), regular orbits could be stabilized by nominally extremely small field variations, which still changed the system dramatically.

21.3 Nonresonant Parametric Modulation

573

f[MHz] Figure 21.5 Scenario obtained in subsidiary absorption (UJVJCI'K — 9.26GHz, H — 1.90fcOe, Pin = H.bdB) on modulation of the microwave frequency. Timedependent FMR signals [a.u.] and power spectra [dB] for increasing modulation depths AU/2TT = 0, 1.4, 2.0, 3.6, and 5.8MHz. Note that the stabilized orbits are of larger amplitude and cannot originate from the unmodulated chaotic attract or.

574

21 Chaos Control in Spin Systems

21A Occasional Proportional Feedback Controlling chaos, in a strict sense, means moving from irregular to regular behaviour, not through a global change of the system as demonstrated in the preceding section, but by slight, local perturbations on the internal dynamics. Ott, Grebogi, and Yorke (OGY) first realized that a chaotic attract or has embedded within it an infinite number of unstable periodic orbits, which, in principle, could be stabilized by conventional techniques of control engineering. According to this tradition the vanishing of the control signal is considered to be a general criterion for successful control. OGY proposed a method [12] to achieve control by means of a feedback technique, where small time-dependent perturbations are made on one of the system parameters. If the trajectory goes near such a hyperbolic orbit, it approaches the orbit along the stable manifold, and leaves it again along the unstable manifold. Close to the orbit, the attracting and repelling phases evolve exponentially in time, so that the motion can be slow enough to be efficiently affected by weak external perturbations. Although the OGY method and its applications have widely been discussed in preceding chapters, we briefly recall its main steps in order to define our notation and to make modifications, necessary for fast experimental systems, more transparent.

21.4.1 The OGY Concept Unstable periodic orbits (UPOs) can be reconstructed by means of the recurrence time method [29] and characterized by their topological properties [30]. In order to stabilize a specific orbit by slight corrections on an externally accessible control parameter p, one has to know the exact position of this orbit as a function of p and the evolution of trajectories in its neighbourhood. Technically, this is achieved by reconstructing the trajectory (of e.g. a 3D flow) from an experimental time series by means of time-delayed coordinates. The problem is simplified by applying a Poincare section perpendicular to the UPO, thus converting the 3D flow to a 2D discrete map (Fig. 21.6). This way, the UPO is mapped to a hyperbolic fixed point £F(PO), whose stability has to be analysed. Stable and unstable eigenvalues Xs and Xu and the respective eigendirections es and eu can be determined from the evolution of subsequent intersection points. Such analysis yields a linearized prediction of the system dynamics in the neighbourhood of £F(po), where the evolution matrix is expressed in terms of the stable and unstable eigenvalues and eigendirections ( eSyU and e* u denote the covariant and contravariant eigenvectors, respectively): £n+i " ZF(PO) ^ (es\se* + eu\ue*u) • [£n - £F(po)] .

(21.8)

Next, one considers a small perturbation of the control parameter p, assuming that mainly the position of the fixed point is affected, but essentially not the eigenvalues

21.4 Occasional Proportional Feedback

575

Figure 21.6 Controlling scheme of Ott, Grebogy, and Yorke. The Poincare section maps the unstable periodic orbit to the fixed point £^(po) with its stable and unstable directions es and eu . The dashed arrows indicate the effect of a small change of the control parameter p.

and eigendirections. The corresponding shift g of the fixed point is determined by (21.9) up p — po The basic idea of OGY is to shift thefixedpoint and the corresponding course of the trajectory for a short time in such a way that after one cycle the next intersection point £ n+1 (p =fi po) ends up on the stable manifold of the original fixed point an< €F(PO) > 3 then to switch off the perturbation again. This way the subsequent intersection points £ n + 2 , £n+3 ,••• exactly approach the fixed point along its stable manifold without being repelled, i.e. after a well-targeted perturbation the intrinsic dynamics of the system shifts the trajectory towards the UPO. The value of this perturbation is obtained by combining eqs. (21.8), (21.9), and the condition that £ n + i — £F(PO) be orthogonal to the unstable manifold [12]: (21.10) 9'e*u (A slightly different expression which is asymptotically equivalent for A u « l has been given by Dressier and Nitsche [31].) Once the system has approached the orbit, the still necessary corrections due to linearization and noise can be maintained by very small perturbations. Though originally developed for discrete maps, this concept can also be applied to low-dimensional continuous flows, but in general is limited to a single unstable direction. Extensions to control chaos with several unstable directions have been discussed in the literature [10].

576

21 Chaos Control in Spin Systems

21.4.2 Experimental Control by an Analog Feedback Device Although the OGY method should apply to real experimental systems as well, in practice its application is restricted for the following reasons: (i) Experimental systems often show high-dimensional chaos, i.e. there is more than one unstable direction, (ii) The measured signal is disturbed by noise; this may either prevent the control to work at all, if in the case of strong noise the system is strongly pushed away from the neighbourhood of the fixed point, or at least reduces the sensitivity of the feedback in the case of weak noise, (iii) The characteristic time scale of real systems is often too fast. For the spin system investigated, typical cycle times are in the order of microseconds, whereas the numerical phase space reconstruction and calculation of the feedback signal requires at least some milliseconds. The first problem could be overcome by selecting chaotic signals of sufficiently low dimensionality. Dimensional analyses show that this is nearly impossible for the coincidence regime, but easy to obtain for subsidiary absorption. The dashed area in Fig. 21.2 marks the corresponding control parameter range where marginal chaos was observed. The chaotic signal to be controlled was of the same type as in the previous section, i.e. characterized by a correlation dimension D2 = 2.1 db 0.1, a single positive Lyapunov exponent Ai = 0.04 (/is)" 1 , and a mean cycle time of about 2 jis. Since it was impossible to make numerical real-time predictions for a time scale of /J,S we had to modify the OGY algorithm in a way so that it could be processed by an intelligent analog feedback device. For reconstructing the attract or we used analog time derivatives instead of time delay coordinates. The Poincare plane and the location of intersection points were determined by analog window discriminators for the signal and its first and second time derivatives triggering a track-and-hold amplifier (Fig. 21.7). The amplitude of the control signal was not determined from a preceding stability analysis of the periodic orbit. Instead, we used a feedback signal which was proportional to the deviation of the momentary signal U(tn) from the set-point Uref given by the U-coordinate of the centre of the phase space window: Ufb(tn) = A-[U(tn)-Uref}.

(21.11)

By variation of the windows settings we moved the corresponding probe volume through the phase space trying to find the neighbourhood of some unstable periodic orbit where the given feedback would result in more or less appropriate control. A similar concept called " occasional proportional feedback? (OPF) was applied by Hunt [13] (see also the chapter by Hunt) for controlling a diode-capacity resonator. He used a stroboscopic mapping for the Poincare section and only one window in order to control non-autonomous, periodically driven systems. The multiple window technique applied in our device allows this concept also to be extended to autonomous systems, such as spin-wave chaos, which are not externally synchronized by periodic excitation. 2 There is complete correspondence of eq. (21.11) to 2

Note that the external microwave excitation of our spin system acts on a time scale of 0.1 ns whereas spin-wave autooscillations occur typically at some /xs, so their time scale is fully separated from the external drive.

21.4 Occasional Proportional Feedback

577

microwave generator

uref

A[U ref -U(t n )]

u(tn) track & hold amplifier

resonator

U(t)

static magnetic field H

TTL

detector window generator

amplifier

ref

|J

Figure 21.7 Analog feedback device for controlling the chaotic FMR signal. Top: Experimental set-up. Bottom: Reconstruction of phase space by means of analog derivatives. The small probe volume can be moved around by varying the settings of the window generator.

578

21 Chaos Control in Spin Systems

the original result of OGY, eq. (21.10), if we identify the amplification factor A with \u(K-l)~ll{9-e*u) and the deviation U(tn)-Uref with [£ n (po)-£F(PO)]-< . In our experiment the position of the unstable periodic orbit was selected by setting windows for the observed signal U(i) and its first and second time derivative by means of three analog window comparators (Fig. 21.7). The corresponding signal was held by a fast track-and-hold amplifier. The deviation of this value from a given set-point is fed back to change the microwave pumping power. By careful adjustment of the windows and variation of the set-point, we succeeded in stabilizing periodic orbits by means of a perturbation which is less than 10~3 of the actual pumping power Pin. The effect of control is illustrated in Fig. 21.8, where we have compared the chaotic signal before and after switching on the feedback. The controlled signal shows a very regular oscillation, and the corresponding phase space trajectory, in fact, consists of a single orbit slightly smeared out by noise. To our knowledge this was the first feedback control of spin-wave chaos stabilizing the inherent periodic dynamics of a strange attractor [32].

21.5 Time-Delayed Feedback Control Control techniques using time-delayed output signals are well established and known for decades in applied mathematics and engineering. They have been rediscovered for the purpose of chaos control [14] as very useful and convenient methods which require neither detailed knowledge of the system nor sophisticated reconstruction techniques, and are easily implemented in experiments. Although these techniques have successfully been applied in a variety of chaos control experiments [33, 34, 35, 36, 37], the control mechanism was poorly understood from a theoretical point of view, and only recently some progress in the explanation of general features has been made [15, 38]. A detailed report of current theoretical understanding is given in the chapter by Just. Here, we focus on the application to spin systems and confine ourselves to summarizing a few theoretical results necessary to explain our findings. The aim of our investigations was the stabilization of unstable periodic orbits embedded in a chaotic attractor. The following reasoning does not apply to the stabilization of fixed points where the external force control [14] seems to be more appropriate. In spin-wave turbulence this technique was reported by Ye et al. [39]. Before reaching the intended fixed point a sequence of periodic orbits is observed, which, however, by construction cannot be UPOs of the unperturbed system although sometimes being close to them.

21.5.1 Principles of Control We consider a nonlinear system which is described by the following set of differential equations: x = /(*),

(21.12)

21.5 Time-Delayed Feedback Control

579

v [MHz]

v [MHz] Figure 21.8 Result of the OPF-type control. Top: Chaotic time signal, spectrum, and reconstructed attractor without control. Bottom: Regular oscillations, control signal, spectrum, and stabilized orbit after switching on the control.

580

21 Chaos Control in Spin Systems

Figure 21.9 Unstable periodic orbit £(£) and a neighbouring trajectory in phase space. A and UJ denote the real and imaginary parts of the Floquet exponent, T the period of the unstable orbit.

where x = (XI,...,XN) are the system variables (for spin systems e.g. three components of uniform magnetization and three more components for every excited spin wave participating in the dynamics). At least one of them, x\ (the transverse uniform magnetization) is accessible in experiment. Explicit knowledge of / is not required. Let £(£) = £(t — T) denote the unstable periodic orbit (with period T) to be controlled. The controlling scheme for this orbit consists of adding a direct or parametric forcing which is permanently directed towards the position of the UPO. The simplest choice for such a control force would be a negative feedback F(t) = -K[x(t)-t(t)],

(21-13)

added to the r.h.s. of eq.(21.12). Since the exact position of the UPO is a priori unknown, £(t) has to be substituted by some other term retaining, at least qualitatively, the direction and strength of this feedback. One could, for instance, simply replace £(t) by x(t - T) which is the position of the trajectory exactly one cycle T before. A more general form for a delayed feedback control forcing is given by F(t) = -K[g(x(t)) - g(x(t - r))],

(21.14)

where g(x) can be any "physically meaningful" scalar (e.g. the diode signal resulting from the transverse uniform magnetization) or vector functional. The delay time r has to be adjusted to the period T of the desired UPO. The amplitude K is a measure of the control strength. If the control was successful we have x(t) =£(£), which means by construction that x{t) = x(t — T) and implies the vanishing of the control signal F(t) independently of the specific form of g(x). The method is especially well suited for periodically driven systems, since r = T is exactly known in this case, but it can also be applied to autonomous systems. In the latter case the average cycle time of the chaotic signal can serve as a first estimate for T. More sophisticated approaches to T are described in the chapter by Just, sec. 4. In his pioneering paper [14] Pyragas considered a forcing by only one accessible

21.5 Time-Delayed Feedback Control

581

-1 0

1 2

3

4

5

Figure 21.10 Real part of the controlled Floquet exponent for UJ = n/T. achived for Kc < K < Kmax-

Control is

variable, g(x) = #i, but the following ideas can be extended to general controlling, including multi-component or parametric forcing [15], various kinds of filtering or even control forcing based on memory effects ("extended time-delay autosynchronisation") [37]. For our spin-wave experiment, applying the feedback again to the microwave input power, it is sufficient to consider a single-variable direct forcing, since the microwave field couples only to the transverse component of the uniform magnetization and enters the equations of motion as a constant driving force [7]. Theoretical ideas concerning the efficiency of control, the critical amplitude KC1 and the dependence of the transient time on K, r, and A can be obtained from a linear stability analysis. This way, analytical relations between the largest Floquet exponents of the uncontrolled system A + iu, A > 0 (see Fig. 21.9) and of the controlled system A(K,r) +iQ,(K,T) have been derived, A + ift = A + ILJ + (xl + ix")K (l -

+ O(K2)

(21.15)

The complex parameter \' + ix" contains all information concerning the feedback force on the system. It turns out that the most important conclusions neither depend decisively on the specific system to be controlled nor on the specific choice of the feedback. They can be summarized as follows: • Stabilization is achieved when A becomes negative. The critical amplitude Kc where A changes sign (A = 0) obeys A = -X'KC[1

-

(21.16)

which cannot be satisfied for ftc — 0. This implies that time-delayed control can only be achieved for orbits with nonzero torsion (Qc ^ 0).

582

21 Chaos Control in Spin Systems

microwave generator F(t)

resonator

( t \y static magnetic ^

field H detector

U(t-T)

Figure 21.11 Experimental set-up for delayed feedback control.

• The critical amplitude is minimal if neighbouring trajectories flip during one cycle (flc = TT/T) . In this case x(t) and x(t — T) are located on opposite sides of the orbit, and the feedback signal is maximal. This is generally guaranteed when chaos has evolved through a period doubling bifurcation. • The dependence of A on amplitude K determines the transient time, i.e. the efficiency of the control. In the case of an uncontrolled flipping neighbourhood (Fig. 21.10, u = TT/T) the minimum at Kopt indicates optimal control strength. In the region Kc < K < Kopt the frequency of the control signal is given by ft = TT/T, while for K > Kopt a splitting into two frequencies occurs, which is connected with a re-increase of A. • If A becomes positive again at a certain value Kmax the controlled orbit is destabilized and control is no longer possible. A Hopf bifurcation occurs, giving rise to an additional frequency component in the spectrum. • If the uncontrolled Lyapunov exponent A is too large, the minimal cusp may already occur before A has changed its sign. In this case control cannot be achieved. However, it might be possible to reach or extend the control regime by modification of the feedback, e.g. through multi-component forcing [15] or multiple time-delay [37]. A disadvantage of the delayed feedback control is the fact that the UPO to be stabilized is selected only by the choice of the delay time, so in the case of several UPOs having the same cycle time, one generally cannot predict which one of them will be stabilized. It is also open whether control can be achieved through every accessible system variable. On the other hand, because of the continuous control

21.5 Time-Delayed Feedback Control

583

v [MHz]

Figure 21.12 Suppression of a period-2 orbit with delayed feedback control (parallel pumping, v = 9.39GHz, Pin = 13.3dJB, H = 1613Oe ). L.h.s. top to bottom: stable period-2 orbit (K = 0), controlled period-1 orbit (K = 0.2), feedback induced torus (K = 0.5). R.h.s.: corresponding phase space representations. Note that the stabilized UPO (dark) is located close to the starting P2 orbit, while for large K an attractor widening occurs.

signal, the method is rather insensitive to noise.

21.5.2 Application to Spin-Wave Chaos Application of delayed feedback control to spin-wave chaos in YIG spheres meets the following problems: (i) We are dealing with an autonomous system, so the period of the orbit to be stabilized is not exactly known, (ii) The spectrum of coupled modes in YIG spheres is in general so complicated that we have no appropriate model to work with. Therefore, we cannot predict which system variable would be the best for applying the feedback and are limited to "trial and error". The experimental set-up which we used for our experiment (Fig. 21.11) was basically the same as in our OPF experiment. We chose the microwave input power as the feedback variable, which means to apply a direct force on the uniform mode rather than a parametric feedback. The control device consisted of a cascade of electronic delay lines with a limiting frequency of about 3MHz and several operational amplifiers acting as preamplifiers, subtractors or inverters. The device allowed to apply a control signal of the form F(t) = ±K[U(t) - U(t - T)}

(21.17)

584

21 Chaos Control in Spin Systems

0.5

l.o v [MHz]

Figure 21.13 Suppression of chaos with delayed feedback control (subsidiary absorption, v = 9.39GHz, Pin = 8.5dB, H = 1865Oe ). L.h.s. top to bottom: chaotic attractor (K = 0), stabilized period-1 orbit (K = 0.37). R.h.s.: corresponding phase space representations. Note that the stabilized periodic orbit (dark) is embedded in the chaotic attractor.

with r-range 10ns...21^s. The feedback variable U = g(x\) is the diode signal resulting from the transverse uniform magnetization. To illustrate the theoretical predictions, as a first step we considered a stable period-2 orbit (Fig. 21.12, K = 0), which was generated through a period doubling, leaving an unstable period-1 orbit with flipping neighbourhood (u = 7r/T). This unstable orbit was selected for control. The delay time r = 2.09/xs was evaluated from the very sharp and dominating peak in the spectrum. Turning on the feedback and increasing the control amplitude K, we observed a changeover to period-1 (Fig. 21.12, K — 0.2), while the period-2 component was suppressed by more than 20dB. The vanishing control signal (below a noise level of about 1% of the diode signal) indicated successful control. On further increase of K, the orbit was destabilized again. A widening of the attractor occurred, accompanied by a Hopf bifurcation which resulted in an additional broad peak at about 1.53MHz (Fig. 21.12, K = 0.5). Supporting our theoretical expectations, there is a Kwindow of successful control which is limited at low K-values by a flip bifurcation and at high K-values by a Hopf bifurcation. The following control experiments on marginal chaos did not fully reach this quality of regular signal, but clearly prove our theoretical expectations. We looked for a parameter range where chaos evolves via a period doubling, leading to a nipping neighbourhood (u = TT/T), but was followed by two Hopf bifurcations. A proper starting value for the cycle time r = 2.08//5 was obtained from the unperturbed spectrum, (Fig. 21.13, K = 0). We tried to improve this value by applying the sophisticated iterative procedure

21.6 Conclusions

585

described in sec. 4 of the chapter by Just, but the system was not sensitively affected by that. The unperturbed spectrum again shows a noisy, but pronounced period-2 component. Applying a moderate feedback amplitude (K = 0.37), the irregular behaviour is largely suppressed (Fig. 21.13). The period-1 peak becomes rather narrow, the period-2 fluctuations are decreased by about 15
21.6 Conclusions We have shown that ferromagnetic samples driven by a strong microwave field are of particular interest for studying concepts of nonlinear dynamics in real solids. The reasons are manifold: (i) Magnetic systems represent intrinsically nonlinear systems, whose nonlinearities originate from well-known interactions, (ii) Their nonlinearities give rise to spatio-temporal pattern formation and chaos, (iii) Their chaotic behaviour can be controlled by means of very weak external time-dependent perturbations. Nevertheless, in experiment one often meets the problem that the interesting phenomena occur on rather inconvenient time and length scales. This makes the application of sophisticated control techniques rather difficult or even impossible. In our FMR experiments on a YIG sphere suppression and control of chaos was achieved (i) by applying a fast parametric modulation to the microwave pumping field, (ii) by implementing a modified OGY method through an analog feedback device, and (iii) by applying a time-delayed feedback signal to stabilize regular dynamic behaviour. Control in a strict sense was achieved by the second and third method, whereas in the first case the system itself was clearly changed by the external forces. For the parametric modulation method such a change of the system is the basic mechanism which makes this method work. By analytical as well as numerical calculations, we could show that the effect of strong modulation is equivalent to the change of the modulated system parameter. Accordingly, with increasing modulation amplitude, the system follows a bifurcation route "out of chaos", which is exactly reverse to that observed on direct variation of the respective parameter. The OGY method and time-delayed feedback control, on the other hand, aim at the stabilization of existing unstable periodic orbits and should basically not affect the system in a global way. The original OGY technique is in general too intricate to be applied to fast experimental systems. Present computers are still too slow to perform complex control schemes on a timescale of microseconds or even faster. A possible way out of this problem is to model the original scheme or part of it through fast electronic circuits. For our spin-wave experiments we applied a

586

References

modified scheme of occasional proportional feedback and succeeded in controlling an unstable periodic orbit of frequency v = 470kHz. Because of its simplicity, universality, and robustness against noise, time-delayed feedback control represents a very promising method for practical use in fast technical systems. Nevertheless, its successful application depends on the proper choice of the delay time. In non-autonomous experimental systems like a nonlinear diode resonator, where the delay time r is exactly defined by the period of the external drive, time-delayed control works very well and allows the stabilization of higherperiodic orbits (P2, P4, P5). By proper adjustment of the feedback parameters, we even succeeded in "tracking" the stabilized orbit far beyond the next bifurcations into parameter regimes of very different regular behaviour [40]. In autonomous systems the proper choice of r might become a problem or require at least more sophisticated techniques of evaluation. The essential criterion, however, is related to the torsion of trajectories near the orbit to be stabilized. Control is most efficient if the trajectory flips during one cycle, i.e. when chaos has evolved through period doubling bifurcations, which is rather seldom in spin systems. Here, in general, the situation is more complicated due to other competing bifurcations and a higher number of degrees of freedom. Our successful control may be taken as an indication that the time-delay feedback technique is a very powerful and universal tool for future applications, even in very complex nonlinear systems.

Acknowledgements Many of the ideas presented here result from stimulating discussions with W. Just, J. Holyst, and Yu. Kivshar. Our experimental results include previous contributions by Th. Bernard, R. Henn, F. Rodelsperger, and Th. Weyrauch. We are grateful to K. Mayes for carefully reading the manuscript. This project of SFB 185 "Nichtlineare Dynamik" Frankfurt/Darmstadt was partly financed by special funds of the Deutsche Forschungsgemeinschaft.

References [1] R.W. Damon, in Magnetism,Vol.l, edited by G.T. Rado and H. Suhl (Academic Press, New York, 1963), p. 551 [2] R.W. Damon, Relaxation effect in the ferromagnetic resonance, Rev. Mod. Phys. 25, 239 (1953) [3] N. Bloembergen, S. Wang, Relaxation in para- and ferromagnetic resonance, Phys. Rev. 93, 72 (1954) [4] E.S. Schlomann, J.J. Green, U. Milano, Recent developments in ferromagnetic resonance at high power levels, J. Appl. Phys. 31, 386 S (1960)

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[6] V.E. Zakharov, V.S. L'vov, and S.S. Starobinets, Spin-wave turbulence beyond the parametric excitation threshold, Sov. Phys.-Usp. 17, 896 (1975) [7] H. Benner, F. Rodelsperger, and G. Wiese, in Nonlinear Dynamics in Solids, edited by H. Thomas, (Springer, Berlin, 1992), p. 129 [8] V.S. L'vov, Wave Turbulence under Parametric Excitation: Applications to Magnets, (Springer, Berlin, 1994) [9] Nonlinear Phenomena and Chaos in Magnetic Materials, edited by P.E. Wigen (World Scientific, Singapore, 1994) [10] T. Shinbrot, Progress in the control of chaos, Adv. Phys. 44, 73 (1995) [11] Yu.S. Kivshar, F. Rodelsperger, and H. Benner, Suppression of chaos by nonresonant parametric perturbations, Phys. Rev. E 49, 319 (1994) [12] E. Ott, C. Grebogi, and Y.A. Yorke, Controlling chaos, Phys. Rev. Lett. 64, 1196 (1990) [13] E.R. Hunt Stabilizing high-periodic orbits in a chaotic system: the diode resonator, Phys. Rev. Lett. 67, 1953 (1991) [14] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170, 421 (1992) [15] W. Just, T. Bernard, M. Ostheimer, E. Reibold, and H. Benner, Mechanism of time-delayed feedback control, Phys. Rev. Lett. 78, 203 (1997) [16] F. Rodelsperger, Chaos und Spinwelleninstabilitdten, furt, 1994), in German

(Harri Deutsch, Frank-

[17] see e.g. H.G. Schuster Deterministic Chaos, 2nd rev. edn. (VCH, Weinheim, 1988) [18] G. Gibson and C. Jeffries, Observation of period doubling and chaos in spinwave instabilities in yttrium iron garnet, Phys. Rev. A 29, 811 (1984) [19] F.M. de Aguiar and S.M. Rezende, Observation of subharmonic routes to chaos in parallel-pumped spin waves in yttrium iron garnet, Phys. Rev. Lett. 56, 1070 (1986) [20] D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys. 20, 167 (1971) [21] J. Becker, F. Rodelsperger, Th. Weyrauch, H. Benner, and A. Cenys, Intermittency in spin-wave instabilities, submitted to Phys. Rev. E

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[22] C. Grebogi, E. Ott, F. Romeiras, and J.A. Yorke, Critical exponents for crisisinduced intermittency, Phys. Rev. A 36, 5365 (1987) [23] F. Rodelsperger, A. Cenys, and H. Benner, On-off intermittency in spin-wave instabilities, Phys. Rev. Lett. 75, 2594 (1995) [24] A. Krawiecki and A. Sukiennicki, On-off intermittency and peculiar properties of attractors in a simple model of chaos in ferromagnetic resonance, Acta Phys. Pol. 88, 269 (1995) [25] see e.g. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin (1983) [26] F. Rodelsperger, Yu.S. Kivshar, and H. Benner, Route out of chaos by hf parametric perturbations in spin-wave instabilities, J. Magn. Magn. Mater. 140-144,1953 (1995) [27] A. Azevedo and S.M. Rezende, Controlling chaos in spin-wave instabilities, Phys. Rev. Lett. 66, 1342 (1991) [28] M. Ye, D.W. Peterman, and P.E. Wigen, Controlling chaos in a thin circular YIG film, J. Appl. Phys. 73, 6822 (1993) [29] D.P. Lathrop and E.J. Kostelich, Characterization of an experimental strange attractor by periodic orbits, Phys. Rev. A 40, 4028 (1989) [30] G.B. Mindlin, H.G. Solari, M.A. Natiello, R. Gilmore, and X.-J. Hou, Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction, J. Nonlinear Sci. 1, 147 (1991) [31] U. Dressier and G. Nitsche, Controlling chaos using time delay coordinates, Phys. Rev. Lett. 68, 1 (1992) [32] R. Henn, F. Rodelsperger, and H. Benner, Controlling unstable periodic orbits and hyperbolic fixed points in spin-wave turbulence, Proc. 26th Congress Ampere on Magnetic Resonance, Athens 1992, p.371; R. Henn, F. Rodelsperger, and H. Benner, Controlling unstable periodic orbits due to OGY in a spin-wave experiment, ICM '94 Proceedings in: J. Magn. Magn. Mater. 140-144, 1935 (1995) [33] K. Pyragas and A. Tamasevicius, Experimental control of chaos by delayed self-controlling feedback, Phys. Lett. A 180, 99 (1993) [34] S. Bielawski, D. Derozier, and P. Glorieux, Controlling unstable periodic orbits by a delayed continuous feedback, Phys. Rev. E 49, R 971 (1994) [35] A. Kittel, J. Parisi, K. Pyragas, and R. Richter Delayed feedback control of chaos in an electronic double-scroll oscillator, Z. Naturf. A 49, 843 (1994)

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[36] T. Pierre, G. Bonhomme, and A. Atipo, Controlling the chaotic regime of nonlinear ionization waves using time-delay auto synchronization method, Phys. Rev. Lett. 76, 2290 (1996) [37] D.W. Sukow, M.E. Bleich, D.J. Gauthier, and J.E.S. Socolar, Controlling chaos in a fast diode resonator using extended time-delay auto synchronization: Experimental observations and theoretical analysis, Chaos 7, 560 (1997) [38] W. Just, J. Mockel, D. Reckwerth, E. Reibold, and H. Benner, Delayed feedback control of autonomous systems, preprint [39] M. Ye, D.W. Peterman, and P.E. Wigen, Controlling chaos in thin YIG films with a time-delayed method, Phys. Lett. A 203, 23 (1995) [40] Th. Bernard, Chaoskontrolle nach Pyragas: Delayed Feedback Control, Diploma thesis, Technische Hochschule Darmstadt (1995), in German

22 Control of Chemical Waves in Excitable Media by External Perturbation O. Steinbock and S. C. Miiller Otto-von-Guericke-Universitat Magdeburg, Institut fur Experimentelle Physik, Universitatsplatz 2, D-39106 Magdeburg, Germany.

22.1 Introduction A wide variety of chemical systems is capable of supporting travelling fronts and wave trains [1,2]. One of the first published examples is probably the occurrence of a front in the permanganate oxidation of oxalic acid, where the purple solution becomes .colorless at a sharp boundary that moves with a constant velocity [3]. Other examples include the iodate arsenous acid reaction [4] or the acid-catalysed reaction between chorite and tetrathionate [5]. All these reactions involve an autocatalytic step which gives rise to nonlinear terms in their kinetic equations. The other important ingredient for a chemical system showing wave propagation is the diffusive coupling between adjacent areas. The mathematical description of these two features leads to partial differential equations that are commonly known as nonlinear reaction-diffusion equations [2,6]. A classic and simple example is the Fisher equation [7] (22.1) where c,k,D denote a concentration, a rate constant, and a diffusion coefficient, respectively. This equation was proposed by Fisher in 1937 to describe the spatial spread of a favoured gene in a population [7], but the equation could as well describe a simple chemical system. The general similarity between some classes of chemical and biological media is one of the important motivations for the current activities in this field of research [8,9]. The Belousov-Zhabotinsky (BZ) reaction [1] is one of the chemical systems that has proven to share a variety of dynamical features with certain biological media [9]. Zhabotinsky, Zaikin, Busse, Hess and Winfree reported the observation of twodimensional wave patterns in thin liquid layers of this reaction solution [10-13]. Since then our understanding of the complex chemistry of this system has continuously increased and a broad spectrum of intriguing phenomena related to chemical wave patterns has been reported. One focus of our research is the external perturbation of these wave patterns. The present Chapter discusses different approaches for control of chemical spiral rotation, where the Belousov-Zhabotinsky reaction serves as the experimental model system. Certain features of this system, such as

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22 Control of Chemical Waves in Excitable Media by External Perturbation

the presence of ionic species, organic radicals or its photosensitivity are used for the purpose of applying the desired external perturbations.

22.2 Spiral Waves and the Belousov-Zhabotinsky Reaction The most striking wave patterns known from (quasi) two-dimensional reactiondiffusion media are structures consisting of expanding concentric circles (target patterns) and rotating spiral waves [14]. Figure 22.1 shows six examples of spiral waves in quite different systems ranging from the heterogeneous catalysis on platinum to slime mold colonies and the mammalian heart [6,8,9]. The occurrence of wave patterns that are not only similar in their geometry but also in their dynamic evolution is caused by similarities in the reaction- diffusion equations which describe the specific systems (e.g., their excitable kinetics). In contrast to the simple Fisher equation (1), which mainly generates propagating fronts, excitable systems (Fig. 22.1) can only be described by sets of N > 2 coupled reaction-diffusions equations. For N = 2 one can distinguish a fast propagator variable u (e.g., the concentration of an autocatalytic species) and a slower control variable v. The typical scenario of wave propagation can then be sketched as follows: A local perturbation causes a localized fast increase of u giving rise to a diffusive flux of u into the adjacent region. If the u-level increases above a certain threshold it triggers its own production (autocatalysis) and the system switches from an excitable to an excited state. At the same time a slow increase of the control variable v takes place and the system switches into a refractory state in which u is small and v decreases slowly. During this phase the system recovers and cannot be excited again, until v reaches sufficiently low values. The excitation wave that results from this mechanism propagates with constant amplitude and velocity. Its spatial concentration profile shows typically a steep front and a much smoother back. The velocity V of wave trains in reaction-diffusion systems, however, depends also on the frequency / [14]. Typically one finds monotonically decreasing dispersion relations V(/), where V saturates for low frequencies. Due to the existence of the refractory phase (wave back), the frequency of wave trains is limited by a maximal value that reflects the specific dynamics of the medium. In two (and three) spatial dimensions the front curvature also influences the local propagation velocity. It can be shown that the normal wave velocity N obeys in a good approximation the eikonal equation N = V - DUK

(22.2)

where V, K, and Du denote the velocity of the planar wave, the local front curvature, and a proportionality factor, respectively [14]. It has also been shown that Du is approximately the diffusion coefficient of the autocatalytic propagator species. Equation (2) quantifies the effect of increased diffusive fluxes at concave front segments (K < 0) and decreased fluxes at convex segments (K > 0). Notice, that

22.2 Spiral Waves and the Belousov-Zhabotinsky Reaction

593

Figure 22.1 Six examples of spiral waves in excitable media. (A) Population of spirals with different rotation periods and wavelengths in the catalytic COoxidation on a platinum surface, visualized with photoemission electron microscopy (PEEM). Image area: 450 400 mm2, rotation period of spirals with intermediate wavelength 20 s (from: Nettesheim et al. [44]). (B) Spiral wave in the Belousov-Zhabotinsky reaction. The distribution of light intensity reveals the front of oxidized cerium catalyst at 1 = 344 nm as a black band moving through a reduced solution layer of 0.7 mm thickness (bright background). Wavelength: 2.1 mm; rotation period 40 s. (C) Spiral Ca2+-wave pattern observed in Xenopus laevis oocytes (wavelength 60 mm, period 3 s). IP3-mediated Ca2+-release is detected by confocal laser scanning microscopy (courtesy of Goldbeter and Clapham). (D) Aggregation of social amoebae in the cellular slime mold Dictyostelium discoideum observed in darkfield-optics. In bright areas cells move chemotactically towards the spiral core, while in dark bands no directed cell migration is found.Wavelength 2.5 mm; rotation period 7 min. (E) Spiral- shaped front in neural tissue: These 'spreading depression' waves on chicken retina are visualized by white light scattered in zones of increased turbidity. The waves are moving through the otherwise transparent medium at a speed of 2.2 mm/min. Image section: 9 10 mm2. (F) Clock-wise rotating wave in a slice (20 20 0.5 mm) of isolated canine cardiac muscle. Visualized by use of a potentiometric dye (with fluorescence excited at 490 nm; measured at 645 nm). Rotation period: 180 ms (from: Davidenko et al. [43]).

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22 Control of Chemical Waves in Excitable Media by External Perturbation

below a critical curvature Kcrit = Du/V no wave propagation can occur. Of particular interest is the dynamics of the spiral tip, which is the open wave end in the center of the pattern [15]. The tip location is defined by the point of highest curvature of the isoconcentration line at the average level of concentration. In the simplest case, the spiral tip describes a circular trajectory with a fixed center. This rigid rotation leads to a circular area that is known as the spiral core [16], in which only small concentration changes occur. The core remains essentially in an unexcited state. Another mode of spiral wave rotation gives rise to tip trajectories that are roughly hypocycloids or epicycloids. This behavior is called meandering. Figure 22. 2A gives a typical example for a hypocycloidal motion showing a snapshot of a spiral wave in a chemical medium with its superimposed tip trajectory (white curve). The perhaps most intensively studied model systems for experiments on chem-

(M) 0.4-

—I

1

100

Figure 22.2

1 200

time

(min)

(A) Snapshot of a rotating spiral wave in the Belousov-Zhabotinsky (BZ) reaction. The bright and dark regions indicate high and low concentrations of the oxidized catalyst, respectively. The trajectories of the spiral tip (overlaid white curve) is similar to a hypocycloid. Such non-circular trajectory are typical for spiral meandering. In this particular experiment a photosensitive ruthenium complex ([Ru(bpy)3]2-f-/3+) is catalyzing the reaction. Image area: 3.8 3.0 mm2 (Figure taken from: Steinbock et al. [38]). (B) Spiral tip trajectories monitored as a function of time for four different Cecatalyzed BZ solutions. The value cA corresponds to the concentration of H2SO4. The parameter plane is divided into three domains showing: stable spiral waves (1); planar waves, but no spirals (2); and no wave propagation at all (3) (Figure taken from: Nagy-Ungvarai et al. [19]).

22.2 Spiral Waves and the Belousov-Zhabotinsky Reaction

595

ical wave propagation is the, already mentioned, Belousov-Zhabotinsky (BZ) reaction [1,17]. It consists of the bromination of certain organic compounds, such as malonic acid (CH2(COOH)2) in a sulfuric acid solution employing appropriate redox catalysts. A frequently used catalyst is the redox couple ferroin/ferriin which leads to striking color differences (red/blue) along the profile of chemical waves. These differences can be readily detected by two-dimensional spectrophotometry [16]. In this technique the absorption at a given wavelength (usually 490 nm for maximal contrast) is detected as a function of space and time by using video cameras. Sequences of video images are digitized and then available for computer analysis. In this reaction, the dynamics of spiral tips has been, in fact, a major focus of research. The system shows spiral waves performing rigid rotation as well as meandering [15,18]. A detailed study for the closed Ce(IV)-catalysed BZ medium has been carried out by Nagy- Ungvarai et al. [19]. Their data reveal a complex dependence of the structure of the tip trajectories on the initial concentration of sulfuric acid (cA) and on the elapsed reaction time (Figure 22. 2B). For a given concentration value one observes a continuous variation of the dynamics that typically leads to increasingly larger trajectories. Also notice, the change from hypocylces to epicycles that occurs for CA ~ 0.2M after about 70 min of reaction time. At the limit of spiral wave stability one observes a transition from tip rotation to straight tip propagation. The latter state is characterized by small wave segments that travel through the system without any signigficant changes in their shape. In the course of the reaction these segments begin to shrink and eventually vanish, thus, indicating a transition to a parameter region where excitation waves have no stability at all. Many of the features of spiral waves in the BZ system are captured by the Oregonator model [14,17]. The Oregonator is one of the most widely used models for numerical simulations of this reaction-diffusion system. It is based on the specific chemical mechanism of the BZ reaction, which has been discussed in great detail elsewhere [1,6]. The model considers the spatio-temporal evolution of three chemical species: (a) The autocatalytic species HBrO2, (b) the oxidized state of the catalyst (e.g., Ce(IV) or ferriin), and (c) the inhibitor bromide. In the TysonFife reduction (adiabatic elimination of the bromide concentration) it leads to the following differential equations: du

at

.

™ ^

u v

S = ~>

_i f

r

o

fv(u ~

(u + g)

(22-3)

where u and v are proportional to the concentration of HBrC>2 and the oxidized catalyst, respectively. These variables play the role of the fast propagator (u) and the slow control variable (v). The parameters g, e, and / denote parameters derived from the reaction kinetics, initial concentrations, and the reaction stoichiometry.

596

22 Control of Chemical Waves in Excitable Media by External Perturbation

22.3 External Control 22.3.1 Chemical Parameters and Oxygen-Inhibition Chemical oscillations in well stirred BZ solutions reveal pronounced dependencies on the initial concentrations of sulfuric acid, bromate, bromide, and malonic acid [1]. An increase in sulfuric acid concentration, for example, usually speeds up the oscillations and the oscillation frequency might increase by a factor of ten or more. Molecular oxygen is a chemical parameter that is often ignored in the discussion of experiments on spatio-temporal pattern formation in BZ systems. Dissolved oxygen reacts with organic radicals that are formed during the oxidation of malonic acid and bromomalonic acid by the oxidized catalyst [20]. While the detailed chemistry of these reactions is not yet understood, there are several studies reporting an overall trend of oxygen-induced inhibition of oscillations and wave patterns. This means for spatially homogenous systems that oscillation periods are usually increased or oscillations suppressed. A remarkable consequence of this oxygen-induced inhibition has been recently

2

3

4

5

6

Drop radius R / mm

Figure 22.3 The period of chemical oscillations T in small droplets of the ferroincatalyzed BZ reaction is shown as function of the drop radius R. The presented data is obtained from experiments carried out under aerobic reaction conditions. A pronounced oxygen-induced inhibtion of chemical oscillations is found for small droplets, since the surface-to-volume ratio increases with decreasing values of R. Under the given experimental conditions a critical radius of Rcrit = 1.5 mm exists below which no oscillations are observed. The corresponding experiments (nonoscillatory droplets) are indicated by the symbol " at an arbitrary T value (Figure taken from: Steinbock and Miiller [21]).

22.3 External Control

597

reported for small BZ droplets [21]. The drops were pipetted into a Petri dish containing inert silicone oil, where they sank to the bottom and flattened immediately. The resulting droplets had a circular base with radii being in the range of 0.1 - 6.0 mm. Under the experimental conditions specified in the captions of Figure 22. 3 one observes a pronounced dependence of the period of chemical oscillations on the drop radius R. In the case of oxygen-saturated oil the period increases with decreasing radii and below Rc « 1.5 mm no oscillations were found at all (Figure 22. 3). This inhibtion is controlled by the surface-to-volume ratio of the droplets, which increases with decreasing radii and determines the strength of oxygen-influx.' Hence, small droplets are subject to strong oxygen-inhibition (i.e., long oscillation periods), while large droplets are only slightly affected. The absence of oscillations for R< Rc indicates a bifurcation from oscillatory to excitable behavior. In spatially extended media, such as thin layers of the reaction solution or thin BZ gel systems, oxygen is either decreasing the propagation velocity of waves or suppressing wave propagation completely [22]. More surprisingly, experiments have shown that thin layers (thickness^ 1 mm) of the aerobic BZ medium, that usually behave as a quasi two- dimensional system, can undergo vertical stratification [23,24]. This stratification gives rise to two (even thinner) sublayers of

B

Figure 22.4 Under certain experimental conditions and in the presence of oxygen thin layered BZ gel systems can undergo a vertical stratification. The top view (A) shows the formation of two pattern forming layers, where wave propagation proceeds without strong interaction. The white (bottom layer) and the black arrow (top layer) indicate an example of these "crossing" waves. In the course of the reaction the chemical patterns begin to couple giving rise to sawtooth-shaped fronts (B). Interval between photographs: 40 s (Figure taken from: Zhabotinsky et al. [23]).

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22 Control of Chemical Waves in Excitable Media by External Perturbation

excitable medium that develop independent wave patterns showing only weak interaction. A typical example for the resulting patterns is shown in Figure 22. 4. It should be reemphasized that the visual impression of crossing waves is misleading, since the patterns evolve in two separate layers. The stratification of thin BZ gel systems is strongly related to an oxygen concentration gradient that is formed by the interaction of diffusive inflow from the atmosphere into the gel system and the consumption of oxygen within the BZ system. Furthermore, one of the BZ intermediates (the volatile bromine) is leaving the gel by diffusion into the atmosphere. Both compounds, oxygen and bromine, act as inhibitors of wave propagation. We are just beginning to understand the microscopic details of this interesting interaction. The described phenomena seem to lead to a non-monotonous profile of excitability that induces the observed stratification [24]. The relevance of bromine diffusion has been also noticed in the experiments on BZ droplets described above [21]. In the case of oxygen-free oil surrounding the aqueous BZ drop, one finds a T(R) dependence that differs significantly from the one shown in Figure 22. 3: The diffusive flux of bromine from the droplet into the oil leads to a decrease of the concentration of the inhibitor bromide. The extent of this indirect removal of inhibitor is controlled by the surface-to-volume ratio, quite similarly to the case of oxygen. In the anaerobic case, however, the oscillation period of small droplets is decreased, since bromide removal acts as an activation process. The authors believe that the role of molecular oxygen in the context of the BZ reaction opens a variety of quite promising research activities, in particular for control experiments. The involved free radical chemistry is far from being understood and implies challenges for future kinetic and mechanistic investigations. Furthermore, oxygen is an intriguing chemical parameter that gives rise to unexpected modes of self-organization in the BZ reaction such as the described stratification of thin BZ gel systems. As we will see in the following, it may also assist other experimental approaches on external control and perturbation exploiting the photosensitivity of reacting compounds.

22.3.2 Control by Electric Fields Numerous chemical species in the complex BZ reaction mechanism are of ionic nature. The central species are the bulky iron complex ferroin that carries a positive charge of two or three depending on its oxidation state and the small negatively charged bromide ion. The autocatalytic species HBrC>2 is another important actor taking part in the chemical events that lead to wave propagation. This species however is electrically neutral. The question we would like to discuss in this section is: What happens to propagating waves and rotating spirals if an external electric field is applied? In 1981 Feeney, Schmidt and Ortoleva performed experiments in which they applied parallel electric fields (25 « 10 - 50 V/cm) to spatially extended BZ media [25]. They observed an increased velocity of waves propagating towards the positive anode, while waves propagating towards the cathode were decelerated. Sevcikova

22.3 External Control

599

Figure 22.5 A pair of spiral waves in the ferroin-catalyed BZ system is perturbed by a constant electric field. Field lines are parallel and oriented vertically with the anode located at the bottom side of the figures. The electric field is inducing a spiral drift towards the anode and a strong deformation of the Archmedian spiral geometry. (A) Snapshot of the pair of drifting spirals. (B) Trajectory of the corresponding spiral tips. Notice, that the drift direction is also influenced by the sense of spiral rotation (Figure taken from Steinbock et al. [29]).

and Marek continued this work [26] and found more recently that at higher field strength waves can reverse their propagation direction or split in a fairly complex fashion [27,28]. Figure 22. 5 illustrates the effect of an externally applied electric field on spiral waves in the ferroin-catalysed BZ reaction. In these experiments a constant electric field (E = 0-6 V/cm) was applied to the BZ gel system via two parallel electrodes which were realized as simple salt bridges to avoid the contamination of the BZ medium by products of undesired electrochemical reactions. A central finding is that spiral waves are drifting towards the anode [29]. Figure 22. 5A shows a typical snapshot of a pair of drifting spiral waves (anode oriented parallel to the bottom side of the figure). The overall drift towards the anode has an additional component perpendicular to the field. The direction of this perpendicular drift depends

600

22 Control of Chemical Waves in Excitable Media by External Perturbation

on the chirality of the spirals, as shown in Figure 22. 5B. Although the trajectory of both spirals point in -x direction, one finds that the clockwise rotating pattern is also pulled to the left, while the counterclockwise spinning wave is pulled to the right. Switching off the electric field stops the drift immediately. Changing the polarity of the field causes a drift back towards the initial locations. For typical experimental conditions as those of Figure 22. 5 the drift velocity of spiral tips is found to be in the range of 0 to 0.3 mm/min. Note, that the drift does not occur along a straight line, but is rather characterized by a continuous trajectory with successive loops. Although present, these loops are not fully resolved in the experimental data displayed in Figure 22. 5B. Another interesting phenomenon that has been observed in experiments on electric- field induced spiral drift is the deformation of spiral geometry [29]. While spiral waves in unperturbed systems have usually an Archemedian shape (i.e., constant pitch), the drift of its tip is generating variations of the wavelength. The deviations from the unperturbed wavelength reach a maximum in the back of the drift direction (compare Fig.22. 5A). Apparently the increased wavelength is due to a Doppler effect. Furthermore, the shape of the drifting spiral tip can vary significantly during different phases of its rotation. Depending on the direction of its relative orientation to the electric field vector, the curved tip is either elongated or strongly curled. The perturbation of chemical wave patterns by electric fields is not only an interesting field of research in itself, but can also be used for the external control of wave propagation. An intriguing example for exploiting spiral drift is discussed in the following. Since, it is possible to induce spiral drift, one should be able to send a pair of spirals on collision course. Figure 22. 6 gives a sequence of four snapshots illustrating the outcome of a spiral wave collision [30]. Snapshot (A) shows the initial wave pattern consisting of a pair of counterrotating spirals that have nearly identical size and phase. A constant electric field is applied to the BZ gel system with the anode located parallel to the right side of the subfigure. The field is now pulling the spirals towards the anode and is decreasing the relative distance between the spiral tips (Figure 22. 6B). At a certain critical distance ( < wavelength) the spiral tips annihilate, thus removing the spinning pacemakers from the system (C). The resulting unexcited area in the former region of tip rotation triggers a new pattern of low frequency (D), since the bulk dynamics of this particular BZ system is, in fact, not excitable but oscillatory with rather long oscillation period. These intrinsic oscillations of the BZ bulk were earlier suppressed by the high-frequency spiral waves. Now, where the spirals have vanished, the system creates an autonomous pacemaker according to the local phase gradients created by the last spiral rotation. The collision of a spiral pair has therefore lead to spiral annihilation and eventually to the creation of a target pattern with its pacemaker located in the spiral collision region. Additional experiments revealed that the electric field strength and the initial geometry (i.e., phase and symmetry) determine whether spiral annihilation can occur. If the initial symmetry" line between both spirals is not parallel to the electric

22.3 External Control

A/

601

:

Figure 22.6 Evolution of spiral wave annihilation due to an electric-field induced drift. The electric field is oriented with the anode to the right and the cathode to the left. Opposite perpendicular motions of spiral tips in the initial structure (A) reduce their relative distance (B). Annihilation occurs when the separation distance of the tips is below a certain critical size (C). In a truely excitable system, the remaining circular waves would continue to propagate outwards leaving behind a "quiet" region without chemical acticity. Since this particular BZ medium can show autonomous oscillations, we were able to observe the birth of a non-rotating trigger wave (D) in the central region of spiral wave annihilation. The location of its pacemaker is determined by the local phase information created by the last spiral rotation (Figure taken from: Schtze et al. [30]).

602

22 Control of Chemical Waves in Excitable Media by External Perturbation

field, annihilation becomes more unlikely. Furthermore, a critical field strength of about 2 V/cm was found below which no annihilation occurred. Disregarding the trivial case where the spirals pass each other at large distance, low electric fields can induce another unexpected scenario. Experiments showed that under these conditions colliding spiral waves can interact without annihilation. While one of the spiral tips is proceeding its drift along the expected straight line, the other tip is forced to deviate significantly from its straight course. Recent numerical simulations also indicate that under ideal conditions, two spiral can form a bound state that keeps the tips at a constant average distance (of the order of one wavelength) and moves them towards the anode along a trajectory that is parallel to the electric field vector [31]. Notice, the striking similarities between spiral waves and particles: A single, one- armed spiral has a topological charge (either -fl or —1) that is determined by its sense of rotation. It is a well known fact that the total sum of topological charges in an excitable medium (without boundaries) is conserved. Therefore, only spirals having a different sense of rotation can annihilate. Depending on the sign of its topological charge the drifting spiral tip is experiencing different contributions to the vector of drift velocity. Prom a phenomenological point of view this behavior is quite similar to moving electrically charged particles which are subject to a magnetic field. Furthermore, interacting spirals can form a bound state that has a topological charge of 1 — 1 = 0. This spiral pair is drifting with a velocity that has no component perpendicular to the field; similar to an uncharged atom that is experiencing no magnetic forces that would alter its velocity. It will be interesting to see, how far the concept of particles can be fruitfully used for understanding the dynamics of spiral waves in excitable systems. The microscopic driving force of the observed phenomena is electromigration of ionic species. The underlying electric force can result in a local change of the concentration c that is given by the equation ^ = M^Vc,

(22.4)

where E is the electric field vector and Vc the gradient vector of the local ion concentration. The parameter fi is the ionic mobility of the charged species defined as the proportionality factor between the velocity of ions and the applied electric field. Hence, the value ofjiE corresponds to the migration velocity of the particular ion in an electric field E. By adding the right hand term of equation (4) to the set of reaction-diffusion equations describing the chemical system, one obtains a mathematical model that is suitable for the numerical simulation of experimental data. On this basis, numerous numerical studies have been carried out to reproduce and understand the effects of electric fields on pattern formation in chemical systems. Generally, these investigations are in good agreement with experiments, although certain quantitative features of numerical studies are not quite satisfying yet. Bromide ions seem to be the key species of electric-field effects in the BZ reaction [25,26,29]. Electric fields are forcing the inhibitor bromide to migrate towards the anode following the mathematical description discussed above. A BZ wave

22.3 External Control

603

front travelling towards the cathode is now experiencing an electric-field induced flow of bromide that is slowing down its propagation. On the other hand, a front travelling towards the anode is always expanding into regions of lowered inhibitor concentration and therefore propagating with an increased velocity. The presented drift of spiral waves is directly related to this effect.

22.3.3 Control by Light While the perturbation of spiral rotation by electric fields is an example for perturbations by a vectorial quantity, the following section describes a powerful approach for the control by a scalar quantity, which is the intensity of an external light field [32]. The metallo-organic complex [Ru(bpy) 3 ] 2+ / 3+ has proven to be an extremely useful alternative to the classic catalysts, ferroin and Cerium, since the Ru(bpy)3+catalysed BZ reaction is photosensitive [32,33]. By illuminating the system with visible light one can stimulate the production of additional bromide, which is inhibiting the excitable behavior of the system. The maximum effect is established at a wavelength of about 454 nm via the formation of a photochemically excited ruthenium complex [34]. The detailed chemistry of the light-induced inhibition is not understood yet. Nevertheless, working models exist that were derived from the Oregonator model of the BZ reaction (cf., eqn. 3) by adding an additional source of bromide, where the rate of bromide production f is assumed to be simply proportional to the light intensity [35]:

at —

' =

u-v,

L

(" +
The initial goal of our experiments with the light-sensitive BZ reaction was to develop tools for the controlled creation of spiral waves [36]. An argon ion laser of relatively high intensity (0.8 W attenuated by a neutral density filter OD 2.0) is used to irradiate small circular areas (typical diameter 0 . 1 - 2 mm) of the BZ medium. These areas are now inhibited in the sense that locally wave propagation becomes impossible. The time required for reaching efficient inhibition is short (about 2 s) and the effect is reversible. If a solitary chemical wave is propagating across the inhibited spot, it breaks apart and two open wave ends are generated. After switching off the laser these wave ends develop to a pair of rotating spirals [36]. Hence, the argon laser can be used as a highly controllable tool for generating spirals. Furthermore, it is possible to pin spiral tips to laser-inhibited spots. In this context the laser spot acts as an artificial spiral core that forces a circular wave rotation around the boundary of the inhibited spot. By these means, the spiral rotation period and the wavelength of the chemical pattern can be adjusted to desired values. Figure 22. 7 shows a pair of spiral waves in the photosensitive BZ reaction where this laser control is carried out. The unperturbed spiral (left) rotates with a characteristic period of 26 s and has a wavelength (pitch) of 1.3 mm. The perturbed spiral (right) rotates around an unexcitable laser spot having

604

22 Control of Chemical Waves in Excitable Media by External Perturbation

.^i

, '

*

'

' '"

'

Figure 22.7 Argon-ion lasers allow an efficient manipulation of spiral waves in the photosensitive BZ reaction. This example shows a pair of spiral waves, where the left spiral is unperturbed and rotates freely. The right spiral spins around a small laser spot (diameter 1.2 mm) that creates an unexcitable disk (artificial core). This constant perturbation induces an increase in rotation period and wave length (Figure taken from: Steinbock and Miiller [36]).

a diameter of 1.2 mm. The perturbation leads to an increase of period (49 s) as well as wavelength (3.4 mm). In the subsequent stages of the experiment the area covered by the perturbed spiral decreases continuously. Eventually, the spiral on the left conquers the whole observation area leaving a defect at the position of the laser spot. A series of experiments confirmed that the period of spiral rotation (and also the wave velocity) increases monotonically with the diameter of the laser spot constituting the artificial rotation center (artificial core). An additional function of the argon laser allows to shift the center of spirals through the system. If a spiral tip is pinned by the laser spot, one can slowly move the BZ probe and prevent further rotation [37]. It is crucial that the probe is translated with a speed that is roughly equal to the propagation velocity of planar waves. Under this condition the tip is experiencing a permanent obstacle and follows the relative motion of the laser spot. We found this technique to be an useful tool for removing undesired spirals from the observation area and thus preventing the interaction of spirals that might otherwise influence the measurement. In addition, we found that a spiral which is moved to the physical boundary of the BZ system (e.g., the boundary of a Petri dish) transforms to a defect rotating around the entire circumference of the probe. Using this procedure one can accumulate numerous spirals of identical topological charge in the central region of the system and further use the laser beam for creating multi-armed spirals [37].

22.3 External Control

605

A quite different approach for exploiting the photosensitivity of the rutheniumcatalysed BZ reaction focusses on spatially homogeneous perturbations that are modulated in time [38]. We investigated a system in which spiral tips are describing a roughly five-lobed trajectory if subject to an average (comparatively low) light intensity (compare, Figure 22. 2A). For the minimum and maximum intensities applied in the following example, we observed four- and six-lobed trajectories, respectively. Periodic modulations of the light intensity within these bounds cause dramatic changes in the dynamics of spiral rotation. The small variations of the trajectory shape that occur during each period of modulation accumulate with time. As a result, the modulated perturbation forces the tip to follow trajectories that differ significantly from those observed at constant intensities. The shape depends strongly on the modulation period Tm. The underlying effect, that gives rise to the observed phenomena, is the entrainment of the initial quasiperiodic tip motion. In contrast to nonlinear oscillators where entrainment has been investigated in great detail, we now have to understand the response of a spatio-temporal system (having an infinite number of degrees of freedom) to periodic perturbations. Figure 22. 8 shows five examples of tip trajectories of spiral waves traced in a BZ system under periodic modulation of light intensity. The trajectories of Figure 22. 8 B-D are members of an entrainment band of hypocyloids with one lobe corresponding to one external period. The number of lobes continuously increases from three to more than twelve with increasing values of Tm. The one-to-one phaselocking between the number of lobes and the number of modulation cycles has been observed only in a well-defined interval of modulation periods (Tm = 20 - 35 s). For smaller periods a quite different behavior occurs, as shown in Figure 22. 8 A. This trajectory is a deformed five-lobed curve with exactly one lobe described during two modulation periods. Figure 22. 8E illustrates one possible mode of spiral tip dynamics at slow modulation showing a surprising trajectory with alternating distances between neighboring lobes. In this small frequency range the spiral tip describes one pair of lobes during one external modulation. For periods between those of (D) and (E) we observed irregular motion with epicyclic segments of the trajectories. In all examples of Figure 22. 8, however, the tip motion is synchronized (or entrained) by the external rhythm. Numerical simulations of reaction-diffusion systems are useful supplements for experimental investigations, since they allow a thorough scan through broad parameter regions in a tolerable time [38]. V. Zykov has carried out extensive computer simulations in order to achieve a more complete picture of entrainment of meandering spiral waves [39]. The calculations are based on the modified Oregonator model (eqn. 5 ) describing the photosensitive BZ medium. Figure 22. 9 shows some tip trajectories obtained numerically by changing the modulation period Tm as well as the modulation amplitude A. The function $ is given by the equation $(*) = 0.01 + A8in{2nt/Tmod).(6)

(22.6)

The value To denotes the wave period at the center of the unperturbed spiral. Additional parameters are: e = 0.05, # = 0.002, and / = 2.0. Notice, that due to the complex motion of a meandering spiral wave different excitation periods are de-

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22 Control of Chemical Waves in Excitable Media by External Perturbation

p a

0

Figure 22.8 Sequence of spiral tip trajectories measured under a sinusoidal modulation of light intensity with a period Tm of: 17.0 s (A), 26.2 s (B), 30.4 s (C), 34.5 s (D), and 52.2 s (E). The shading of the trajectories indicates the current intensity of illumination and clearly shows the one-to-one phase locking of the lobes to the external rhythm (B-D). The traces in (A) and (E) obey a 1:2 and a 2:1 phase locking, respectively. Period of unperturbed spiral (0.93 mW cm-2) as measured in its symmetry center: TO = 24.5 s. Scale bar: 0.2 mm (Figure taken from: Steinbock et al. [38]).

tected at the center (To) and at infinite distance (Too). For a n-lobed hypocycloid trajectory the periods obey the equation: To = T^n - l ) / n . From the numerical results shown in Figure 22. 9 one can see that the entrainment band of trajectories that obey a one-to-one phase locking is located around the period Tm = TQ. The trajectories of this band correspond to the experimental observations shown in Figure 22. 8 B-D. Figure 22. 9 depicts three additional entrainment bands where the number of lobes and the number of modulation cylces have the constant ratios 1:3, 1:2, and 2:1. Notice, the good agreement between the simulated curves in the 1:2 and 2:1 band with the experimental data shown in Figure 22. 8 A,E, respectively. Similar to the experimental observations the simulations also reveal irregular trajectories in between the entrainment bands. The corresponding motion in these parameter regions shows no phase locking with respect to the rhythm of external modulation. The photosensitivity of this particular class of BZ systems also allows the ex-

22.3 External Control

607

2/1

To

Too 4 - 0

Figure 22.9 The experimental results shown in Figure 8 are in good agreement with numerical simulations. Tip trajectories calculated as a response to sinusoidal modulation of bromide production are shown in the plane of modulation period Tm and modulation amplitude A. The center of each tip trace is arranged at the actual parameter pair (Tm, A) used for calculations. Their size was varied to allow a better resolution of details. Dashed lines indicate boundaries of entrainment bands related to different ratios n:m , where n is the number of lobes per m periods of modulation. TO and T are the excitation periods of the unperturbed spiral as measured in its symmetry center and at a point far away from the center, respectively. The thick black line indicates parameters for which resonance drift was found (see text for further details) (Figure taken from: Zykov et al. [39]).

perimental realization of single- and multi-channel feedback control [40,41]. In the simplest case, a spatially homogeneous light pulse is applied to the excitable medium at the instants of time that correspond to the passage of a wave front through a particular detection site. The application of such a "one-channel" feedback results in two new dynamic regimes of spiral wave rotation. These will be briefly discussed in the following for the example of a spiral that, in the absence of feedback control (unperturbed case), describes a simple four- lobed meandering pattern. Figure 22. 10 (bottom row) shows three examples of spiral tip trajectories in the presence of such a single-channel feedback. The detection site is marked by the symbol " and turns out to become the symmetry center of the asymptotic trajectory. Such asymptotic patterns are called entrainment attractors according

608

22 Control of Chemical Waves in Excitable Media by External Perturbation

0.500.40-

JL * 01

030

"

0.20 : 0.100.00-

Figure 22.10 Feedback-control of spiral waves in the photosensitive BZ reaction. The entrainment attractors (A-C) are observed for different time delays t: 0 s (A), 3 s (B), and 5 s (C). Thick segments of the trajectories indicate the application of light pulses. These are triggered by the passage of a wave through the detection site (indicated by the symbol "). The upper part shows the mean value R* of inner and outer radius of the attractor as a function of t. Scale bar: 0.5 mm (Figures taken from: Grill et al. [41]).

to Grill et al. [40,41]. The examples (A) to (C) were obtained from experiments, where the time delay r between front passage through the detection site and light pulse was varied. The resulting trajectories are, in fact, simple hypocycles, but the number of lobes increases with the additional control parameter r. Figure 22. 10 (top) shows the mean value R* of inner and outer radius of these tip attractors as a function of the delay time. The entrainment attractors of Figure 22. 10 are only observed if the detection point is located close to the center of the initial, unperturbed trajectory. For larger distances a quite different scenario is observed as shown in the two examples of Figure 22. 11 (bottom row). The structure of these resonance attractors reveals a higher complexity and their mean radii R* as a function of the delay time r follow a nearly linearly decreasing function (Figure 22. 11; top). Due to the large distance

22.3 External Control

609

between spiral tip and detection point, the average period of the triggered stimuli is close to T^ , which is the resonance period for the case of non-feedback periodic modulation. The entrainment and resonance attractors observed in one-channel feedback experiments agree qualitatively with phenomena oberved under harmonic perturbations. Entrainment, in general, occurs for modulations with Tmod « To (compare, Figure 22. 8, 9), while resonance is the charactristic response for perturbations with TmocL « T^ , regardless whether the perturbations stem from feedback or non-feedback control. Both approaches utilize characteristic facets of external control of spatially extended excitable systems, which might prove to be useful tools for applications in biology [8,9,42] or biomedicine [9,43]. In this context, a certain advantage of feedback control is given by the stability of the chosen dynamics against 1.8 1.6 1.4 1.2 1.0 Q:

0.8 0.6 0.4 0.2 0.0 5

10

15

20

25

30

T[S]

B Figure 22.11 Resonance attractor for different time delays t . The upper part shows the mean value R* as a function of time delay t. Two examples of the resulting trajectories (i.e., attractors) are given in the lower part: t == 7 s (A) and t = 25 s (B). Thick segments of the trajectories indicate the application of light pulses. Scale bar: 0.5 mm (Figures taken from: Grill et al. [41]).

610

22 Control of Chemical Waves in Excitable Media by External Perturbation

transient changes in the characteristics of the medium (i.e., transient changes of the rotation period of spirals). The presented experimental results of this section demonstrate that the photosensitive BZ reaction is an excellent model system for the investigation of excitable reaction-diffusion systems by means of external control and perturbation. First of all, spiral waves can be generated in a reproducible fashion and their location in the system can be chosen at will. Secondly, artificial cores, created by unexcitable laser-irradiated spots, can be exploited for the control of the rotation period and the wavelength of spiral waves. These cores can also host numerous spiral arms and are therefore stabilizing multi-armed wave patterns. We have also discussed some central aspects of periodic forcing and feedback-control of excitable systems that give rise to phenomena such as entrainment, phase-locking, resonance, and multifrequency patterns. It should be noted, that other authors have demonstrated the potential of the light-sensitive BZ media for the use of image processing [33]. In this context, the BZ gel system acts as an artifical retina that is capable of reversing or smoothening photographies which are projected onto the system.

22.4 Conclusions The application of external perturbations or constraints to excitable systems opens new avenues for the investigation of chemical self-organization. The BZ reactions can be perturbed in a controlled fashion by numerous techniques. The examples discussed in this Chapter involved (1) chemical parameters, such as oxygen, (2) electric fields acting on ionic species, and (3) light stimuli giving rise to local and global perturbations. Additional approaches have been discussed in the current literature. Some of them focus on the realization of spatial constraints [44] or even heterogeneous media that give rise to surprising wave patterns [45]. All of these research activities have the potential to yield important insights into the rules governing biological self-organization, since living systems are strongly influenced by periodic rhythms and constant gradients in their natural environment (such as circadian rhythms). Furthermore, the techniques developed for the external perturbation of chemical wave propagation might proof to be valuable tools for the analysis and control of spatio- temporal chaos and turbulence [2]. Until today, control of chaos in chemical systems has been demonstrated only for spatially homogeneous systems. This control was achieved by small variations offlowrates in continuously-stirred tank reactors (CSTRs) for the example of the BZ reaction [46]. The application of these basic ideas to spatially extended systems will not only require further theoretical studies on this complex issue, but also require the use of a broad spectrum of experimental methodologies. These might involve such powerful approaches as the control of photosensitive systems by spatio-temporal variation of the applied light intensity.

References

611

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23 Predictability and Local Control of Low-dimensional Chaos R. H. Doerner Physikalisches Institut der J. W. Goethe-Universitat, Robert-Mayer-Strasse 2-4, D-60325 Frankfurt am Main, Germany

23.1 Introduction For more than a decade the chaos theory has been the object of intensive research. Great efforts have been made in order to understand the underlying rules of nonlinear systems and have led to impressive results. In this article we want to deal with two topics of general relevance in this field, the predictability of chaotic motions and techniques for the control of chaotic motions. We will demonstrate that the predictability of chaotic motions can vary with the state space coordinates of a system, sometimes leading to sophisticated and rather esthetic patterns. The unstable periodic orbits of the system play an important role for the formation of these patterns. Further we will point out some problems arising during the control of systems with strongly varying predictability and show how to solve them with a variant of the well-known OGY method [Ott, 1990]. This local control method uses only local properties of the system dynamics rather than invariant manifolds of periodic orbits. Hence it enables us to stabilize even aperiodic solutions of the equation of motion of the system. Finally, we present a method to improve the knowledge about a dynamical system and hence to improve predictions of its motion by observing it during chaos control. The observed deviations between the goal dynamics and the measured motion, and the applied control signal are used to calculate iteratively better and better approximations of the system dynamics. Our example systems will be a driven pendulum (both, numerically and in a laboratory experiment) and the Lorenz equations. Both systems are dissipative and in certain ranges of their system parameters their motion is governed by strange attractors. Their equations of motion can be written in a general form using the vectorflowfieldas x = F(x)

(23.1)

with the solution (called the flow) x(t)=f*(x(0)).

(23.2)

616

23 Predictability and local control of low-dimensional chaos

More specifically the equation of the pendulum reads: (23.3) where 6 means the angular deflection, 7 the damping coefficient, a, LJ and ip = ut the amplitude, the frequency and the phase of the driving sinusoidal torque and b an additional constant torque. The Lorenz equations usually are written [Lorenz, 1963, Sparrow, 1982]: X = -a(X-Y) Y = RX-Y-XZ

Z

= XY

(23.4)

-bZ

Here a, R and b are constant parameters and X, Y and Z are the variables of the system.

23.2 A definition of predictability The attempt to predict the temporal evolution of dynamical systems is a very general one. Such predictions are more or less restricted depending on the kind of the system and on the time interval for which the motion should be forecasted. The degree of restriction might be quantified by a suitable measure called predictability. It has been pointed out in [Grassberger, 1989] that one has to distinguish between the difficulty and the possibility of making predictions. In order to illustrate this, consider four experimenters each one observing another time series. The first experimenter is measuring an acoustic time series. After some time he recognizes, that the series stems from the ringing of a church bell belonging to a very accurate clock. Then he is able to predict the series for almost arbitrary long times with high precision and low effort. A second experimenter looks at a series of numbers being generated by a calculator. He finds out that the numbers are the first digits of iterations of the logistic map. His effort, necessary to predict the further iterations, increases linearly with the desired number of iterations. The precision of his results becomes less exponentially (if he does not see all digits that the machine is calculating with). A third observer knows that his time series comes from an irregularly shaped membrane vibrating chaotically. He has to struggle solving partial differential equations. His effort increases only linearly, too, but his task is much more difficult than the one of the second experimenter. The last time series consists of integer numbers following each other with a time lag of one second. Slowly their mean value increases. The observer finds that the next number will equal the number of prime factors of the number of seconds that have passed since a specific point of time in the past. At the beginning of the series a prediction of the next number is very easy, but with increasing time the effort

23.2 A definition of predictability

617

for prime factor decomposition grows. Some time it is too hard to manage the calculation within a second. Then the observer cannot predict the next number before it arises although the series is fixed for all times. In experiments 2 and 3 the possibility of making predictions is restricted due to chaotic behaviour. In example 1 and 4 the possibility is not restricted but in 4 the growing difficulty restricts the predictions. In this article we deal with the possibility of predictions, especially restrictions due to chaotic motion, when we are talking about predictability. It is obvious that the possibility of making predictions of the chaotic motion of a non-linear dynamical system is restricted since state space volumina are stretched. If the initial state x(0) of the system that is to be forecasted can be measured only with finite precision eo, one has to consider the evolution of a state space ball B(x(0),eo) centered at x(0) with radius eo in order to estimate this possibility. The more this ball of uncertainty is stretched by theflowof the system B H-> fT(B) during the time interval T, the smaller becomes the probability that a prediction is successful. Whether or not a prediction is to be called "successful" depends on the prediction error er that is tolerated. So we come to a straight-forward definition of predictability as the probability

g(x(0), T, eo, ex)

of a successful prediction

depending on the initial state x(0) of the system, the predicition time interval T and on the errors e0 and er> It is given as the fraction of fT(2?(x(0),eo)) that is contained in the ball B(x(T),er). One may expect the predictability of the motion of a chaotic system to be constant in the state space of the system. An experiment at our first test system, the driven pendulum, tells us that the opposite is true. Our experimental setup is an aluminum rod with one end attached at a right angle to the axis of a motor that applies a sinusoidal torque. At the other end a small weight is fixed. At the axis of the motor the angular velocity 6 of the pendulum is measured using a generator unit that provides a voltage proportional to the velocity. We obtain the angular position of the pendulum by continuous integration of this signal. In order to prevent cumulative errors during the integration we set the integral to zero each time that the pendulum passes a sensor at its bottom position. For a more detailled description see [Heng, 1994]. The pendulum is driven by a fixed amplitude a and a fixed frequency u in the chaotic regime. Each time the phase out equals a multiple of 2?r, i.e. when the trajectory of the pendulum crosses the Poincare section given by ut = 0 (mod2?r), we measure its angular position and its angular velocity. These data are inserted as initial conditions into the equation of motion to calculate a prediction of the next piercing of the trajectory and of the one after the next. The predicted positions then are compared to the positions at which the pendulum really crosses the Poincare sections 1, resp. 2, driving periods later. This procedure is repeated many (about 5000) times. Then we divide the Poincare section by a grid of 100 x 100 cells and color each cell according to the mean prediction error that has occured in the predictions that have been made for trajectories starting from the cell. We plot the

618

23 Predictability and local control of low-dimensional chaos

mean prediction error rather than the predictability since it does not depend on the choice of a tolerable error. With the mean prediction error the predictability depending on a given tolerable error can be estimated easily. The top row of Fig. 23.1 displays the resulting experimental predictability portraits of the pendulum. Since the pendulum was running on its attractor, not the whole Poincare section is colored but only the areas near the set of points where the section is cut by the attractor. It can clearly be seen that the mean prediction error neither is constant nor it is randomly distributed. Rather there are zones of decreased predictability forming a sophisticated pattern in the state space.

23.3 Effective Lyapunov exponents In this section we present a quantity that is closely related to the predictability as it has been defined above. This quantity can be calculated with low effort at the same time as the prediction is calculated. Consider a set of initial conditions in the state space forming a small ball with radius eo and center x(0). For a strictly damped system the volume V(t) of the ball decreases in time according to the theorem of Liouville: V(t) oc exp(- 7 *)

(23.5)

with the damping coefficient 7. On the other hand, due to the sensitive dependence on initial conditions, the ball is stretched in (at least) one direction. In linear approximation, the ball evolves into an ellipsoid with increasing length but decreasing volume:

V(x(0),t) = F(x(0),0)exp ( l > $ (x(O),t)<) ,

(23.6)

where A^ (x(0), t) t denotes the logarithm of the ratio of the i-th. semi-axis of the ellipsoid and the radius of the initial ball. The sum of the exponents equals the negative of the damping coefficient $(x(0),*) = - 7 .

(23.7)

They are called effective Lyapunov exponents [Grassberger, 1988] and they determine the divergence and convergence of adjacent trajectories along the axes of the state space ellipsoid evolving in time. With increasing time the effective Lyapunov exponents converge to the Lyapunov exponents for almost all initial conditions x(0) in the basin of attraction. For finite times, however, they can differ from these and, as we will see, sometimes they do so drastically. For our further investigations only the largest of the effective Lyapunov exponents is interesting. Therefore for the sake of convenience we will call it simply the effective Lyapunov exponent in the following and we will abbreviate it by A^.

23.3 Effective Lyapunov exponents

= 1 driv. per.

619

t = 2 driv. per. < error >

3.0

•0.0

1.0

0.0 -71

0

±TI

0

n

angular deflection 6 Figure 23.1 Top row: experimental predictability portraits of the pendulum for forecasting time intervals t = 1, respectively 2, driving periods. The color of a point indicates the mean prediction error that has occured in predictions of trajectories starting near this point. Points in the white area are not visited by the pendulum running on the attractor. Bottom row: The largest effective Lyapunov exponent of the pendulum, calculated from the equation of motion. The forecasting time intervals equal the intervals in the top row. Obviously the maxima of the effective Lyapunov exponent match the zones where large prediction errors occur.

620

23 Predictability and local control of low-dimensional chaos

In order to calculate the effective Lyapunov exponent for a given initial condition and a given time interval, one first has to determine the linearized flow along the belonging trajectory segment. A small difference vector between two initial conditions evolves according to

c = DF(x)c,

(23.8)

where D denotes differentiation with respect to x, with the solution e(*)=Df'(x(0)).£(0).

(23.9)

The linearized flow Df*(x(0)) is obtained by integration of eq.(23.8) d times simultanuously with the equation of motion. Here d means the state space dimension. A suitable choice for the initial conditions are the standard basis vectors because their images under Df* are just the columns of the matrix Df*(x(0)). The linearized flow maps a ball or an ellipsoid in the state space onto another ellipsoid. In general, the directions of the preimages of the axes of the ellipsoid are different from the directions of the axes. The action of the linearized flow on the ball can be imagined as split into two parts. Firstly the ball is rotated, secondly it is stretched, resp. compressed, along orthogonal directions. This view corresponds to a polar decomposition of the linearized flow Df* into an orthogonal matrix Q and a positive symmetric matrix P . Df* = Q P

(23.10)

Prom the eigenvalues (ev) of P the effective Lyapunov exponents are obtained via: A^ = ^logevW(P).

(23.11)

This approach is equivalent to the definition of Lyapunov exponents via the Oseledec matrix [Eckmann, 1985], since /2

= P.

(23.12)

(Note: Because of its numerical stability the QR-decomposition [Eckmann, 1985] is the favourable method to calculate Lyapunov exponents. Note that the eigenvalues of the upper triangular matrix R are different from the eigenvalues of P for finite time intervals. Hence the QR-decomposition is not suitable to calculate effective Lyapunov exponents.) Remember our definition of predictability. In the presence of large stretching rates a small ball JE?(x(0),eo) evolves into a narrow, long ellipsoid with a large semi-axis eoexp(A$(x(0),T)r).

(23.13)

The remaining semi-axes all are shorter than eo or equal to it (in the presence of one positive Aeff). With a sufficient small tolerable prediction error er now the predictability becomes in the limit of a small uncertainty of the initial state: lim

(0) , T ) r ) .

(23.14)

23.3 Effective Lyapunov exponents

621

Here C means a factor depending on the dimension of the state space. With this result in mind we can use the terms "low predictability" and "large effective Lyapunov exponent" like synonyms. From eq. (23.3) we calculate the effective Lyapunov exponent of the pendulum on the same Poincare section that we have used in the experiment above. The parameters of the equation are determined by fitting them to the experimental setup: 7 = 0.05, a = 0.58, LJ — 0.66. The time intervals for the integration equal the prediction time intervals in the experiment, i.e. 1, resp. 2 driving periods. The result is shown in the bottom row of Fig. 23.1. Since the algorithm is not restricted to initial conditions on the attractor, this time the whole Poincare section is colored. Again we find sophisticated patterns in the state space. As we have expected, the zones of large values of the effective Lyapunov exponent A^ff match very well the zones of low predictability that we have found experimentally abbve. Often one has no information about a non-linear system but a time series of a chaotic signal. Then the possibility to reconstruct a state space using delaycoordinates is of great importance [Takens, 1981]. In recent yfears a variety of methods has been proposed to extract the spectrum of Lyapunov exponents from a reconstructed attractor [Wolf, 1984, Sano, 1985, Eckmann, 1986, Briggs, 1990, Bryant, 1990, Brown, 1991]. Here we follow a scheme proposed in [Brown, 1991] and demonstrate that also the effective Lyapunov exponents can be determined this way. Our test system is the Lorenz system with the common choice of the parameters R = 28, cr = 10,6 = 8/3. First we calculate by integration of eq. (23.5) a series X(n) of N = 2 - 105 values of the X-coordinate. The temporal distance between two successive points is r = 0.1 in dimensionless units. We add white noise to these values with an amplitude ax = 0.025. Then we construct vectors using each d = 4 successive values of the series y(n) = [ X ( n ) , X ( n - l ) , X ( n - 2 ) , X ( n - 3 ) ] . (23.15) The dimension d = 4 is sufficient to embed the dynamics of the Lorenz system [Abarbanel, 1990]. Next we determine the r = 10 nearest neighbors of each y(n). An effective algorithm for this purpose can be found in [Eckmann, 1986]. When the next neighbors are found it is sufficient to consider only the 2-vectors y(n) = [X(n),X(n — 1)] since we want to calculate only the largest effective Lyapunov exponent [Brown, 1991]. Let c*(0) denote the distance between y( n ) a n ^ its i-th nearest neighbor at (discrete) time t = 0. This distance is mapped by the flow f*. to the distance €*(£) between y(n + t/r) and the image of its formerly i-th nearest neighbor (do not confuse with the nearest neighbors at time t). For the evolution of the j-th coordinate of el we use the ansatz:

4 («) = E DfiAio) + E 2^k€i{0)em

+

••••

(23 16)

-

Now we fit the matrices Df* to the "experimental" data using a least squares fit. Then we determine the effective Lyapunov exponents from the Df* via eq.(23.11).

622

23 Predictability and local control of low-dimensional chaos

*

I

t

-2

-30

Figure 23.2 The largest effective Lyapunov exponent of the Lorenz system for two forecasting time intervals. The top row shows the attractor in the original state space, the bottom row shows the attractor reconstructed from a scalar time series.

The results are shown in the bottom row of Fig. 23.2 for time intervals t = 10r = 1 and t = 20r = 2. Clearly zones of low predictability become visible. Due to the topological similarity of the reconstructed Lorenz attractor and the original attractor the figures can be compared easily with predictability portraits that have been calculated from the equation of motion. These portraits are displayed in the top row of Fig. 23.2. As we have seen, the effective Lyapunov exponent can be calculated simultanously with the calculation of the prediction. So it is possible to estimate the reliability of the prediction at the time when the prediction is made. In a situation where the moment at which a forecast has to be made, can be chosen, this knowledge may be used to reject predictions for states with low predictability. This technique can lead to drastic reductions of the mean prediction error as we have demonstrated in [Doerner, 1994].

23.4 Unstable periodic orbits

623

23.4 Unstable periodic orbits During observation of the motion of a dynamic system like the driven pendulum in its chaotic regime one often gets the impression that the system has decided to calm down and to move on a periodic cycle. Twice or more times the system follows almost the same piece of trajectory but sooner or later this almost periodic phase is left again. This observation corresponds to the well known fact that embedded within a strange attractor there is a set of periodic orbits. Generally, the number of orbits with a short period is small but it increases exponentially with the length of the period. The motion on all these orbits is unstable, that means an arbitrary small disturbation is enough to make the system leave the orbit at an exponential rate in time (on the average). In this chapter we will show how to determine unstable periodic orbits (UPOs) and discuss the local dynamics in their vicinity. There are (at least) two different approaches to determine the location of UPOs of a dynamic system. Which way to choose depends on the degree of knowledge one has about the system under consideration. The first possibility is to extract them directly from a time series. For this purpose a reconstruction of the state space using delay coordinates is necessary. This technique founds on the embedding theorem of Takens. In recent years much effort has been done to optimize reconstruction techniques, especially to find the appropriate embedding dimensions and delay time lags. For details see e.g. [Liebert, 1989, Buzug, 1992a, Buzug, 1992b, Gao, 1993, Kember, 1993, Gao, 1994], a list far from being complete. Secondly, for systems with known equations of motion, the UPOs can be calculated directly by Newtons method for example. To do so, consider the motion of the system reduced to a Poincare map. If a point in a Poincare section is mapped almost onto itself under the first or higher iteration of the Poincare map, it is a fine first guess for Newtons method. Let xo be this point, xi be its image, then calculate the sequence T l T x n + 1 = x n - [Df (xn) - l] ~ (f (xn) - x n ) .

(23.17)

Here 1 denotes the unity matrix. If f T (x n ) - x n converges to zero, then an UPO with period T is found. We have applied this method to the equations of motion of the driven pendulum and to the Lorenz equations. In Fig. 23.3 we present ten UPOs of the pendulum equation (23.3) that we have calculated like described above. The period of these UPOs equals one driving period. We have found no further UPOs with this period. As mentioned above the volume of a state space vicinity contracts during its temporal evolution. This is valid, too, for a state space vicinity that is pierced by an unstable periodic orbit. Therefore the repulsion of trajectories must be overcompensated by attraction of others. The local dynamics near an UPO is dominated by two subsets of the state space, the stable and the unstable manifold of the UPO. These manifolds are defined [Guckenheimer, 1983] as the sets containing all trajectories tending to the UPO for T -» -oo or T -» oo, respectively. The

624

23 Predictability and local control of low-dimensional chaos

-7C

0

±7C

0 ±71 0 ±% angular deflection 0

0

+7C

Figure 23.3 The ten shortest UPOs of the pendulum, projected onto the Poincare section cut = 0 (mod 2?r). The dots mark their intersections with the Poincare plane. UPOs a and j are librations about the stable and the unstable fixed points of the non-driven pendulum, respectively. Orbits b and c are librations about these both points, jointly. The orbits d, / , and h are rotations to the left. Due to the symmetry of the equation of motion they all have symmetric counterparts, the UPOs e, g, and i.

dimension of these subsets is lower than the dimension of the state space. Near an UPO a trajectory that is not part of its stable manifold (i.e. almost all trajectories), is attracted to the UPO along its stable manifold and is repulsed along its unstable one. Associated with each UPO there is a set of Lyapunov exponents. One of them equals zero [Haken, 1983], the sum of the other equals the dissipation coefficient as it does for the global exponents. But depending on the system and on the UPO the value of the non-vanishing exponents, hence the divergence rate of adjacent trajectories, can differ dramatically from the global Lyapunov exponents. We will see that this fact leads to the formation of sophisticated patterns of varying predictability in the state space like we have found them above.

23.5 The origin of predictability contours Above we have seen that effective Lyapunov exponents are a suitable tool in order to quantify the predictability of chaotic motions. The temporal fluctuation statistics of these exponents have been discussed before, see e.g. [Abarbanel, 1991, Fujisaka, 1983, Nicolis, 1983, Haubs, 1985]. Note that in some papers effective Lyapunov exponents are called local Lyapunov exponents, whereas elsewhere local Lyapunov exponents mean the limit limA^(x, t). In this chapter we will focus t-+o

our attention on the spatial distribution of predictability, i.e. the geometry of isopredictability surfaces in the state space. We will demonstrate that these surfaces

23.5 The origin of predictability contours

625

Figure 23.4 Schematic illustration of the linearized flow at an unstable periodic orbit. A ball is mapped onto an ellipsoid. In general, the eigenvectors of the flow, s, u, ... are different from the semi-axes of the ellipsoid a, b,

closely follow the stable manifolds of the most unstable periodic orbits or the most unstable fixed points of the system. Let x^ be a point on an UPO of period TUPO. For a calculation of its Lyapunov exponents we have to consider only one period of the motion. For an illustration see Fig. 23.4, where we sketch the flow around the UPO in two dimensions. We show an (infinitesimally small) neighborhood of the UPO at two successive piercings x/r through the same Poincare section. The linearized flow Df T |x F maps a unit circle around xp to an ellipsoid around x^ with semiaxes a and b. The lengths of these axes give the effective Lyapunov exponents for one period:

lnlal

(23.18)

Generally, these axes are no eigenvectors of Df T |x F - They have been stretched, respectively shrunken, maximally in the first period, but they won't be stretched or shrunken by the same factors during the following periods. The eigenvectors of Df |x F , here denoted as u and s are the only vectors that will be stretched, respectively shrunken uniformly, i.e. by the same factor, during each of the following periods. Therefore the Lyapunov exponents of UPOs can easily be calculated by: »(xp)

=

ln|u| 1

In |s|

•lnev«(Df T | XF ).

(23.19) (23.20)

The subspaces, spanned by u and s, are tangential to the unstable, respectively stable, local manifolds of the UPO at x/r. Note that an unstable orbit of a strictly

626

23 Predictability and local control of low-dimensional chaos

dissipative system has (at least some) real eigenvalues [Guckenheimer, 1983]. The Lyapunov exponents of an unstable fixed point xp can be calculated even more easily. They equal the roots of the characteristic polynomial det[DF(xF)-Al].

(23.21)

In Table 1 we list the positive Lyapunov exponents of the ten shortest UPOs of the driven pendulum with the same system parameters as above. Note that all of them are larger than the positive Lyapunov exponent of the strange attractor that embeds these UPOs. Considering the dynamics on the stable and the unstable manifolds of an UPO or an unstable fixed point and close to them, one yields the mechanism, that generates the predictability structures. Let us assume an UPO with a A^1^ larger than that of the attractor that it is embedded in. Trajectories, starting on the stable manifold of this UPO, converge to it at a rate given by the associated negative Lyapunov exponents. During their convergence they will sample more and more the same large local contributions to their A^? as the UPO does. Therefore their A^f (£) converge to the effective Lyapunov exponents of the UPO with increasing forecasting interval t. In addition, the X^(t) of the UPO converge to its Lyapunov exponents AupOOur argument is true also for trajectories close to an UPO, as long as they stay close to it. They are attracted along the stable manifold, but at the same time they are repelled along the unstable one. Near the UPO the rate of divergence is smaller than the rate of convergence along the stable direction, since in a dissipative system the sum of the positive Lyapunov exponents has a smaller amount than the sum of the negative ones. So for a certain time interval the trajectories are guided by the UPO and again they sample almost the same local contributions to their Lyapunov spectrum as the UPO does during this interval. Following these considerations we want to conclude: State space portraits for forecasting intervals t in the range of UPO-periods indicate large values of A^ along the stable manifolds of those orbits or fixed points, that possess larger positive Lyapunov exponents than the attractor does. In the limit t -» oo Oseledec's theorem [Eckmann, 1985] holds: p-almost all initial conditions lead to the same Lyapunov exponents A^ (p means the invariant density of the attractor). The exceptions are fixed points, UPOs, and their stable manifolds. In Fig. 23.5 we illustrate this mechanism at the driven pendulum. We plot the effective Lyapunov exponent on the plane ut = 0 (mod 2?r) for increasing forecasting time intervals. Starting with t = 0 in the left top portrait the predictability structure is rather simple: it depends only on the angular deflection. At the top position of the pendulum, i.e. at i? = ±TT the predictability has its minimum, because there adjacent trajectories are separated at the most. In the next portrait (the right one in the top row) the forecasting interval lasts t = 0.25 driving periods. Here the structure of the first portrait has been distorted slightly. Trajectories that start from the first or the third quadrant will reach the critical top position of the pendulum within the forecasting interval. Those starting in the bright zones will

23.5 The origin of predictability contours

UPO a b, c d, e f,g h,i j

attractor

627

A*1' 0.26 ± 0.01 0.30 ± 0.01 0.31 ± 0.01 0.42 ± 0.01 0.57 ±0.01 0.95 ± 0.01 0.160 ±0.002

Table 23.1 Lyapunov exponents of the ten shortest UPOs of the pendulum at the given parameter set. The letters refer to Fig. 23.3

have a low velocity when they reach the top position and so they will spend a rather long time there. Therefore their A^ is maximal. For longer prediction time intervals the structure becomes more and more sophisticated. The main branches are visible already for t = 0.5 driving periods. In the next portraits these branches are split and new, finer structures arise. The last part of Fig. 23.5 shows pieces of the stable manifolds of some of the ten short UPOs of the pendulum. The points where these UPOs pierce the Poincare section are marked by circles. Obviously these pieces match very well zones of maximum effective Lyapunov exponent. The stable manifolds of different UPOs tend to come very close to each other, therefore it is difficult to attach each isopredictability surface to a specific UPO. The best correspondence between zones of low predictability and the stable manifold of an UPO we have found for the orbit j . This orbit j has another interesting property. If you reduce 7 and a in eq. (23.3) simultanously to smaller and smaller values (i.e. 7, a -> 0, 7/0 = const.), the projected circle in Fig. 23.3 contracts to the point (TT,O). In two dimensions this point is the location of the unstable fixed point of the Hamiltonian pendulum. At the same time the stable manifold of the UPO j converges on every Poincare section to the separatrix of the Hamiltonian pendulum. UPO j is the only orbit that we could trace this way, the other UPOs vanish at finite 7, a. These both facts emphasize the extraordinary importance of the point at the top position for the dynamics of the driven pendulum. In the Lorenz system with the standard paramters i? = 28, cr = 10,6 = 8/3 we find a slightly different mechanism for the generation of predictability structures. Its UPOs all consist of one or more loops around each of the two unstable fixed points C± in the centers of the two wings of the Lorenz attractor [Franceschini, 1993]. We abbreviate a loop around C+ by P and a loop around C_ by N. The Lyapunov exponents of the UPOs that we have investigated do not differ too much from the Lyapunov exponent of the attractor. We list them in Table 2. Here it is another prominent solution of the equations of motion that possesses an extraordinarily large Lyapunov exponent, it is the unstable fixed point 0 = (0,0,0). Therefore at the Lorenz system the isopredictability zones fol-

628

23 Predictability and local control of low-dimensional chaos

-3 -71

0

±71

0

n

angular deflection 6 Figure 23.5 The evolution of predictability structures in the state space of the driven pendulum on the Poincare section cut = 0 (mod 27r), starting with a prediction time interval t = 0 in the left top part and evolving up to an interval * = 1.5 driving periods in the bottom left part. Slowly the characteristic pattern arises and becomes finer and more and more structured. The right bottom figure shows parts of the stable manifolds of six short UPOs of the pendulum. These parts are calculated by integration of the equation of motion backwards in time with initial conditions very close to the UPOs. The intersection of the UPOs with the Poincare section are marked by dots. Obviously the predictability structures follow closely these manifolds.

23.6 Chaos control in the presence of large effective Lyapunov exponents

UPO PN PPN, PNN PPPN, PNNN PPNN 0 attractor

629

0.997 ± 0.0005 0.964 ± 0.001 0.922 ± 0.002 0.971 ± 0.001 12.0 0.905 ± 0.002

Table 23.2 Lyapunov exponents of the fixed point at the origin and of the 2,3,4-UPOs of the Lorenz attractor at R = 28. Note the huge A^ of the fixed point at the origin.

low the stable manifold of this point and are not correlated with the location of the UPOs nor of their manifolds. For a detailled discussion of the stable manifold of O and its dependence on the system parameters see [Sparrow, 1982] and [Jackson, 1985a, Jackson, 1985b]. In the latter two you find suggestive diagrams of the manifold with the spiral structure that also occurs in the predictability portraits in Fig. 23.2. In [Doerner, 1996] we present a simplified model of the dynamics near an UPO. From this model we can deduce some geometric properties of isopredictability surfaces that can be observed in Fig.23.5. We find that for increasing forecasting time the length of the intersection of an isopredictability surface and a Poincare surface increases linearly whereas its width decreases exponentially. Further, we investigate at the Lorenz system the dependence of predictability on the parameter R. We find that the stable manifold of 0 affects the predictability of chaotic transients even if the system parameters are in the non-chaotic regime and no strange attractor exists.

23.6 Chaos control in the presence of large effective Lyapunov exponents The idea of controlling the chaotic motion of a non-linear system has been stated first by Hubler and Luscher [Huebler, 1987, Huebler, 1989a, Huebler, 1989b]. They use an open control loop adding a previously calculated driving force to switch the chaotic motion to a regular one. The periodic motions that they achieve are no solutions of the original system, because the system is changed. A completely different approach is the well-known method of Ott, Grebogi and Yorke (OGY) [Ott, 1990]. The controlled trajectories are nearly solutions of the undisturbed system, because the algorithm balances the trajectory along an UPO of the system. To this end a control signal is calculated each time that the trajectory pierces a chosen Poincare section. This control signal then is applied

630

23 Predictability and local control of low-dimensional chaos

to the system until its trajectory passes the Poincare section again. In order to calculate the control signal you need firstly a model of the local dynamics near the UPO and secondly a precise measurement of the current state of the system (the point where it passes the Poincare section). Any error of this measurement will be amplified exponentially during one Poincare period causing the control to fail if the largest effective Lyapunov exponent near the UPO is large enough, i.e. if the UPO is highly unstable. As we have seen above, all the UPOs we have found at the driven pendulum are very unstable. A state space element near UPO j in Fig. 23.3 is stretched by factor around 5000 during one driving period. Since we use a 12bit analog-todigital converter to determine the angular deflection and the angular velocity of the pendulum, a typical error of the measurement is blown up to the size of the state space during only one period of this UPO. In order to overcome these problems and to control even very unstable UPOs we have introduced an extension of the OGY method, the local control method [Huebinger, 1993, Huebinger, 1994], that we will briefly review here. Since the local control method makes use only of local properities of the flow it is even possible to control arbitrary aperiodic trajectories that are solutions of the equation of motion.

23.6.1 The local control algorithm The idea is to apply a new control signal not only at one Poincare section but at N > 1 sections, so that the uncertainty about the current system state cannot grow too much. To this end the goal of control must be reformulated. The OGY algorithm forces the difference vector between the UPO and the real trajectory onto the stable manifold of the UPO. Generally, a trajectory moving on this manifold does not converge to the UPO at every point. Rather it may locally diverge from the UPO for short times. But on the average over one period of the UPO the trajectory moves towards the UPO and the motion is trapped by the control. Locally in the state space at every moment an infinitesimal ball centered around the current system state is mapped in the very next moment onto an ellipsoid. Generally, the longest semi-axis of this ellipsoid is longer than the radius of the initial ball. This holds at least in some areas of the state space that are visited by the trajectory frequently. We call the direction of the longest semi-axis the local unstable direction. The shortest semi-axis of the ellipsoid is shorter than the radius of the ball, if the system is dissipative everywhere in its state space. This direction we call the local stable direction. Note that the local stable and the local unstable direction are always orthogonal. In general they are different from the tangent vectors of the invariant manifolds of the trajectory. Let Szn be the difference vector between the real trajectory and the desired trajectory in the n-th Poincare section S n . A short time later, when the trajectories cross the (n + l)-th Poincare section E n + 1 , it becomes: = An8zn + wn6pn.

(23.22)

23.6 Chaos control in the presence of large effective Lyapunov exponents

631

Here A n means the Jacobian of the Poincare map fj : E n h+ E n + 1 , w n = df£/dp means the local sensitivity of that map to changes of the control parameter. Now, the local control demands that the projection of the current system state onto the unstable direction must decrease with every control step: e£ + l t (Jz n + 1 = (1 - p) . e J j W ,

0 < p < 1,

(23.23)

where e£ denotes the unit vector of the unstable direction and p a constant rate of decay. Inserting (23.23) in (23.22) leads to

We simplify this expression by using the polar decomposition A n = Q n P n = Q"(^2 e "e"' + ^™e"e™ ). For short enough time steps, i.e. for large enough N, we can set Q n ss 1,/u™ « 1 and e™+1 « e" and we obtain the control amplitude Spn

« - p ^ l ,

(23.25)

This algorithm does not need properties of a periodic orbit like a stable manifold, instead only local properties of the flow are used. Therefore it can be applied to control any solution of the equation of motion, even aperiodic trajectories.

23.6.2 Experimental results We apply the method to the pendulum introduced above with the parameters 7 = 0.052, a = 0.66, a; = 0.58, that we have determined via a flow vector field analysis [Cremers, 1987] of a measured time series. The offset b is set to zero. We establish N = 64 equidistant Poincare sections E n perpendicular to the time axis, i.e. cut = 0(mod27r). We abbreviate their temporal distance in the following by At. With the equation of motion (23.3) the matrices A n become: At

^

(23.26)

As control parameter we select the offset torque b. In numerical experiments we have found structural stability against small changes of this parameter. With the control parameter b the sensitivity w n can be approximated for small At by w n « (0, At). The effect of a small offset Sb is an additional angular acceleration within the interval At. The periodic and aperiodic trajectories to be controlled are calculated from the equation of motion of the pendulum. First we control UPOs of different lengths. The coordinates of these UPOs in the 64 Poincare sections are stored in the memory of a small transputer network. Also the unstable directions at these points

632

23 Predictability and local control of low-dimensional chaos

3 2

OClt

1

3 &0

•

•

A

;

-V '---

>

Q

(4) (3)

^

(5)

(6)

(2)

-1

W-' a)--

§ -2

• •

• •

.

p (4)

: (2)

'

^

•;

(3)

-3

10

15

20

25

min

angular deflection

time t

Figure 23.6 On the left side we display the angular velocity measured at the pendulum setup in the Poincare section ut = 0 (mod 2?r) during a control experiment. At time t — 0 the control is switched on. The motion, that was previously chaotic, almost immediately follows the goal dynamics. Every five minutes this goal dynamics is changed to another UPO. We choose UPOs lasting one, two or three driving periods, therefore one, two or three horizontal lines appear in the diagram. Projections of the UPOs onto the 0, 0-plane are shown on the right side.

are stored. The network is connected to the pendulum via an analog-to-digital converter to measure its state and a dialog-to-analog converter with amplifier to apply the periodic torque and the control signal. Each time, when the trajectory of the pendulum crosses a Poincare section £ n , we calculate a control signal Spn. The calculated control signal is applied to the pendulum if \Sbn\ < 0.2a, otherwise it is set to zero. As we have found empirically, a suitable value of the decay rate in our setup is p = 0.15. The proper choice of this constant depends on the number of Poincare sections one uses for the control. For an increasing number, i.e. a decreasing distance, the parameter should be smaller. In Fig. 23.6 we display on the left side a series of the velocity of the pendulum the Poincare section out = 0 (mod 2TT) during a control experiment. In the beginning the control is switched off and the pendulum runs chaotically. At time t = 0 the control is switched on. After a few cycles the motion of the pendulum follows closely the UPO. Every 5 minutes another UPO is chosen. Some of these UPOs have a length of 2, respectively 3, driving periods. Here 2, respectively 3, horizontal lines appear in the diagram. Once controlled, the motion stays stable for hours or

23.6 Chaos control in the presence of large effective Lyapunov exponents

633

days. The pendulum is placed on a simple desk and does not need to be decoupled carefully from external perturbations. Only rough pushes can cause the control to fail. On the right side of Fig. 23.6 we show projections of the UPOs that have been controlled in this experiment. Besides these we found all other UPOs that we have calculated from the model to be controllable at the pendulum setup, too.

4

6

time (driving periods)

Figure 23.7 A comparison of the angular velocity of the predicted trajectory (solid line) and the velocity measured at the experimental setup (dots). In part (a) the pendulum runs freely and its motion diverges from the prediction significantly after about 2 seconds. In part (b) the control is applied using the predicted trajectory as goal dynamics. The motion of the pendulum stays close to the prediction. Fig. 23.7 shows the result of a second experiment that can be realized only due to the specific feature of the local control to be independent of periodic properties of the target trajectory. This target trajectory now is no UPO but an arbitrary solution of the equation of motion of the pendulum. Here we have chosen the trajectory starting at 6 = 0,0 = 0,ut = 0. In the upper part of the figure you see

634

23 Predictability and local control of low-dimensional chaos

the calculation superposed by a series of measured states of the pendulum (dots) starting at the given initial conditions. Obviously this motion diverges from the calculation after about two seconds significantly due to the sensitive dependence on the initial conditions (and due to the finite mismatch of the model and the real pendulum). In the lower part the experiment is repeated with active control. Now the pendulum follows the calculated chaotic trajectory. The figure displays the trajectory for only about 10 seconds but we are able to balance the motion of the pendulum on the calculated trajectory for many hours. Note that a time series recorded at the pendulum during this control looks like an ordinary free-running chaotic motion. Hence it seems to be unpredictable. The experimenter applying the control, however, knows the time series for arbitrary times. How this hided information may be used for cryptographic purposes, e.g., is described in [Martienssen, 1995]. Here we have applied the control to a system with a known dynamics using the equation (23.3) which models the experimental setup up to minor deviations. In [Huebinger, 1994] the local control is successfully applied to a driven bronze ribbon. Here the UPOs and linear approximations of the local dynamics near the UPOs have been extracted from a time series but still the state space coordinates are known. Recently, in [Schenck, 1996], the feasibility of the local control has been demonstrated experimentally using only information from a time series in time-delay coordinates.

23.7 Adaptive orbit correction in chaos control In this section we will discuss a method to improve both, prediction and control of chaotic motion, by iterative approximation of the UPOs of the system. This application demonstrates very well the strong relation between control and prediction. It has been introduced in [Doerner, 1995]. The OGY method, as well as the local control requires a more or less precise knowledge of the local dynamics of the system near the UPO that is to be controlled. Even a small mismatch between the dynamics of the model that the control signal is calculated from, on the one hand, and the dynamics of the real system, on the other hand, will lead to a larger necessary control signal. Especially in the presence of large effective Lyapunov exponents a small mismatch may even prevent control. In general, the UPOs that one wants to control are either calculated from an approximate equation of motion describing the system, or they are extracted from a time series (using a method like in [Auerbach, 1987, Dressier, 1992, Schenck, 1996]). In both cases these approximate UPOs have only finite precision and will differ slightly from the unknown real UPOs of the system. Due to this difference the mean control signal (Sp) will not vanish during control [Schwartz, 1992]. Further the system will not follow exactly the approximate UPO, not the real UPO, but another orbit that we will call the driven UPO. Here we will demonstrate how to use the non-vanishing mean control signal and the coordinates of the driven UPO in order to find successively better approxima-

23.7 Adaptive orbit correction in chaos control

635

tions of the real UPO. During the iteration process the mean control signal will decrease significantly to an almost vanishing amount. Further, we will demonstrate that the method is able to provide a stable control in a slowly (with respect to the system dynamics) changing environment.

23.7.1 Orbit correction in the Henon map Consider a low-dimensional system in its chaotic regime with a Poincare section E. A real UPO of the system pierces this section in XF(PO) = fj_(xF(po),Po),

(23.27)

where fj_ means the Poincare map belonging to E. The UPO coordinates are known only with finite precision and we assume zp to be a suitable approximation of xp. The control parameter p is set to its nominal value po- The quantities that are necessary to apply the OGY control method to the system are zp(po), the Jacobian A(po)

= «

Df ± (x F (p o ),Po) Dfj_(zF(po),Po)

(23.28) (23.29)

and the sensitivity of the system to control parameter adjustments op ~

d~v

•

(23 31)

'

Then with the OGY algorithm we obtain the control signal SPn = -J^v.6zn,

(23.32)

V • W

where v denotes the contravariant unstable eigenvector of A(po), Xu its unstable eigenvalue, and 6zn a small distance between the current system trajectory and the approximate UPO that is to be controlled. Assume that the control is successful and that there is no noise within the control loop. Due to the difference between the real UPO and the approximate one, one observes that the control signal converges to a non-vanishing value <5pooThe intersection of the driven UPO and the Poincare section E converges to the point ZQQ. In general this point differs from zp and from x^. This last difference, SZQO = ZQO —xp is mapped to itself during one Poincare period according to + Spoow.

(23.33)

Therefore, from <Jzoo = ( l - A ) - 1 - < J p o o w ,

(23.34)

636

23 Predictability and local control of low-dimensional

chaos

one yields the real UPO xp — z ^ - Sz^. In the presence of noise, dpoo and ZQO have to be replaced with the time averages (dpoo) and (ZQO). We apply the method to the Henon map

Xn+1 = Yn + \-aXl

(23.35)

yn+1

= bXn

(23.36)

a = 1.4 and b = 0.3.

(23.37)

with

We choose b to be the control parameter, the other parameters are:

and the fixed point

XF\_( ( & - 1 - v ^ T p T 4 ^ ) / 2 a \ _ / 0.63135... YFJ-\b(b-lV(&-l)2+4a)/2a J ~\ 0.18940... We simulate inaccurate knowledge of the system by changing the components of A, w and thefixedpoint by about one percent, and we simulate noise in the control loop by adding random numbers from the interval [-0.001 ... 0.001] to the iterates Xn,Yn before the calculation of the next control signal. Fig. 23.8(a) shows the control signal for this map controlled with the OGY control together with the adaptation scheme. The control signal is plotted versus the discrete time. We start with the inaccurate control quantities and achieve a control signal with nonvanishing mean. After 1000 iterations the correction of the fixed point is calculated resulting in a control signal with a significantly less mean. Note that the error in A and w is not corrected. A correction is calculated a second time after 2000 iterations. A small effect of this correction is visible: the mean control signal does not longer significantly differ from zero. Further corrections at every 1000 iteration show no more visible effects.

23.7.2 Orbit correction in a changing environment In a second numerical experiment at the Henon map we show the effect of adaptation during OGY control in a slowly changing environment. We modulate the parameter a sinusoidal with an amplitude Aa = 0.02 and a period of 5000 map iterations. Fig. 23.8(b) shows the resulting control signal without adaptation, Fig. 23.8(c) with adaptation after every 50 iterations. Obviously the modulation of Sp following the drift of a(t) is reduced significantly. The adaptation enables us to track the slowly oscillating fixed point, the errors in A and w again seem to be of no importance for the success of the method.

23.7 Adaptive orbit correction in chaos control

637

o.i 0.05 0 -0.05 -0.1

(a)

0.1

«§•

13 ° ' 0 5

2 -0.05 c o «

-0.1 0.1 0.05 0 -0.05 -0.1

(c) 1000

2000

3000

4000

5000

control step n

Figure 23.8 Numerical control experiment at the Henon system. We display the control signal that has been applied using the OGY method. In part (a) slightly "wrong" coordinates are inserted into the control formula. After each 1000 control steps the adaptation scheme is applied to calculate a new approximate UPO. In part (b) the parameter a is varied sinusoidally with amplitude 0.02 and a period of 5000 iterations. Without adaptation the control signal Sp shows large variations. When the adaptation is turned on (c), the variations almost vanish.

23.7.3 Experimental orbit correction at the driven pendulum Now we will extend our adaptation method to an experimental system with large effective Lyapunov exponents, the driven pendulum that we have introduced above. Since we have to use the local control (see section 23.6) method to control this device, we need to correct the UPO coordinates not only in one Poincare section, but in AT > 1. The real UPO pierces these sections in x£, n = l,...,iV. Again these intersections are known only with finite precision and assumed to be zjp. And again this leads to a non-vanishing mean control signal Sp1^, n — 1,..., iV, and a driven UPO z ^ differing from both, the real UPO and the approximate one. Assuming again that the most important error in the control quantities is the difference between x£ and z-p we write the map from the n-th Poincare section onto the (n -f- l)-th one as: A

n c~n

- A 07.

i

n c

n

-f- W Op

/OO

AC\\

(Zo.Qv)

638

23 Predictability and local control of low-dimensional chaos

with = z ^ - xj Since the orbit is periodic, the (N + l)-th Poincare section equals the first and therefore, one obtains a set of N linear equations: -A

Silo

1

0 1

0 0

-A' \ 0

SpNwN Sph

-A2

0

V o

-1

1 - A ""

1

V (23.41)

We solve these equations by a standard method and again we achieve an improved approximation of the UPO coordinates with Xp = z ^ — Sz1^. To control the pendulum we choose the torque b as control parameter. Prom the equation of motion we calculate approximate UPOs, the Jacobians A n , their unstable directions e™, and the sensitivities to the control parameter w n . The control signal is calculated and adjusted at N = 64 Poincare sections. Since the equation of motion cannot describe the experimental setup exactly, the approximate UPOs differ from the real UPOs. Therefore during control a periodic component of the control signal becomes clearly visible (see the uppermost part of Fig. 23.9). We determine the sequences (Spn) and (z n ), n = 1,..., N, by averaging several records of Spn and z ^ for each n. Then we apply the adaptation formula Eq. (23.41) to these data and calculate a new approximate UPO. We repeat this procedure several times. In Fig. 23.9 the control signal for four periods of a controlled UPO is plotted. In the first the uncorrected UPO is controlled, the modulus of the (almost periodic) control signal Sp1^ has an average amount of 0.045 which is about seven percent of the amplitude of the driving torque. With each application of the adaptation scheme from row to row the amount of the control signal decreases. In the last row no periodicity can be recognized any longer. The average amount has decreased down to 0.0035, less than a tenth of the signal in the first row. At our pendulum we have applied the method to a number of different UPOs with lengths from one to three driving periods. For all UPOs we have found convergence of the method. Further we have simulated a slowly changing environment tilting the pendulum slowly by several degrees. Even then the method is able to track the UPOs.

23.7 .4 Interaction of prediction and control, outlook The UPOs are of significant importance for the evolution of a nonlinear system in the chaotic or in the transient chaotic regime. The set of all UPOs with periods not exceeding a given cutoff provides a concise approximate representation of a system with chaotic dynamics. The quality of this cylcle expansion usually increases quickly with growing cutoff period. For example the multifractal properties of strange attractors can be approximated systematically in terms of UPOs [Auerbach, 1987, Auerbach, 1988, Gunaratne, 1987, Artuso 1990a, Artuso, 1990b].

23.7 Adaptive orbit correction in chaos control

639

-0, 0,

-0, 0,

13 - 0 .

•s o «g» o .2 o -o.

.2

*m

m*

-0, 0,

-0

64

128

192

256

control step n Figure 23.9 Real-world experiment at the pendulum setup. In the uppermost row we display the control signal that has been applied during the local control of UPO b. The UPO coordinates have been calculated from the equation of motion of the pendulum. Obviously this signal is nearly periodic. The following rows show the control signal after successive iterations of the correction algorithm. During five iterations the signal decreases by about one order of magnitude and the periodic component of the signal almost vanishes.

640

23 Predictability and Local Control of Low-dimensional Chaos

Also all information that is necessary to make short time predictions of the motion of the system is contained in this set of UPOs. Pawelzik and Schuster [Pawelzik, 1991] have demonstrated, how to predict chaotic motion using UPOs. To control an UPO means to use it as a local predictor predicting the motion of the system near the orbit and then to balance the motion along the predicted trajectory segment. We have seen that the method presented here yields best results if the adaptation scheme is applied repeatedly. In each step a new, better approximation of the UPO coordinates is calculated. With a better approximation of the UPO at hand the quality of predictions of the motion in its vicinity increases and the control adjustments become more effective. Hence the control signal decreases. So the local predictions benefit from the data measured during control and vice versa the control benefits from improved predictions. If a complete set of UPOs up to a given length is "treated" with our method it can be used to calculate predictions for arbitrary chaotic trajectories of the system following the suggestions in [Pawelzik, 1991]. Such predictions should be superior to predictions based on a first approximation of the UPOs. In section 23.5 we have seen that the UPOs generate predictability portraits. With the corrected UPOs one may improve a predictability portrait that has been calculated from a model. To that end one has to move each structure by the experimentally determined coordinate shift of the generating UPO. Consider a set of approximate UPOs calculated from a model describing a system approximately. Assume that the adaptation method is applied to these UPOs. In general we expect that the model now can be improved by fitting it to the new, corrected UPOs. For the pendulum we have made several attempts introducing terms of higher order in sin#, 6 etc.. We have used the downhill simplex method [Press, 1992] to fit the new extended pendulum equations to the UPO data but only with poor results. Each of the fitted models is able to follow closely one or another real UPO but no one fits closely to all UPOs. The adaptive orbit correction has also been applied successfully to another real world experiment, the driven bronze ribbon [Dressier, 1995]. The adaptation scheme suggested here provides an interesting opportunity for all applications of chaos control in science and technology, where the control parameters are determined only with finite precision or where the system parameters are slowly drifting. The method is able to track UPOs in a changing environment, when the system is controlled by the OGY method as well as by the local control.

Acknowledgments I want to thank W. Martienssen and B. Hiibinger for years of productive cooperation and U. Dressier, S. Berndt, U. Linketscher, T. Ritz, and A. Schenck for always stimulating discussions. S. Thomae and S. Grofimann have given important contributions to the problem of predictability. The work on predictability has been supported by the Deutsche Forschungsgemeinschaft via the "Sonderforschungsbereich 185, Nichtlineare Dynamik". The work on control has been supported by the Verein Deutscher Ingenieure, VDI.

References

641

References [Abarbanel, 1990] Abarbanel, H. D. I., Brown, R., and Kadtke, J. B. (1990), Phys. Rev. {41A}, 1782. [Abarbanel, 1991] Abarbanel, H. D. I. , Brown, R., and Kennel, M. B. (1991), J. Nonlinear Sc. {1}, 175. [Artuso, 1990a] Artuso, R., Aurell, E., and Cvitanovic, P. (1990), Nonlinearity {3}, 361. [Artuso, 1990b] Artuso, R., Aurell, E., and Cvitanovic, P. (1990), Nonlinearity {3}, 325. [Auerbach, 1987] Auerbach, D., Cvitanovic, P., Eckmann, J.-P., Gunaratne, G., and Procaccia, I. (1987), Phys. Rev. Lett. {58}, 2387. [Auerbach, 1988] Auerbach, D., O'Shaughnessy, B., and Procaccia, I. (1988), Phys. Rev. {37A}, 2234. [Briggs, 1990] Briggs, K. (1990), Phys. Lett. {151 A}, 27. [Brown, 1991] Brown, R., Bryant, P., and Abarbanel, H. D. I. (1991), Phys. Rev. {43A}, 2787. [Bryant, 1990] Bryant, P., Brown, R., and Abarbanel, H. D. I. (1990), Phys. Rev. Lett. {65}, 1523. [Buzug, 1992a] Buzug, T. and Pfister, G. (1992), Phys. Rev. {45A}, 7073. [Buzug, 1992b] Buzug, T. and Pfister, G. (1992), Physica {58D}, 127. [Cremers, 1987] Cremers, J. and Hiibler, A. (1987), Z. Naturforsch. {42a}, 797. [Cvitanovic, 1988] Cvitanovic, P. (1988), Phys. Rev. {38A}, 1503. [Doerner, 1994] Doerner, R., Hiibinger, B., and Martienssen, W. (1994), Phys. Rev. {50E}, R12. [Doerner, 1995] Doerner, R., Hiibinger, B., and Martienssen, W. (1995), Int. J. Bifurc. and Chaos {5}, 1175. [Doerner, 1996] Doerner, R., Hiibinger, B,, Martienssen, W., Grossmann, S., and Thomae, S. (1996), to be published. [Dressier, 1992] Dressier, U., and Nitsche, G. (1992), Phys. Rev. Lett. {68}, 1. [Dressier, 1995] Dressier, U., Ritz, T., Schenck zu Schweinsberg, A. , Doerner, R., Hiibinger, B., and Martienssen, W. (1995), Phys. Rev. {51E}, 1845. [Eckmann, 1985] Eckmann, J.-P. and Ruelle, D. (1985), Rev. Mod. Phys. {57}, 617.

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[Eckmann, 1986] Eckmann, J.-P., Kamphorst, S. O., Ruelle, D., and Ciliberto, S. (1986), Phys. Rev. {34A}, 4971. [Pranceschini, 1993] Franceschini, V., Giberti, C, and Zheng, Z. (1993), Nonlinearity {6}, 251. [Fujisaka, 1983] Fujisaka, H. (1983), Prog. Theor. Phys. {70}, 1264. [Gao, 1993] Gao, J. and Zheng, Z. (1993), Phys. Lett. {181A}, 153. [Gao, 1994] Gao, J. and Zheng, Z. (1994), Europhys. Lett. {25}, 485. [Grassberger, 1988] Grassberger, P., Badii, R., and Politi, A. (1988), J. Stat. Phys. {51}, 135. [Grassberger, 1989] Grassberger, P. (1989), Phys. Scr. {40}, 346. [Guckenheimer, 1983] Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag. [Gunaratne, 1987] Gunaratne, G. H. and Procaccia, I. (1987), Phys. Rev. Lett. {59}, 1377. [Haken, 1983] Haken, H. (1983), Phys. Lett. {94A}, 71. [Haubs, 1985] Haubs, G. and Haken, H. (1985), Zeitschr. Phys. {59B}, 459. [Heng, 1994] Heng, H., Doerner, R., Hiibinger, B., and Martienssen, W. (1994), Int. J. Bifurc. and Chaos {4}, 751. [Huebinger, 1993] Hiibinger, B., Doerner, R., and Martienssen, W. (1993), Z. Phys. {90B}, 103. [Huebinger, 1994] Hiibinger, B., Doerner, R., Martienssen, W., Herdering, M., Pitka, R., and Dressier, U. (1994), Phys. Rev. {50E}, 932. [Huebler, 1987] Hiibler, A., Reiser, G., and Luescher, E. (1987), Helv. Phys. Acta {60}, 226. [Huebler, 1989a] Hiibler, A. (1989), Helv. Phys. Acta {62}, 343. [Huebler, 1989b] Hiibler, A. and Luescher, E. (1989), Naturwissenschaften {76}, 67. [Jackson, 1985a] Jackson, E. A. (1985), Phys. Scr. {32}, 469. [Jackson, 1985b] Jackson, E. A. (1985), Phys. Scr. {32}, 476. [Kember, 1993] Kember, G. and Fowler, A. C. (1993), Phys. Lett. {179A}, 72. [Liebert, 1989] Liebert, W. and Schuster, H. (1989), Phys. Lett. {142A}, 107.

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[Lorenz, 1963] Lorenz, E. N. (1963), J. Atmos. Sci. {20}, 130. [Martienssen, 1995] Martienssen, W., Hiibinger, B., and Doerner, R. (1995), Ann. Physik {4}, 35. [Nicolis, 1983] Nicolis, J. S., Meyer-Kress, G., and Haubs, G. (1983), Z. Naturforsch. {38a}, 1157. [Ott, 1990] Ott, E., Grebogi, C , and Yorke, J. A. (1990), Phys. Rev. Lett. {64}, 1196. [Pawelzik, 1991] Pawelzik, K. and Schuster, H. (1991), Phys. Rev. {43A}, 1808. [Press, 1992] Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992), Numerical Recipes in C, Cambridge University Press. [Sano, 1985] Sano, M. and Sawada, Y. (1985), Phys. Rev. Lett. {55}, 1082. [Schenck, 1996] Schenck zu Schweinsberg, A., Ritz, T., Dressier, U., Hiibinger, B., Doerner, R., and Martienssen, W. (1996), to be published in Phys. Rev. E. [Schwartz, 1992] Schwartz, I. B. and Triandaf, I. (1992), Phys. Rev. {46A}, 7439. [Sparrow, 1982] Sparrow, C. (1982), The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag. [Takens, 1981] Takens, F. (1981), Lect. Notes Math. {898}, 366. [Wolf, 1984] Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A. (1984), Physica {16D}, 285.

24 Experimental Control of Highly Unstable Systems Using Time Delay Coordinates U. Dressier and A. Schenck zu Schweinsberg Daimler-Benz AG, Research Institute Frankfurt, Goldsteinstrafie 235, D-60528 Frankfurt, Germany

24.1 Introduction Since the late 1980th controlling chaotic systems using methods of nonlinear dynamics has become a very popular area of research in the nonlinear dynamics community. The problem of controlling chaos was first addressed by Liischer and Hiibler who proposed an open-loop control to force a chaotic system to a desired goal dynamics by adding a specially designed continuous driving force to the system [Hiibler, 1989]. To calculate this generally aperiodic force in advance a global model of the system has to be available or must be constructed. Another approach is the feedback control of Ott, Grebogi, and Yorke (OGY) which we address in this paper. Ott, Grebogi, and Yorke proposed 1990 to stabilize unstable periodic orbits (UPOs) embedded in a chaotic attractor by tiny time-dependent parameter perturbations [Ott, 1990a]. This work has triggered immense research activities to apply feedback control to chaotic systems (see [Shinbrot, 1993, Shinbrot, 1995, Chen, 1993] and references therein). One reason for the attractiveness of the OGY idea is that it is in principle possible to obtain all control values of the feedback loop from a careful analysis of a scalar measurement series. No global model is required, solely the local dynamics in the vicinity of the unstable orbit has to be extracted from the measurement data. With respect to the applicability of the OGY-control approach to real-world experiments there have been various variations and extensions, e.g. the tracking approach in [Schwartz, 1992] to cope with slowly varying parameters or the simple OPF feedback control [Hunt, 1991] to control very fast experimental systems. In this paper we want to address and combine the two following modifications of the OGY-control. First, in the original OGY-control approach the control frequency was limited to the frequency of the piercings of the continuous trajectory through the Poincare section of the UPO, as OGY reduced the stabilization of a continuous UPO to the stabilization of the corresponding UPO of the Poincare mapping. For driven systems, which we consider in this paper, where the Poincare

646

24 Experimental control of highly unstable systems using time delay coordinates

section is naturally taken as section of constant phase of the driving, the maximal control frequency is thus the driving frequency. If now the instability of the UPO is very high then the amplification of measurement noise can spoil the feedback control when controlling only once per driving period. Therefore, for experiments with large instabilities of the UPO quasicontinuous extensions of the original OGYcontrol have been introduced in [Reyl, 1993] with the minimal expected deviation method (MED) and in [Hiibinger, 1993, Hiibinger, 1994] with the local control method (LC). The, in principle, arbitrarily high control frequency is obtained by taking N equally spaced Poincare sections E n per period T of the driving as control stations. To achieve stability of the UPO a quasicontinuous OGY-control thus has to work with the linearizations of the mapping p(n>n+1) = <j>*£ ? At = T/N, which maps a state from the Poincare section E n to the next section S n +i. Now in the original OGY-method the control requirement is based on the eigendirections of the linearized Poincare mapping. For the quasicontinuous control methods the control requirement has to be changed as the eigenvalues of the linearized mapping p(n,n+i) c a n h a v e complex eigenvalues. Therefore the MED and LC method use control requirements which are suited for complex eigenvalues too. The second modification which we address comes in when one uses time delay coordinates [Packard, 1980, Takens, 1981] for the reconstruction of the attractor. In this case the OGY-feedback formula has to be modified in the way that also preceding values of the control parameter have to be considered to obtain the actual value of the control parameter [Dressier, 1992]. The reason for this modification is that for delay coordinates the Poincare mapping does not only depend on the actual control parameter but also on all preceding ones which were changed during the time window rw = (d — l)r of the delay vector (x(t),x(t — r ) , . . . ,x(t — (d — l)r)). For the original OGY-method, controlling once per driving period and a control requirment based upon the unstable eigendirection of the UPO, this has been studied in numerical simulations in [Dressier, 1992, Nitsche, 1992, Romeiras, 1992, So, 1995] and in a magnetoelastic beam experiment in [Ding, 1996]. The main part now of this paper is devoted to the combination of the just presented modifications of OGY, i.e. to the quasicontinuous control using time delay coordinates, and to its application in a mechanical experiment - a periodically driven bronze ribbon - where the control vectors have to be extracted solely from the analysis of a single scalar measurement signal. For a quasicontinuous control, with a control frequency being the sampling frequency ££, the mapping p(n>n+l) = ^ w[\\ depend on the last w = £(d - 1) parameter changes with £ being the time delay in units of the sampling time and d the embedding dimension. Thus we have to deal with a dependence on the control parameter which can be expressed as z n + 1 = p(n>n+!) (z n ,p n ~™,... ,pn~l,pn). Now, to obtain a control formula one could use the extended state space approach introducing as extended state vector y n = (zn,pn~w,... ,p n ~ 1 ) as has been done in [Romeiras, 1992, So, 1995, Ding, 1996] for the original OGY-method. The attractiveness of the extended state space approach for time delay coordinates is that at

24.1 Introduction

647

first glance one can use the same control requirement as for the physical state space without a formal change. For the OGY-approach using eigenvalues and eigendirections of the UPO as done in [Romeiras, 1992, So, 1995, Ding, 1996] this also works in practise as the eigenvalues and eigendirections in the delay space and the extended space are related in a very simple manner [So, 1995, Ding, 1996]. However, for the quasicontinuous control methods considered in this paper, i.e. MED or LC, it turns out that the use of the extended state space formulation leads to a scaling dependence on the units of the control parameter. Therefore, for the quasicontinuous control methods considered here we prefer another possibility to derive a control formula. We require stabilization not for z n + 1 but only for zn~*~w+1 with Spn ^ 0 and Spn+1 = ... = 5pn+w = 0. This control requirement coincides with the second modification of the original OGY-method in [Dressier, 1992] given for time delay coordinates. While for the original OGY-method these two approaches, i.e the extended state space and the modified control requirement, are equivalent this is no longer the case for the quasicontinuous control approach. In our paper we discuss this point. Finally, we compare the performance of the two quasicontinuous control approaches using time delay coordinates in a bronze ribbon experiment. For the control experiments, all control vectors needed for control are extracted from the analysis of the measurement signal. How this is done will be explained and discussed in detail. Furthermore we extend the adaptive orbit correction [Doerner, 1995] to time delay coordinates to correct the position of the UPO used in the feedback formula during control. In the experiment the modified control requirement will show best result while the extended state space approach is not suitable for the quasicontinuous MED and LC method. Before we start we want to mention that the control experiment presented in this paper is not the first realization of a quasicontinuous control using time delay coordinates. Reyl et al. apply the MED method in an NMR-laser experiment using time delay coordinates [Reyl, 1993]. However they do not consider the dependence on preceding values of the control parameter. De Korte et al use the local control method and time delay coordinates to control an experimental driven pendulum [de Korte, 1995]. In their experiment they only consider the dependence of one preceding parameter value as they use only 4 control stations per period of the driving and a time delay window (d — l)r which is smaller than T/4. Furthermore in their control experiment they apply the original control requirement of the local control, i.e. stabilization in the next section, without modifying the control requirement as we do. Our paper is now organized as follows. In section 24.2 we give a short reminder on the original formulation of the OGY-control method. In section 24.3 we briefly recall the two extensions of the OGY-method which we address in this paper, the quasicontinuous control for highly unstable systems and the OGY-method for time delay coordinates. In section 24.4 we formulate the quasicontinuous control for time delay coordinates. In section 24.5 we then describe the experimental setup of the bronze ribbon experiment which is used to test the performance of the quasi-

648

24 Experimental control of highly unstable systems using time delay coordinates

continuous control using time delay coordinates in an experiment where the control vectors have to be extracted solely from the analysis of a scalar measurement signal. How this is done in general and exemplarily in this experiment is described in detail in section 24.6. Finally in section 24.7 we apply the quasicontinuous control using time delay coordinates to the bronze ribbon and report some results on tracking the unstable periodic orbits of this experiment. Wefinishwith a summary and some conclusions.

24.2 The OGY control scheme In their famous control paper Ott, Grebogi, and Yorke present their control method for two-dimensional Poincare sections and one unstable eigendirection [Ott, 1990a]. In this section we follow this description. Extensions to higher dimensional systems and more than one unstable eigendirection may be found in [Romeiras, 1992, Ott, 1990b, So, 1995, Ding, 1996, Warncke, 1994]. To stabilize an unstable periodic orbit lying in a chaotic attractor for p = p0 Ott et al. propose to monitor the system in a Poincare section E. Whenever the intersection z n of the orbit with the Poincare section gets close enough to the UPO zp(po) = P( Z F(PO)JPO) € S the control parameter p is adjusted to a new value pn such that the next intersection z n +i, given by z n +i = P(z n ,p n ), falls on the local stable manifold of zp(po)- The parameter perturbation Spn = pn— Po is restricted to a maximal possible perturbation Spm3iX. To calculate the necessary parameter perturbation Spn one uses the linear approximation of the Poincare map P, near zF(po) and po 8zn+l = A(Po) • Szn + bSpn,

Szn e R2

(24.1)

with Szn = zn - zF(po), Spn = pn- po, and A(p0) = DzP(zF(po),Po)- The vector b = ^ £ ( Z F , P O ) measures the sensitivity of the system to parameter perturbations. An alternative way to describe the effect of the parameter perturbation Sp is to look at the resulting shift of the fixed point ZF(PO

+ Sp) - zF(p0) = gSp

(24.2)

Using this notation and linearizing P around z F (p n ) the dynamics near can alternatively be described as: <Szn+i = g6pn + A(pn) • (Szn - gSpn)

(24.3)

With A(pn) = D Z P(z F (Pn),Pn). Denoting with fu the contravariant unstable eigenvector of A(po) (i.e. fu es = 0 and e s being the stable eigendirection) and with Au the unstable eigenvalue the control requirement, (zn+i shall fall on the stable direction), can be expressed as

24-3 Extensions of the OGY-control method

649

fu • (Jz n+ i = 0. Together with (24.1) or (24.3) this renders the equivalent control formulas *Pn = - ^ i - f u - * z n t •b

(24.4)

or $Pn = -7T 777 fu * Szn (24.5) (Au - 1) fu • g assuming A(p0) « A(pn). With this assumption the vectors g and b are related by b = (11 - A) g.

(24.6)

Note that with Eq. (24.1) the dynamics of the system in the vicinity of a periodic orbit is described by a linear mapping. Being at this point of the analysis the pole placement technique which is well known to the control engineering community can also be used to calculate a feedback control formula [Romeiras, 1992]. If one uses as eigenvalues of the controlled problem in the pole placement formulation //1 = As and /i2 = 0 one obtaines the OGY-control formula (24.4).

24.3 Extensions of the OGY-control method 24.3.1 Quasicontinuous control for highly unstable systems The quasicontinuous control concept was introduced to cope with experimental systems of high instability [Hiibinger, 1993, Hiibinger, 1994]. With highly unstable we refer to systems whose UPOs have either large unstable eigenvalues (i.e. of the order of 10) or large effective Lyapunov exponents \eff(zp,T). The effective Lyapunov exponent Aeff(zp,T) (for the return time T) of an UPO zp in one Poincare section is related to the largest singular value (i.e the maximal possible stretching rate) /i(zp,T) of the linearization Dz F P = DzFT around zp of the Poincare map P. This relation is given by [Grassberger, 1988, Doerner, 1991] Aeff(zF,r):=|logMzF,r).

(24.7)

In contrast to the eigenvalues of an UPO its singular values and thus its effective Lyapunov exponents does depend on the chosen Poincare section. As the effective Lyapunov exponents affect the predictability of a future state [Doerner, 1991] the large effective Lyapunov exponents of the UPOs found in our experiment have severe consequences for the determination of the control values as well as for the performance of the feedback control itself. Concerning the control performance of the feedback control it was already pointed out in [Hiibinger, 1993] that the amplification of noise by large effective Lyapunov exponents can spoil the feedback control when the control parameter is adjusted only once per return time T of the Poincare map. The reason for this is that an error e in determining the actual state near a fixed point zp in the

650

24 Experimental control of highly unstable systems using time delay coordinates

Figure 24.1 Illustration of the Poincare map P^ and the flow mapping p( n ' n + 1 ) for driven systems. The flow mapping p( n - n+1 ) describes the evolution of the system from one Poincare section £ n to the following £ n +i (zn »-> z n + 1 ), while the Poincare map P ^ describes successive intersections of a trajectory rn+N with the Poincare section E n (z71 ) •

Poincare section can be amplified in the worst case during one return time T by = exp(Aeff(zF,X')T). If the amplified measurement error exceeds the size of OGY's control parallelogram [Ott, 1990a] which is determined by the maximal allowed parameter perturbation, the feedback control fails. Thus for systems with large effective Lyapunov exponents one cannot wait one period to adjust the control parameter. In order to raise the control frequency and to adjust the control parameter say N times per period T of the driving a quasicontinuous control introduces N successive Poincare sections S n . For driven systems with driving period T, where the Poincare sections are naturally taken as section of constant phase of the driving, this leads to an adjustment of the control parameter every At = T/N. Let pn = po + Spn be the value of the control parameter p when the system goes from section E n to S n +i. Thus the flow mapping p( n ' n + 1 ) — 0^* which maps a state z n in E n to a state z n + 1 in £ n +i can be expressed as /X(ZF,T)

z n+l

_ p(n,n+l)

(24.8)

Fig. 24.1 gives an illustration of the action of the Poincare map P used in the OGY control method and the flow mapping p(n»n+!) given by Eq. (24.8) in the case of a driven system with two-dimensional Poincare sections. The starting point of a quasicontinuous OGY-control is now the linearization of the flow mapping p(n'n+*) around the UPO z£ and p 0 , 8zn+l = An • Szn n

n n

1

bn6pn

(24.9) n

(n

with A = D Z nP( ' + )( z p,p 0 ) and b = ^ r P Now already in two dimensions the linearization An of the mapping P n can have complex eigenvalues. Therefore, OGY's control condition that the next state

24-3 Extensions of the OGY-control method

651

Figure 24.2 Idea of the local control method. At time step n the parameter perturbation Spn is chosen such that the difference vector 6zn is mapped by An onto the stable direction v£+1 which will maximally shrink under the action of A n+1 .

z n + 1 falls on the stable eigendirection of An, which is formalized as f™ • Jz n + 1 = 0 (see Sec. 24.2), cannot be used anymore. The LC method and MED method solve this problem by posing a control requirement which does not depend on the eigendirections of An. In the local control method Hiibinger et al. use the singular value decomposition (SVD) of An in order to formulate a control condition. Let An - Un • Wn • Vn f denote the singular value decomposition of An with the orthogonal matrices Un and Vn having as column vectors uj1 and v? and the diagonal matrix Wn with positive entries wf1, the singular values. Having calculated the SVD the action of An can be described as An • vj1 = wfuf. Thus the orthonormal basis {vf} is mapped on the orthonormal basis {uf} with an additional stretching or shrinking by the singular values w™. Having the singular values ordered by size v^ is the unstable direction of An (here we assume that only one unstable direction exists). In the local control method one requires that the projection of Szn+l on the unstable direction vj1"1"1 is diminished. Thus, for perfect stabilization the control requirement of the local control could be written as n+l t

(24.10)

This idea is schematically illustrated in Fig. 24.2 for a two-dimensional Poincare section. In the minimal expected deviation method Reyl et al use another control requirement. They require that the distance ||(Jzn+1|| to the UPO in £ n +i is minimized. In order to find this minimum the derivative of ||(Jzn+1||2 with respect to 8pn is calculated and its zeros have to be determined. This calculation leads to the condition b n t -(5z n + 1 =

(24.11)

652

24 Experimental

control of highly unstable systems using time delay coordinates

Thus, also the control requirement of MED can be expressed in the way that the projection of <5zn+1 on a special direction vanishes. If one compares (24.10) and (24.11) the control requirement of the LC method and the MED method only differs with respect to the special choice of this direction. To proceed with the derivation of an explicit control formula let us call this direction for a moment h n . As was said the most pretentious control requirement would be h n * • $z n+1 = 0. In the paper of Reyl et al. this maximal control requirement is used for N = 4 control stations per period. But if the number N of the control stations increases the time of the control action At = T/N decreases. If the maximal control requirement were retained the small control time At would lead to a large control signal Spn. In order to avoid large control signals for high control frequencies Hiibinger et al. propose to weaken the control requirement. In the LC method they introduce a decay factor (1 — p) in the control requirement [Hiibinger, 1993]. We follow this idea but formulate the reduced control condition in a slightly different way which leads to better control results. We require that the application of the control signal Spn diminishes the projection of Jz n + 1 on h n by a factor of (1 — p) compared to the one which Szg^=0 would have if no control were applied, i.e. nt

=

( 1

_

p )

hn

t

.

Inserting Eq. (24.9) in (24.12) gives the explicit control formula for the quasicontinuous control

Spn = p

- ^r^-Szn

(2413)

with h n = v" + 1 for the local control (LC) and h n = b n for the minimal expected deviation method (MED). With Eq. (24.13) we have the general feedback formula for quasicontinuous control in physcial state space if the linearization of p( n ' n +!) are given by (24.9). Note that in the derivation of the control formula (24.13) it has never been used that the successive points Zp are intersections of an UPO with E n . If one does not count n as n modulo N the quasicontinuous control method allows to follow every aperiodic orbit.

24.3.2 The OGY-control method for time delay coordinates The other modification of the OGY control approach which we want to apply in this paper comes in when time delay coordinates [Packard, 1980, Takens, 1981] z n = (x(tn),x(tn — T), ... ,x(tn - (d - I)?"))1" are used for the reconstruction of the attractor. Here x is an accessible measurement signal which is related to the physical state z of the system by some scalar measurement function h, i.e. x = h{z). r is the time delay and d the embedding dimension. In this case for active OGY-control the Poincare map P depends not only on the actual value pn of the control parameter but also on all preceding ones which were active during

24-3 Extensions of the OGY-control method

653

the time window [tn - (d - l)r, tn] of the delay vector z n [Dressier, 1992]. This is expressed as z n + i = P ( z n , p n _ r , . . . ,p n -i,Pn)- We demonstrate this dependence in the context of quasicontinuous control in Sec. 24.4.1. For the original OGY-method with a control frequency 1/T usually only one preceding parameter value have to be considered because the lag window (d - l)r is in most cases smaller than the driving period T. To derive a feedback formula for this case, i.e. z n + i = P (z n ,p n ,p n _i), one can start with the linearization of this equation, Jz n +i = A - Szn + b 1 Spn-i + b°Spn. Now to obtain a stable control formula the OGY control requirement fu«<Szn+i = 0 is modified in [Dressier, 1992, Nitsche, 1992] in the way that that the system stabilizes not in the next but only in two control steps, i.e. fu • 5z n+2 = 0, for an appropriately chosen parameter perturbation Spn ^ 0 and <5pn+i = 0. This yields a control formula that can be written as Spn = Kx • Szn + K2 6pn-!

(24.14)

with \2 f

\ f

K1

Kl

and K = -A u f u -b° + fu.bl > = -A u f u -bo + f u -b 1 - ( 2 4 1 5 ) Another possibility to derive a feedback formula for time delay coordinates is to use the extended state space approach which was first proposed in [Romeiras, 1992] together with a pole placement technique and lateron worked out and realized in numerical experiments in [So, 1995]. Ding et al. report successful control of a magnetoelastic beam experiment using time delay coordinates and an extended state space approach [Ding, 1996]. In addition to the use of the extended state space approach for time delay coordinates the work in [Romeiras, 1992, So, 1995, Ding, 1996] also generalizes the OGY-control requirement in the way that also systems with more than one unstable eigendirections can be controlled with a single control parameter. While in [Romeiras, 1992, So, 1995, Ding, 1996] the extended state space approach for OGY is formulated for general parameter dependences, i.e. z n +i = P(z n ,Pn-r, - • • 5Pn-i?Pn)5 we give here only the extended state space formulation for r = 1 and the case of one unstable eigendirection. In section 24.4.2 we give for general r the formulation of the extended state space approach in the context of quasicontinuous control. In the extended state space approach one includes the preceding relevant values of the control parameter in the system's state to form an extended state vector y n = (z n ,p n _i). The linearized dynamics can then be rewritten as

Syn+l = C • 6yn + d6Pn

(24.16)

with C =

=(?)•

654

24 Experimental control of highly unstable systems using time delay coordinates

The advantage of the extended state approach is that also for time delay coordinates one could use the old control formalism since the linearized dynamics have the same form as in the original OGY-method (compare Eqs. (24.1) and (24.16)). For the original OGY-method with one unstable eigendirection it can be demonstrated that the resulting control formula 6pn = K • 6yn is equal to (24.14) with K = (Ki, K2) if one uses in the extended state space the OGY-control requirement (i.e. that the next state y n +i falls on the stable eigendirection or its equivalent in the pole placement formulation).

24.4 Quasicontinuous control using time delay coordinates 24.4.1 Local dynamics in the time delay embedding space Before we start to give the control formulas for quasicontinuous control with time delay coordinates we want to give in this subsection a derivation of the parameter dependence of the mapping p( n ' n + x ) — ^t m ^e time delay space when control is applied every At = T/N where At is the sampling rate of the measurement process. To start with our argumentation let us denote with E n the N Poincare sections of constant phase of the driving in the physical state space. Note that we use in this section a " for the quantities referring to the dynamics in the physical state space in order to distinguish them from the corresponding ones in the time delay embedding space. Thus we write z n + 1 =p( n ' n + 1 )(z n ,p n ) for the flow mapping (24.8) in the physical state space which maps a state from E n to S n +i or more generally

for the flow mapping from E n to En+A:Now for the reconstruction of the attractor of our periodically driven system we want to use time delay coordinates. For a driven system it is appropriate to retain as additional coordinate the phase of the driving. Doing so we can introduce N successive Poincare sections E n , n — 1,..., JV in the embedding space which exactly correspond to the sections E n in the orginal state space. For an embedding dimension of d+1 a point z n G E n is then given by z n = (x n , xn~l,..., xn~^d~1^1) , where xn — x(to +nAi) is an accessible measurement signal and r = £At an appropriately chosen time delay. This measurement process can be mathematically expressed as a scalar function h on the physical state space, i.e. xn = h(zn). For shortness of our formulas we assume here that the measurement does not depend on the phase of the driving. According to Takens' theorem [Takens, 1981, Sauer, 1991], for an appropriately chosen time delay r and a sufficiently high dimension d there exists

24-4 Quasicontinuous control using time delay coordinates

655

a smooth, invertible embedding function $ n , i.e. z n = n (z n ), which maps points z n from the Poincare section E n to points z n in the time delay Poincare section E n . This function which is given by z n = $ n (z n ) = (/i(zn), /i(z n ~*),..., h{zn~^d~1)£))^ is closely related to the dynamics of the system and thus dependent on preceding parameter values when the control is applied. To write the states z n ~^, j — 1,..., (d - 1), as a function of z n the inverse p( n ~^' n ) of the mapping p(n-j7,n) h a v e t 0 b e u s e d Taking into account, that p( n -#» n ) depends on the parameter values pn~j£,... ,pn~l the embedding function $ n can be written as a n function of z as h(zn) \ (n-2£,n) '

",..., pn~l)

nn-2£

(24.17)

with w =

(d-l)t

(24.18)

Thus for quasicontinuous control the embedding function $ n which maps S n in the physical state space to the corresponding Poincare section E n in the time delay embedding space does not only depends on z n but also on the last w = (d - 1)1 parameters. For abbreviation we write z n = 4>n (zn,pn~1) with p n - 1 = (pn ,pn~l) and for the unperturbed dynamics z n = $ n (z n ,po) with po = This dependence of <£n on the preceding control parameters has direct impact on the flow mapping p( n ' n + 1 ) in the time delay embedding space which maps z n to z n + 1 . Starting with z n + 1 = $ n +i (z n + 1 ,p n ), z n + 1 = p( n ' n + 1 )(z n ,p n ), and z n = $~ 1 (z n ,p n ~ 1 ) one can express z n + i as a function of z n as zn+1 -

,pn) .

(24.19)

But this is the flow mapping p(n>n+x) which as a consequence depends on the w = (d - 1)£ preceding values of the control parameter and the actual value p n , i.e. we can write n n+1 )( Z n ,p n ~ 1 ,p n ). (24.20) z n+l zz: p( ' This finally leads directly to the fundamental equation for a quasicontinuous OGYcontrol for time delay coordinates, which is the linearization of p( n ' n + 1 ) around the UPO Zp and the control parameter po = An • 6zn

(24.21) 2=0

with 6zn = zn - zg, An = D Z nP( n ' n+1 ) (zg,po, • • • ,Po), and M ^ ( + 1 )

656

24 Experimental control of highly unstable systems using time delay coordinates

In [Schenck, 1996] this result is supported by giving an explicit relation between the linearizations An = D5 n P ( n > n + 1 ) and b n = ^ - P ( n ' n + 1 ) in the physical state space and the linearizations An and b n>l in the time delay embedding space.

24.4.2 Quasicontinuous control formula for time delay coordinates In this subsection we proceed with the quasicontinuous control in the time delay embedding space when the starting point of the derivation of the feedback formula is the linearization (24.21). We describe two different ways to obtain control formulas when the linearized dynamics in the time delay embedding space is given in the above form. These are the extended state space approach and a modification of the control requirement. While for the original OGY-method with one unstable eigendirection both approaches yield the same control formula (see Sec. 24.3.2) for the quasicontinuous control methods considered here the resulting control formulas differ substantially as we will show. The extended state-space approach We start with the extended state space approach [Romeiras, 1992, So, 1995, Ding, 1996]. An extended state is given by y n = (z n ,p n - 1 ) . Thus with the extended states the preceding parameter dependences p n l — (pn w,... ,p n *) are considered as state variables. Doing so, the linearized dynamics (24.21) can be written as = Cn • Syn + dnSpn

(24.22)

with

f n

C =

b"^-1

An

0

1

... 0 (24.23)

0 0

1 0

and d n = (b n '°,0,...,0,l) + .

(24.24)

Note that Eq. (24.22) has the same form as the linear approximation (24.9) in the physical state space. Therefore the control requirement can be chosen in complete analogy to (24.12). This leads to the control formula t

(24.25) h • dn with h n = v^ + 1 the singular vector corresponding to the largest singular value of C n + 1 for the LC and h n = d n for the MED method. For the local control method n f

(.4 Quasicontinuous control using time delay coordinates

657

we assume that the largest singular value w™+l of C n + 1 is the only singular value which is larger than 1. Prom a formal point of view the problem of quasicontinuous control with time delay coordinates could be regarded as solved. But as was already mentioned in the introduction the at first glance simple and convincing use of the extended states leads for the quasicontinuous control methods considered here to an undesirable dependence on the units of the control parameter p. We demonstrate this by showing that the control formula (24.25) depends on the scaling of the parameter p. To do this we replace all parameters pn by pn jo and all vectors b n 2 by crb n '\ Taking the structure of y n = (z^p72"1)1" and C n , d n in Eq. (24.23), (24.24) into account we obtain in the case of the MED method (h n = d n ) the following expression for Eq. (24.25):

b y " - t A ' n- y f + n g - "-'fr"-'i. 2 b '° • b '° + cr-2

(24.26)

Due to the appearance of a~2 in the denominator this control formula is not invariant with respect to the scaling factor a. A similar result is obtained for the LC method in which the scaling dependence is caused by the singular value decomposition of Cn. The deeper reason for this is that although the equation (24.22) for the linearized dynamics in the extended state space is invariant under a scaling of the control parameter the control requirements (Eqs. (24.10), (24.11) or (24.12)) of the LC and MED method are not invariant under a rescaling of p if they are formulated in the extended state space. For the MED method this can directly be seen inserting d n and 6yn+l in the control requirement (24.11) or (24.12). For the LC method one has to have in mind that the control requirement (24.10) is based on the singular value decomposition (SVD) of C n + 1 . But a rescaling of p can be viewed as a coordinate transformation and, as is well-known, the SVD is not invariant under a coordinate transformation. Thus for both quasicontinuous methods the use of an extended state space shows an undesirable dependence on the units of the control parameter. The control experiments in Sec. 24.7.1 will show that this dependence indeed causes a breakdown of the control for certain choices of the scaling parameter a. In contrast to this the extended state approach in [Romeiras, 1992, So, 1995, Ding, 1996] is combined with a control requirement based upon eigenvalues of the controlled problem [Romeiras, 1992] or the eigendirections of the UPO [So, 1995, Ding, 1996]. As the eigenvalues are invariant under a transformation of coordinates no problem arises with the pole placement technique in [Romeiras, 1992]. The use of the eigendirections of the UPO in [So, 1995, Ding, 1996] for the control requirement does not lead to the above discussed problem either. This is so because the eigendirections of the linearized Poincare mapping A and the associated matrix C in the extended state space obey a very simple relationship [So, 1995, Ding, 1996] which is preserved under a coordinate transformation. If we denote with e" and ef the unstable and stable eigendirections of the d x d-matrix A then the unstable eigendirections of the (d+w) x (d+w)-matrix C are given by (eV, 0,..., 0)* G JRd+w

658

24 Experimental control of highly unstable systems using time delay coordinates

and the stable directions by (e^,0,... ,0)* £ Hd+W and an arbitrarily chosen wdimensional basis of the null-space of Cw [Ding, 1996]. This relationship between the eigendirections of A and C is not affected by a change of scales of the control parameter p. Therefore the use of the extended state space works fine for the control requirements used in [Romeiras, 1992, So, 1995, Ding, 1996]. The modified control requirement An alternative way to derive a feedback formula starting from the linearization (24.21) is to modify the control requirement. For the original OGY-method with time delay coordinates this is described in section 24.3.2 for the special case that only one preceding parameter dependence appears in the linearization. The control requirement was modified in the way that the system had to stabilize only in two control steps, i.e. for z n + 2 , for an appropriately chosen parameter perturbation Spn ^ 0 and 8pn+l — 0. We generalize this modified control requirement for general w and the linearization (24.21) of the quasicontinuous control. Because the parameter value pn influences the trajectory z n + 1 , z n + 2 , . . . till zn+w, we require that the system stabilizes only after (w + 1) time steps for an appropriately chosen parameter perturbation Spn ^ 0 without further control interventions in between, i.e. the modified control requirement is +l =

(1

fizft^1,

_ p) hn t .

= 8pn+2 = - -. = 8pn+w = 0.

(24.27) (24.28)

The condition (24.27) is chosen in complete analogy to (24.12) with tfzjjj^1 denoting the distance vector to Zp+™+1, which zn+w+l would have, if no control at all (i.e. 5pn = 0) were used. The direction h™ depends on w and is chosen below in such a way that the LC method and the MED method are obtained. To derive an explicit control formula, 8zn+w+1 and Szffi^1 in (24.27) have to be expressed as a function of the actual state Szn. This is done by using the linearizations A% = D z »P< n ' n+u;+1 ) (zg,p 0 ,. •. ,po) and bft* = ^ z r P ( n ' n + l i ; + 1 ) (zp,po5 • • • ,Po) and taking the condition (24.28) into account. One obtains w 6zn+w+l = An

. Szn

+

£ h^Sp"'*.

(24.29)

2=0

The matrices A7^ are given by A^ = An+W • • • An. The vectors bj»* can be recursively calculated using b^'* = b n>i and f \

^\

i

for j > i, for j <%.

Inserting (24.29) in (24.27) yields the desired control formula W t . [ An . Xrjfi I \ p ^

hkn'ifinn~i~\

24-5 The bronze ribbon - Experimental setup

659

For the LC method, the direction h£ is given by h£ = v ^ + u ; + 1 where v^+w+l is the singular vector of An+wJtl which refers to the direction of maximal stretching. For the MED method the direction has to be chosen as h™ = b™'° in order to minimize the deviation of the trajectory from the desired orbit in section £ n+w ,+i. Both possible directions lead to a control formula which does not depend on a scaling factor of the control parameter, in contrast to the control formula (24.25) using extended states. The simple reason for this is that not only the linearized dynamics (24.21) but also the control requirement (24.27) with (24.29) is not affected by a rescaling of p. Note that in the formulation of the modified control requirement (Spn ^ 0 and fipn+i _ _ fipn+w = Q^ a n ( j t n e c o n s e c u tive derivation of the control formula (24.30) the parameter perturbation are in principle calculated till Spn+W. In the experimental realization however where noise is always present we calculate Spn at every control step n once again in order to use the possibility to correct the parameter perturbation using the actual measurements of the system. Now we want to test the performance of the two different control formulas (24.25) and (24.30) in a mechanical experiment. For the control experiments all control vectors needed for the control formulas are extracted from the analysis of a scalar measurement signal. Before the determination of the control vectors are reported we first describe the experimental setup.

24.5 The bronze ribbon - Experimental setup The experiment is stimulated by the magnetoelastic buckled beam experiment described in [Moon, 1987]. In our case we have a horizontally cantilevered elastic bronze beam equipped with two small permanent magnets at its free end. The beam is located in an inhomogeneous magnetic field produced by two bigger permanent magnets. The resulting magnetic force F ~ dB/dr, where r is the position of the dipole at the tip of the ribbon, destabilizes the straight unbent position of the ribbon and creates two stable equilibrium positions of the bronze ribbon. A photograph of the experiment is given in Fig. 24.3. To drive the system two coils are placed around the free end of the beam. When the coils are supplied with an ac voltage U(t) the beam starts to vibrate. For the experiment U(t) is chosen as U(t) = E/Asinu;£ + p. The amplitude is kept fixed and p is selected as control parameter for the feedback control. During the time we worked with this experiment we used amplitudes C/A = 0.6 V and UA = 0.7 V. In our experiment p can vary between (—1,1) V and its value is obtained via a 12-bit resolution digital-analog converter from a 486 PC. In Fig. 24.4 the experimental setup is schematically shown. The dynamics of the beam is recorded using a wire strain gauge which measures the curvature near the base of the ribbon. It gives a voltage signal x which has a monoton relation to the deflection of the tip. The voltage is measured in the range of (-0.5,0.5) V and is transferred to the PC using a 12-bit resolution analog-digital converter. The output voltage is sampled at a frequency of 64/T where T = 2TT/UJ

660

24 Experimental control of highly unstable systems using time delay coordinates

Figure 24.3 Photograph of the experiment.

is the period of the forcing term U(t). This implies that we can use maximally 64 Poincare sections for the quasicontinuous control within our experimental setup. In our experiments the period of the excitation is chosen to be T = 1 s. With this driving the ribbon vibrates chaotically when the control parameter is set to p = —0.2 V which will be our parameter po where we want to control the ribbon. In Fig. 24.5 we present a bifurcation diagram of the experiment. For the control parameter p varying from - 1 V to 1V stroboscopic measurements of xn = x(nT) are plotted. Around p = 0 V a broad chaotic region can be detected. With the sampled measurement signal xn = x(to -f nAt), At = T/N the dynamics of the beam is reconstructed using time delay and differential coordinates. In both cases the phase of the driving 6 = t mod T is chosen as naturel state space coordinate. It facilitates taking the Poincare sections considerably. One only has to set 0 = const. For the time delay embedding we use two coordinates (xn,xn~e) with time lag £ == 5. This choice is suggested by the integral local deformation method [Buzug, 1992] and the local exponential divergence method [Gao, 1993] which we applied in order to determine optimal parameters r and d for the time delay embedding. By using differential coordinates (xn,xn) for the reconstruction we have in mind that a one-mode approximation for a damped beam with a free end should describe the beam's equation of motion to first order. The velocity xn is estimated from the measurement signal by numerical differentiation and scaled with a factor 1/10 such that the state space coordinates x and x varied within the same range. Thus we use as units of time [10""1s]. For 64 Poincare sections

The bronze ribbon - Experimental setup

661

wire strain gauge

N Figure 24.4 Experimental setup of the chaotic bronze ribbon. A horizontally cantilevered bronze beam equipped with two small permanent magnets is located in an inhomogeneous magnetic field. Two coils are placed around the free end of the beam and are supplied with an ac voltage U(t) = UA sin ut+p with UA = 0.6 V and T = ~ = 1 s. The offset voltage p is used as control parameter. Measurements are taken with a wire strain gauge at the fixed end of the beam to obtain a voltage signal x related to the deflection of the beam.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

p[V]

Figure 24.5 Experimental bifurcation diagram of the bronze ribbon. Varying the offset voltage of the coils p as control parameter the stroboscopic measurements xn = x(nT) corresponding to (z n )i in the Poincare section Ei are plotted versus p.

662

24 Experimental Control of Highly Unstable Systems Using Time Delay Coordinates

(b)

0.6 0.4 0.2 -

0 •

0.2

via (imJBSh

-0.2 -

I *e -0.2 -

-0.2 -0.6 -

-0.4 •

'

^

mW

-0.8 -0.6

-0.4

-0.2 x(nT)

0

0.2

0.4

-0.6

-0.4

-0.2

0

0.2

0.4

x(nT)

Figure 24.6 Chaotic attractor for the bronze ribbon for po = —0.2 V. 50 000 points z n are shown in the Poincare section Si using (a) differential coordinates, (b) delay-coordinates with d = 2 and r = 5At.

this results in a time difference At « 0.16 [10 1s] between two successive Poincare sections. In Fig. 24.6 the chaotic attractor of the experiment for p0 = -0.2 V is shown in the Poincare section 6 = 0 for (a) differential coordinates and (b) time delay coordinates. A clear deterministic structure is visible. The correlation dimension L>2 in the Poincare section has been determined to be about 1.75.

24.6 Control vectors from scalar measurements In order to apply the quasicontinuous control methods to the experiment described in the previous section, the control vectors have to be extracted from the scalar measurement data. To do this one has to determine for N Poincare sections all quantities which appear in the linerization (24.9) in the physical state space or (24.21) in the time delay embedding space. These are the UPOs Zp, the linearizations An of the mappings p( n ' n + 1 ) and the dependences on the control parameter, i.e. b n for the physical state space or bn'% i = 1,... ,w in the embedding space. Generally there are two possible approaches to extract the control vectors. One may calculate them from a global mathematical model or utilize local fits around recurrent points in the state space. For the bronze ribbon we did not succeed yet to extract a global model from the data. Therefore the control vectors have to be calculated solely from the dynamical behavior near recurrent points in the Poincare section [Lathrop, 1989, Pawelzik, 1991, Ott, 1990a]. In this paper we restrict our discussion to the second way. The approach of modeling the system and extracting the control values is described in detail in [Hiibler, 1987, Hiibinger, 1994]. To extract the control values in the time delay embedding and in the physical state space approach (differental coordinates) nearly the same methods are used. Differences will be marked whenever appropriate.

24-6 Control vectors from scalar measurements

663

24.6.1 Unstable periodic orbits from recurrent points For the quasicontinuous control the intersection points zg of the UPO with each Poincare section E n , n — 1,..., TV, are needed. Therefore m best recurrent points in every section E n are recorded, i.e. we look for pairs of points (z n *,z n * +iv ), zni,zni+N eXn,rii mod N = n, with ||Zn2 - Z n 2 + 7 V |

min

\\zk -

zk+N\

To find the correct grouping of the recurrent points into classes belonging to the same UPO the classification scheme described in [Nitsche, 1992] is adopted. In one selected Poincare section, the classification section S n c , the best recurrent point z n i G E nc is taken as master point of the first class. The second best recurrent point z n2 £ E nc is classified to be in the same class if its distance to z n i is less than a maximum distance parameter e, if not z712 forms the master point of the second class and so on. Having thus obtained a classification in S n c the best recurrent point of the other sections have to be classified according to the classes in E nc . For this purpose we record not only the recurrent points z n ,z n + i V G E n but also z n ' G S n c , n
24.6.2 Linear dynamics of the unperturbed system To determine the linearization of theflowmappings pC 71 ^ 1 ) for an UPO we do not use the already collected best recurrent points as was proposed in [Nitsche, 1992], we rather make a new run looking for the nearest neighbor points of the UPO {zp}, regardless whether they are best recurrent points or not. This effort turns out to be worthwhile since for systems with large unstable eigenvalue Au and even larger quotient Au/As of the global Poincare mapping P ^ ='#£, the best recurrent points z n are mostly placed along the stable direction e™ of zg (see Fig. 24.8 (a)). Thus considering only the best recurrent points for the fit of the linear mappings we would overweight these directions. Nearest neighbor points in contrast give

664

24 Experimental control of highly unstable systems using time delay coordinates

0.3 -

(a)

^

(b)

.—

(c)

^ - .

I

?r -0.3 -

(

-0.6 0.3 1 5" -0.3 •

-0.6 -

(e)

(d)

a

-0.6 -0.3

0

0.3

x(t) Figure 24.7

^ ^

-0.6 -0.3

0

\

0.3

-0.6 -0.3

0

0.3

x(t)

The four unstable period-one orbits ((a) - (d)), the unstable period-two orbit (e), and the unstable period-three orbit (f) of the bronze ribbon which have been detected analyzing best recurrent points for po = —0.2 V. They are shown in the x(t)-x(t — r) plane with r — 5A£.

0.20 X

0.18-J

xxx x

•H-

0.18-

X X

xxx*x

0.160.14-

> *

0.160.14-

x 5

0.12X>

0.10-0.10

Figure 24.8

-0.08

0.10 -0.08

-0.08

(a): Out of 100 000 periods 50 best recurrent points (+), z n G Si, belonging to the class of UPO 1 and their images (x), zn+N G £ i , under the Poincare map P p = (j)J^ are shown. The two arrows are located at Zp and give the stable (filled vector) and unstable (open vector) eigen directions of the linearization Af = D z i Pf with eigenvalues Au « 9 and As « 0.1 obtained using the product (24.38) of the local matrices An. As can be seen the best recurrent points are mostly placed along the stable direction while their images are mapped along the unstable direction due to the large ratio Au/As « 90. (b): The 50 nearest neighbor points (+) z n of Zp (•) are shown. They give information of the full linear mapping around zp.

24-6 Control vectors from scalar measurements

665

also information about other directions (see Fig. 24.8 (b)) and thus give better approximations of the full linear mappings An. To extract the mappings An from the dynamics of the nearest neighbors we record in each section beside the m nearest neighbor points zn> £ S n itself also z n i + 1 . Finally An = Dz*0u* is obtained from a least square fit using the relation

Before we present the determination of the linearization An in the case of differential coordinates we want to mention its approximation for small At. Consider a nonautonomous system z = v(z,t) = v(z,£ + T), z € JR2 where the Poincare sections of the equivalent autonomous system is taken at constant times tn (or constant phases of the driving). It comes right from the definition that for (infinitesimally) small At 0 £ B (z) « z + v(z,t n )At

(24.32)

holds. For the quantities of interest in our control problem this leads immediately to An « U + DZnv(zP,*n)A*

(24.33)

dv bn*^-(z%ytn)AL

(24.34)

and

Specializing further to a nonautonomous system derived from a differential equation of second order, i.e. x — f(x,x,t), and choosing (x,x,t mod T) as state space coordinates, Eq. (24.33) and (24.34) can be rewritten as

and

" - ( gA« ) ' In Fig. 24.9 (a) the linearization An of the flow mappings p(n>n+!) in the physical state space approach determined using Eq. (24.31) are shown for the UPO 1. For each UPO continuous curves of the afj with respect to n are found. Furthermore the calculated An agree nicely with the approximation (24.35) for the An which we expect to hold for small At. Note that a\2 of our experimental An is almost constant, it only varies between 0.15 and 0.18. This is in agreement with At « 0.16 [lO"1^] which we have as time difference between two successive Poincare sections in our experiment, a^ and a ^ are of the order of 1 as expected. The largest variation can be found in a^i which again coincides with the approximation (24.35). Thus the approximation (24.35) gives a good hint for the confidence one can have in

666

24 Experimental control of highly unstable systems using time delay coordinates

(a) U-

(b) 1-5

aj2

\ 0(

1

1'

1

0.5

0-

-0.5

\

//^\\\ // r V «21 \

16

32 n

iQ? 0Js

/

48

0

^'

*n V

•Oi 64

^ . . . ^

"" ^ J l

/

V

-1

3^

16

32 n

48

64

Figure 24.9 Linearization An = (a£) of the flow mappings p( n ' n+1 ) for the UPO 1 of the bronze ribbon: (a) using differential coordinates, (b) using time delay coordinates (d = 2 and r = 5 At). the reconstruction using differential coordinates and in the An obtained from the experimental data. The linearization An of the flow mappings p(n»n+1) in the time delay embedding are shown in Fig. 24.9 (b) for UPO 1, too. In contrast to the corresponding linearizations in the embedding using differential coordinates (Fig. 24.9 (a)) all curves of the a^ show significant structures, not only the component a\\x. ^n [Schenck, 1996] a relation is derived which allows one to calculate the linearizations in the time delay embedding from the corresponding ones in the physical state space. If one applies this relation to the data of Fig. 24.9 (a) best agreement with Fig.24.9 (b) is obtained. Before we proceed with the estimation of the stability properties we want to mention a difference in the determination of the local dynamics between the physical state coordinates and the time-delay coordinates which is important when the chosen dimension d of the Poincare section in the delay space is bigger than the dimension d of the manifold which contains the attractor in the Poincare section. To start we note that for an optimal estimation it is necessary that the difference vectors 5zni = z n ' - zg and <$zn>+1 = z n ' + 1 - Zp+1 are infinitesimal and lie in tangent spaces. But in general 6zn> and <Szn'+1 are not infinitesimal and lie in submanifolds which generally have a curvature in the embedding space. Fig. 24.10 gives an illustration of an two-dimensional submanifold and the corresponding tangent space in a three-dimensional embedding space. Therefore ST.71' and 6zn>+l of the m nearest neighbor points are not restricted to the tangent spaces as they should be in the ideal case of inifinitesimal distances. To eliminate these extra dimensions projections IIz" and IIzn+i are introduced in [Schenck, 1996] which project the differences Sznj and 6zn>+1 onto the corresponding tangent spaces. The projection I I ^ can be estimated using the singular

.6 Control vectors from scalar measurements

667

Figure 24.10 Illustration of the embedding of a two-dimensional manifold into a threedimensional space. The embedded manifold is generally curved in the embedding space and thus the nearest neighbors of a specific point lie not in the tangent space.

value decomposition Un • Wn • V^ of the matrix (Szni,..., 6znm )* of nearest neighbors of Zp. The singular values of this matrix measure the extension of the nearest neighbor points in direction of the corresponding singular vectors which are the columns of the orthogonal matrix Vn. Assuming a sufficiently small neighborhood the nearest neighbor points are mainly spread in the directions of the tangent space. Thus the tangent space is spanned by the directions corresponding to the d largest singular values, where d is the dimension of the tangent space. The remaining singular vectors are normal to the tangent space indicating the directions of the curvature of the manifold. So we obtain for the projections

assuming that the singular values are ordered by size. In the case that the dimension d of the tangent space is not known the method proposed in [Broomhead, 1987, Broomhead, 1989] can be used to estimate d. Broomhead et al. show that the directions of the tangent space can be identified by the scaling behaviour of the corresponding singular values when the diameter of the neighborhood is decreased. After the determination of the projections, the mappings An are estimated using the relation ILn+i • 8znjJrl = An • IIZ« • Sznj

(24.37)

668

24 Experimental control of highly unstable systems using time delay coordinates

instead of (24.31). For the determination of the An in the time delay embedding space shown in Fig. 24.9 (b) these projections have no consequence as d = d in this case. But for d > d this projections are necessary and lead to better estimations of An. This is investigated and shown in detail in [Schenck, 1996]. Not for control but to estimate the stability of the UPOs one needs in addition the linearization A® of the global Poincare mapping P ^ = <j>T . Determining them directly from fitting a linear mapping between 5zn* and 5znj+N turns out to be not appropriate because of the extremely high effective Lyapunov exponents of the system [Hubinger, 1994]. To obtain an estimate of the A® it proves to be much better to use the local mappings An obtained from (24.31) and the relation A£=An+"-1.-.An+1.An.

(24.38)

Using this relation we determined the stable and unstable eigenvalues As and Au of the UPOs as UPO 1 Au = 11 and As = -0.07, UPO 2 Au = -13, As = -0.05, UPO 3 Au = - 1 1 , As = -0.06, and UPO 4 Au = -16, As = -0.03. For the periodtwo UPO 5 we obtain Au = 10, As = 0.06 and finally for the period-three UPO 6 Au = —720 and As = —0.001. Values of the same order are obtained using time delay coordinates as it has to be.

24.6.3 Dependence on the control parameter In this subsection we describe how the dependences b n i , i = 0,1,...,ty, on the control parameter can be obtained in the case of time delay coordinates. If differential coordinates are used the same precedure is possible assuming w = 0. Another approach is given in [Hubinger, 1994] where the displacement of the location of the UPO in the Poincare sections are measured when the control parameter is changed by a fixed amount. Here we record a second time series where the control parameter is disturbed at each time step by a random perturbation Spn G [-0.07 V,+0.07 V]. The resulting time series then consists of a set of data pairs

These perturbations do not change significantly the global dynamics of the system as Fig. 24.11 compared with Fig. 24.6 (b) shows. Once again, out of 50 000 periods we record in each section E n m = 300 nearest neighbor points znj of zg and the successor points z n ' + 1 . Since the dynamics An of the system in the neighborhood of the UPO has already been determined we use the difference between a vector (Szn'+1 and An • 6zn> to determine the dependences bn>w,..., b n>0 . They are obtained by a least square fit using w

(Szn'+1 - An • Szn') = ^bn'Mpn>-i j=0

(24.39)

24-6 Control vectors from scalar measurements

669

-0.8 -0.6

-0.2 0 x(kT)

0.2

0.4

Figure 24.11 Poincare section of the chaotic attract or when the system is disturbed in each time step by a small random perturbation 8pn G [—0.07 V,+0.07 V]. This perturbed attractor should be compared with the unperturbed attractor in Fig. 24.6 (b). or

n z n + i • ((5zn'+1 - An - 5zni) =

if d > I

(24.40)

i=0

The so obtained estimates of the b n '* are much more noisy than the estimated A [Schenck, 1996]. But in our control experiments their accurracy proves to be sufficient for satisfying control results. n

24.6.4 The adaptive orbit correction Before we discuss the control experiments to compare the performance of the quasicontinuous control for time-delay coordinates based on the formulas (24.25) and (24.30) we first want to describe the adaptive orbit correction [Doerner, 1995]. We always use this procedure in order to redetermine the position of the reference UPO zg used in the feedback formula before we start with the actual control experiment in question. As the control experiments show errors in the determination of the reference UPO affect the control performance much more than errors in the determination of An and b n > \ The adaptive orbit correction improves the control performance of every OGY based control method considerably because it helps to deal with drifting parameters which affect the true position of an UPO. As Schwartz et al already reported for the original OGY method an errror in the determination of the UPO leads to a systematic deviation of the averaged control signal from zero [Schwartz, 1992]. For the quasicontinuous control Doerner et al. observed that a difference between the true UPO zg t r u e of the system and the orbit zg used in the feedback control

670

24 Experimental control of highly unstable systems using time delay coordinates

0.03-

-0.03 -0.06 0.12 0-

-0.121

16

32 n

48

64

Figure 24.12 Quasicontinuous control of UPO 1 using Zp extracted from the analysis of best recurrent points. 20 periods of the controlled orbits are recorded. In the upper part the differences Sxn = xn — xp between the desired orbit Xp used in the feedback control and the observed trajectory xn and in the lower part the applied parameter perturbations Spn for successful control are shown versus n for 20 periods of control.

formula leads to an almost periodic trajectory z n « z n+iV and an almost periodic control signal 6pn « 8pn+N (Fig. 24.12). The adaptive orbit correction enables one to calculate a new estimate of Zp t r u e during control by exploiting the periodicity of z n and 6pn. We describe the adaptive orbit correction in the context of quasicontinuous control using time delay coordinates. The basic idea is the following. As the controlled trajectory z n is still close to the true UPO zg true and the parameter perturbations Spn are small, in each section E n , n = 1,..., JV, z n and Spn have to fulfill the linearized dynamics

n+l

— zF,true

_ ~~

- z p itrue ) +

(24.41) i=0

with An = An+N, bni = bn+jV»S and zg true = z ^ e . For perfect periodicity of Spn and z n this set of N vector equations can be solved with respect to zg true to obtain a new guess zg new of the true orbit zg t r u e . In the case of time delay coordinates it is not necessary to solve the whole set of TV vector equations (24.41). Because of the special structure of a time delay vector z n = (x n , xn~e,..., xn~w)^ already the first component of each of the iV equations suffices to determine a whole time delay UPO zg n e w = (zg, new , xg"^ w ,..., s g " ^ ) • Thus for time delay

24-6 Control vectors from scalar measurements

-0.8 -0.6 -0.4 -0.2 0

0.2 0.4

671

0 0.1 0.2 0.3 0.4 05 0.6 0.7 0.8 0.9 I X

x(t)

Figure 24.13 (a) The UPO zp o l d , used as Zp in the feedback control, and the corrected orbit z^new a r e shown in the x{t)-x(t — r) plane, (b) The old value Zp old is not immediately replaced by Zp new in the feedback loop. Instead the weighted average Zp(x) = (1 — x)zF,oid + XzF,new 1S used for the quasicontinuous control beginning with x — 0 a n d increasing \ a ft er 1^0 driving periods by 0.1. Depending on the weighting factor x the magnitude of the averaged control signal (\Sp\) varies. As optimal x w e t a k e ^he one which minimizes the averaged control signal, here Xopt = 0.7.

coordinates one solves the set of N scalar equations d l

~

X

~ ^F,true

=

(

\

Z-> al,i+l \ x

~ ^F,true )

(24.42)

2=0

with respect to xg t r u e to obtain a new estimate Xp n e w J n = l , . . . , i V to build Zp n e w . Of course, this holds only for perfect periodicity of Spn and xn. Because of measurement noise in experiments one averages Spn and xn over some periods

and insert the averages 5p" := jj J2iio 6pn+iN and x^ := -jfc Ylf=o 5xn+iN instead of Spn and xn into the set of equations (24.42). In the experiment we average over M = 4 periods. Nevertheless, due to errors of An and b n>i we cannot expect that the new estimate zp new gives already the correct value of Zp true . Therefore in our experiment we do not immediately replace the old value zp)Oid by Zp new , but use a weighted average zg(x) = X^g new + (l-x)zF,oid> X € [0,1], in the feedback control. Depending on the weighting factor x the magnitude of the necessary control signal (|(Jpn|) (averaged over one period of the driving, i.e. (|<5pn|) = ^ S n = i \&Pn\) v a r i e s This is shown in Fig. 24.13. Starting with % = 0 we increase x slowly until no

672

24 Experimental control of highly unstable systems using time delay coordinates

x(kT)

0.50-0.50.09-

I

^^^~* orbit correction

0.060.03n300

600

900

1200

1500

Figure 24.14 Repeated orbit correction of the period-three-orbit shown in Fig. 24.7 (f) beginning with the orbits extracted from the analysis of recurrent points: In the upper part the stroboscopic measurement x(kT) in the Poincare section Si, in the lower part the averaged control signal (\Sp\) over one period of the UPO is shown for 1500 periods.

further decrease of the control signal can be achieved. Using the orbit resulting from optimal value of x the next correction step is started. After some repetitions the applied averaged control signal (\Sp\) is minimized. This procedure is shown in Fig. 24.14 for the period-three-orbit (Fig. 24.7 (f)). As can be seen with this procedure the adaptive orbit correction leads to a reduction of the averaged control signal by a factor of about 7. Depending on the initial error in the determination of the true UPO we also observed a reduction factor 10.

24.7 Control experiments - The bronze ribbon 24.7.1 Quasicontinuous control of the bronze ribbon with time delay coordinates Before we now report the results of our control experiments we mention the additional control rule which we always use in our control experiments. To stabilize the UPO the control signal Spn calculated from (24.25) or (24.30) is only applied when its magnitude is less than a maximal allowed parameter perturbation <5pmaxIn addition, control is also applied if \\6zn\\ is less than a maximal distance SzmAX but the calculated parameter perturbation exceeds 5pmliX [Hiibinger, 1994]. In this case we only restrict the control to 8pm^ and give the control signal the sign of the calculated Spn. This additional control rule ensures that control is never suspended if the system is close enough to the UPO.

(.7 Control experiments - The bronze ribbon

673

(7

20

200

2000

20000

200000

0.3-'ir'r, ?::-•••-. v

0-

-0.3-0.60

19-,

0.080.04-

n -

0500

1000

1500

2000

2500

k

Figure 24.15 Local control of UPO 1 using extended states with Szm&x = 0.1, Spm&x = 0.12 V, and p = 0.15. Starting with the scaling factor a = 20 every 500 driving periods the scaling factor a is increased by a factor 10. In the upper part the stroboscopic measurement x(kT) in the Poincare section Si and in the lower part the applied control amplitude (\Sp\) averaged over one period of the driving is shown as function of the period k.

To study the performance of the quasicontinuous control in the mechanical experiment we concentrate on the period-one orbit of Fig. 24.7 (a) which we call UPO 1. The results do not vary significantly from the other UPOs. We first try to stabilize UPO 1 using the control formula (24.25) making use of the extended states. As was already discussed in section 24.4.2 for the quasicontinuous control methods (LC or MED) the control formula (24.25) depends on the choice of the units of the control parameter p. To investigate this effect in the experiment we replace all parameters pn by pn/a and the corresponding vectors b n i by ahn>1 and apply the local control using (24.25) for different scaling factors a. In Fig. 24.15 we show the experimental result for the local control method using a = 20, a = 200, a = 2 000, a = 20000, and a = 200000. As can be seen successful control can only be achieved for a = 200 and a = 2 000. Furthermore, the calculated control signal clearly depends on the scaling factor a. We did the same experiment for the MED method using the extended states. As for the LC method only for certain values of cr, however different ones than for the LC experiment, control could be reached. There is no simple rule how to find the right values of a. We observe that the appropriate value depends on the method and even on the parameters d, r of the time delay embedding. The best choice with respect to control cannot be predicted in advance. Thus using MED or LC control in the extended state space requires a blind search in the control experiment for the right units of p.

674

24 Experimental Control of Highly Unstable Systems Using Time Delay Coordinates

1 0.3-

•.•/-«":-;••-••./•.'•

0-0.3-

ftfi

-0.6-

2

3

4

5

6

7

8

9

;.

n

n 19 -,

0.080.0401000

2000

3000

4000

Figure 24.16 Local control of UPO 1 using the modified control requirement with ^ m a x = 0.1, 6pm&x = 0.12 V, and p = 0.15. In the upper part the stroboscopic measurement x(kT) in the Poincare section Ei and in the lower part the applied control amplitude (\Sp\) averaged over one period of the driving is shown as function of the period k. From 0 - 499 the stabilization is demanded after j — I time step, afterwards the value j is increased every 500 periods by 1.

Next, we want to investigate the performance of the quasicontinuous control if one uses the control formula (24.30) which results from the modified control requirement (24.27), (24.28). Here one requires that the system stabilizes only after w + 1 time steps. For the time delay embedding with d = 2 and £ = 5 the window length w = (d — 1)1 is 5 and therefore the stabilization has to be required after 6 time steps. To demonstrate that the modified control requirement (24.27), (24.28) is indeed necessary we apply local control to UPO 1 using a feedback control formula which results if one requires stabilization after j = 1,2,..., till 9 time steps. Fig. 24.16 shows the result of this experiment. It is clearly visible that for j = 1 and j = 2 no successful control is achieved. The UPO can be stabilized for j > 3, but the control signal is minimized for j = 7. A further investigation shows that the averaged distance between the UPO and the trajectory of the system has a minimum for j = 5. Both results confirm our theoretical choice j = w + 1. With this choice we could stabilize all UPOs of Fig. 24.7 using MED or LC in (24.30). In Fig. 24.17 we show an example of a successful control experiment using the LC method. Every 500 driving periods we switch from one UPO to the next one. As can be seen the transient time to stabilize the next UPO is always very short. We did the same experiment for the MED method. There was no qualitative

(.7 Control experiments - The bronze ribbon

b

a)

)

c)

e)

d)

0.3-

£

o-

H

-0.3-

675

,

^.

2500

3000

-0.60.12-

=g 0.08"

0.040n

500

1000

1500 k

2000

Figure 24.17 Local control (modified control requirement) of all UPOs shown in Fig. 24.7. The magnitude of the unstable eigenvalues of the period-one UPO and of the period-two orbit are of the order of 10, while the unstable eigenvalue of the period-three orbit is —720. The control is switched on at k — 0, every 500 driving periods we change the UPO to be controlled. The stroboscopic measurement x(kT) in the Poincare section Ei and the applied control amplitude (\6p\) averaged over one period of the UPO is shown as function of the period k.

difference between the MED or LC method as long as one used the modified control requirement with j = w -f 1. [htp] To investigate the robustness of the LC method and the MED method with respect to noise we introduce noise by adding small noise terms e-r]n to the measurement xn, rjn being identically distributed in [—1,1]. Starting with e = 0 every 100 periods the noise level is increased by 0.005. In Fig. 24.18 (a) the number Ns of stabilizied periods is shown for the LC and the MED method using two and three time delay coordinates versus e. For the MED method with a the three-dimensional Poincare section in the embedding space the control signal averaged over the stabilized periods are shown in Fig. 24.18 (b).

24.7.2 Tracking of the bronze ribbon experiment An immediate and most promising extension of OGY's control method is the tracking idea of Schwartz and Triandaf [Schwartz, 1992]. They propose to first stabilize an UPO lying in a chaotic attractor and then to slowly alter the control parameter p while adjusting the control values of the feedback loop to its new actual values such that control is maintained over the whole tracking process. The tracking idea in combination with OGY's control method has direct impact on the possiblility

676

24 Experimental control of highly unstable systems using time delay coordinates

MED, d = 2

MED, d = 3 O

O.Ol

O.O2

O.O3

O.O4

O.OS

O.O6

noise amplitude £

Figure 24.18 Robustness of the LC and MED method with respect to identical distributed measurement noise. Starting with e = 0 the noise level e is increased every 100 period by 0.005. (a) The number N$ of stabilizied periods are shown for the LC and the MED method using two and three time delay coordinates, (b) The control signal averaged over the stabilizied periods are exemplarily plotted versus the noise level for the MED method using three time delay coordinates in the Poincare section. of real world applications of this type of chaos control. Having a reliable tracking method at hand, one can deal with fluctuating external parameters, which improves the stability of OGY's control for technical systems, and secondly it is possible to extend the range of stability of a system [Gills, 1992]. The feasibility of the tracking idea has been demonstrated experimentally in an electronic circuit [Carroll, 1992], in a laser experiment [Gills, 1992], and in the Belousov-Zhabotinsky reaction [Petrov, 1994]. A slightly different tracking technique which does not need parameter adjustment over a whole parameter regime, is reported in a laser experiment in [Bielawski, 1993]. While all these experiments [Gills, 1992, Carroll, 1992, Petrov, 1994] follow the same tracking idea, they differ in two aspects: First, in the variant of OGY's feedback control method which is applied for control (original OGY [Carroll, 1992], occasional proportional feedback (OPF) [Hunt, 1991, Gills, 1992], a map based algorithm [Petrov, 1994]), and second, in details of the strategy used to determine the correct parameters of the feedback loop for each new tracking step. To explain the main ingredients of a tracking strategy let us write the general form of a OGY-feedback control as Spn = K(p) • (z n - z F ( p ) ) .

(24.43)

The control vector K(p) (which is calculated using A(p) and b(p), see Eq. (24.4)) and the position of the UPO zp(p) are the parameters of the feedback loop of the OGY control.

24-7 Control experiments - The bronze ribbon

677

Figure 24.19 The bifurcation diagram of the bronze ribbon for the tracking regime p = 0 V - 0.7 V. Note that for p - 0.7 V the offset of the voltage exceeds the amplitude UK = 0.6 V of the sinusoidal driving (see Fig. 24.4). In addition, the tracked orbit is plotted in the diagram. As size of the tracking steps we used Ap = 0.01 V.

While a small deviation of the control vector K(p) from its optimal value influences primarily the time one needs for successful control [Romeiras, 1992], an error in the determination of z F (p) results in a systematic deviation of the averaged control signal (Spn) from zero [Schwartz, 1992], which is proportional to the difference between the true fixed point z F)tr ue(p) and the reference point z F (p) used in the feedback loop (24.43), i.e. \(Spn

(24.44)

holds. It is this relation which Schwartz and Triandaf exploit in their tracking procedure [Schwartz, 1992] in order to correct a prediction of zF,true(p)- At each tracking step they vary the reference value zF(p) in the feedback loop until the averaged control signal \(Spn)\ is minimized. In the tracking experiments, we report here, we use the adaptive orbit correction in order to redetermine the position of zg(p) in the quasicontinuous feedback loop at each tracking step. Although it is in principle necessary in a tracking process to adapt beside zg(p) also the control vector Kn(p) (calculated from An and b n or b n ' { ) we neglected this point in a first attempt to track an UPO of the bronze ribbon experiment. We did the tracking experiment for differential coordinates and time delay coordinates. Let us first give the result of the tracking with differential coordinates. For the starting parameter p - 0 V we determined the control values zg(p = 0 V) and n i4»(p = 0V), b (p = 0 V) as described in Section 24.6. With these control values the corresponding UPO of the bronze ribbon is stabilized. The tracking process is then started using as size of the tracking step Ap = 0.01 V. For each tracking step

678

24 Experimental control of highly unstable systems using time delay coordinates

p = 0V

0.6 -

p = 0.2V

p = 0.4V

0.3 -

-0.3 -0.6 - (a)

(b)

(c) p = 0.7V

p=0.6V

0.6 -

p = 0.7V

0.3 0-0.3 -0.6 -

^ \

/

(e)

(d) -0.6 -0.3

X

/

0

x(t)

0.3 0.6 -0.6 -0.3

^ ^ 0

x(t)

J

(f)

^—/h v ( —V

0.3 0.6 -0.6 -0.3

0

0.3

0.6

x(t)

Figure 24.20 The tracked UPO for different parameters p along the tracking process in the zr-x-plane. We start to draw the curve at (x,x) values correponding to the Poincare section Si. In (f) we have, for comparison, plotted the coexisting stable orbit at p = 0.7 V.

the value of Zp (p + Ap) is estimated by using the repeated orbit correction until no further progress in the diminishment of (|£pn|) can be reached. We succeed to track the UPO from p = 0 V t o p = 0.7V using an averaged control signal of no more than 0.05 V. In Fig. 24.19 the bifurcation diagram of the bronze ribbon of the tracking regime where the tracked UPO is included reveals that we are able to track the UPO into a parameter regime where the system has switched to a stable period-one behavior. In Fig. 24.20 it can be neatly seen how the tracked UPO continuously changes its shape in dependence on p. The final UPO at p = 0.7 V (Fig. 24.20 (e)) differs drastically from the coexisting stable orbit (Fig. 24.20 (f)). The success of the tracking is a little bit surprising if one remembers that we did not redetermine An(p) and bn(p) which give the control vector K n (p) and in addition crucially enter the counterpart of formula (24.41) for the adaptive orbit correction in the physcial state space [Dressier, 1995]. For our experiment we explain this by the fact that for differential coordinates the quantities An(p) and b n (p) do not change drastically when p is varied. The approximation (Eq. (24.35) and (24.36)) of An and b n for infinitesimal times A* in Sec. 24.6.2 reveals that indeed some components of An and b n are independent of p. Determining An and b n for different p supports this finding.

24-7 Control experiments - The bronze ribbon

O

3OO

6OO

9OO

12OO

15OO

1SOO

21OO

24OO

679

2700

Figure 24.21 Tracking of the bronze ribbon using time delay coordinates from p = — 0.2 V till p = 0 V with a tracking step Ap = 0.025 V. In each step we wait 100 periods before the adaptive orbit correction is started. Every 300 periods we begin with a new tracking step. In the figure we plot the stroboscopic measurements in the Poincare section Si and the averaged control signal for this experiment. The increasing and decreasing of the control signal due to the parameter change and the orbit correction is clearly visible.

In contrast to this for time delay coordinates there is structure in each component of An (see Fig. 24.6 (b) compared to Fig. 24.6 (a)). A variation of p therefore affects every component of An. The tracking experiment with delay coordinates has thus been not as successful as with differential coordinates. In Fig. 24.21 we show the tracking of UP O 1 from p = - 0 . 2 V t o p = 0V using two time delay coordinates in the Poincare section and MED control. As tracking step Ap = 0.025 V has been taken. After changing p we wait 100 periods before we start the repeated adaptive orbit correction. Every 300 periods a new tracking step is started. In Fig. 24.22 the UPO is shown at the start and the end of the tracking process. Shortly after p = 0 V the tracking procedure failed. For time delay coordinates a redetermination of An and b n>i during the tracking could possibly ameliorate this result. In a first attempt we tried to do this by fitting An and b n '* using the observed 6zn and 6pn of the controlled system together with Eq. (24.21) during control. But till now the results are not yet satisfactory as the accuracy of the so obtained An are not as good as the ones obtained from the unperturbed time series because during control one fits An together with b n ' f . Therefore errors in b n '* can have impact on the accuracy of An. To improve the redetermination of An and b n '* during control further research has to be done.

680

24 Experimental Control of Highly Unstable Systems Using Time Delay Coordinates

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

x(t)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

x(t)

Figure 24.22 The unstable periodic orbit is shown in the x(t) — x(t — r) plane for (a) p = -0.2V and (b)p = 0V.

24.8 Summary and Conclusions In a bronze ribbon experiment we have implemented quasicontinuous versions of the OGY-control method using differential coordinates and time delay coordinates for the reconstruction of the attractor. With quasicontinuous we refer to control methods with in principle arbitrarily high control frequencies which are especially advantageous for highly unstable systems. The bronze ribbon is such an experiment as its UPOs have unstable eigenvalues of the order of 10 (period-one) or even 700 (period-three). In the bronze ribbon experiment we apply as quasicontinuous methods the local control method and the minimal expected deviation method. Both methods achieve a high control frequency by introducing several Poincare sections per period of the driving. As is known for the original OGY-method, for time delay coordinates the feedback law has to be modified in the way that also preceding values of the control parameter have to be included in the control formula. For a quasicontinuous control with high control frequency being e.g. the sampling frequency I/At, there are w = £(d — 1) preceding parameter changes to be taken into account. Here I is the discrete time delay, i.e. r = £At, and d the embedding dimension in the Poincare section. This is an immediate consequence of the fact that in the time delay embedding space the mapping p(n>n+1) = 0j^ from section E n to the next E n +i depends on all preceding parameter values which were changed during the time window rw = (d — l)r of the delay vector. In an explicit calculation we show this dependence.

24-8 Summary and Conclusions

681

Starting with the linearized dynamics including An = DznP( n ' n+1 ) and b n i = *n t ^ ie ^ mp e delay s P a c e there are two ways to obtain an explicit control d y p formula. One is the extended state space approach followed by the standard control requirements of the MED and the LC method, the other is a modified control requirement which requires stabilization not for z n + 1 but for zn+™+1 with Spn ^ 0 and Sp71^1 = • • • = Spn+W = 0. With this modified control requirement one bears in mind that the parameter change Spn influences at time n the system till the time n + w + 1 and 8pn~l only till n + w. While for the original OGY-method both approaches, i.e. extended state space and modified control requirement, are equivalent, for the quasicontinuous control methods considered here (MED or LC) this is no longer the case. In fact, it turns out that the extended state space approach is not suited for the MED or LC method as it leads to a control formula which depends on the scaling of the control parameter p. In the experiment we demonstrate that this scaling dependence causes a breakdown of the control for certain units of p. In contrast, the control formulas based on the modified control requirement show a good control performance in the experiment, even for highperiodic orbits with large instabilities. Furthermore between the LC method and the MED method no qualitative difference could be observed. Both methods will lead to satisfying control results provided that a high enough control frequency has been used. Finally we remark that for control experiments where the control parameter have been determined in advance it is advisable to use an adaptive orbit correction to obtain an optimal reference value for the feedback control. This improves the control performance considerably irrespectively which OGY-control method has been used. In the last part of the paper we also reported how the adaptive orbit correction could be used for tracking the bronze ribbon experiment. This investigation reveals that especially for time delay coordinates beside a correction of the UPO as reference value in the feedback control also the linearized dynamics An and b n i should be adapted during control to its acutal value in order to obtain a satisfying tracking result. Here future research has to be done. In conclusion, with a quasicontinuous OGY control approach the stabilization of highly unstable systems proves to be feasible. The principal advantage of a high control frequency is that measurement noise or errors in the determination of the control vectors do not have time to lead to a failure of the control but can soon be corrected in the next control step. A disadavantage, of course, is that there are much more control vectors to be determined, to be exact, N times more than in a discrete OGY control when only controlling once per driving period. But as we show in our experiment all these control quantities can be extracted with sufficient accuracy from a scalar measurement signal without feeding in anything about the physical equations of the system. This makes quasicontinuous control with time delay coordinates a promising control method for highly unstable or noise contaminated technical systems and should lead to real world applications in the near future.

682

References

Acknowledgments We gratefully acknowledge the inspiring continuous cooperation with B. Hubinger, R. Doerner and W. Martienssen from the University of Frankfurt and T. Ritz during his stay at the Daimler-Benz institute. We thank R. Pitka and M. Herdering for their contribution setting up the bronze ribbon experiment.

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Index

absolute instability 125f adaptive orbit correction in chaos control 640,634ff,669ff advanced control stragies 3 Iff ambipolar diffusion 539 analog feedback device 576ff applications of controlling laser chaos 5O5ff argon-ion lasers 604 Arnold tongues 308, 315, 316 atrial fibrillation in humans 44Iff autonomous continuous-time systems 307 average targeting 154 backward part 389 Belousov-Zhabotinsky (BZ) reaction 121, 591, 592ff Benjamin-Feir limit 125 bifurcation diagram 397 biological networks 405,418 brain tissue 448 bronze ribbon 659, 672 bucket brigade delay line 357 bursting 247,251 butterfly effect 329 capture time 95 cardiac dynamics 428ff cascaded - drive-response synchronization 235f - synchronous system 240 cascading circuit 242f channel capacity 198 chaos - control in the presence of large effective Lyapunov exponents 629ff - in plasma 513ff - modulation 293 - shift keying 293

- synchronization 67,68ff chaotic - categorizer 41 Iff - masking 293 - saddles 8, 181ff - scattering 181, 189ff - sensitivity 157ff - states 542 - systems 427 chemical waves in excitable media 59Iff circle map 307 classical chaotic systems 205ff closed-loop - control 518ff - system 98, 99, 105 CO2laser 489ff complete replacement (CR) drive-response 231 complex Ginzburg-Landau equation 12Iff, 322, 408 conditional - control 498 - Lyapunov exponent 233, 283, 332, 345ff conditions for control 89f, 108 continuous feedback control 522f, 546ff control - algorithm for spatiotemporally chaotic systems 87, 119ff - by electric fields 598ff - by light 603ff - in Fourier space 42Iff - in the complex Ginzburg-Landau equation 129ff - in the presence of noise 98ff - of chaos in electronic circuits 459ff - of highly unstable systems 645ff - of ionization wave chaos 543ff - of patterns 43ff

688

Index

- of periodic orbits 10Iff - of spiral 132f - of the hole solution 129ff - of turbulence 120 - parallelogram 8,650 - vectors 662ff controllability condition 90 controllable regions 50 controlling - chaos in two dimension 4ff - in higher dimensional systems 470ff - motion on fractal basin boundaries 189 - permanent chaos 187f - transient chaos 18 Iff convective instability 123, 125, 475 core instability of the spirals 128 Coriolis parameter 372, 374 coupled - diode resonators 468ff - Henon maps 337, 348 - logistic maps 285, 334ff - map lattice (CML) 44, 46ff, 94, 120 - ordinary differential equations (CODE) 44 - Rossler and Lorenz systems 338f, 349 - systems 318ff coupling thresholds 247f critical noise amplification factor 111 Cuomo-Oppenheim communication 243 cusp catastrophe 537 cycle expansion 214ff defect turbulence 121 definition of the phase 309f delay - coordinates 16f, 382, 375ff - differential equation 406ff - mismatch 34ff delayed feedback control 499ff, 582 desynchronization thresholds 249ff deterministic coupled map lattice (CML) 88 deterministic diffusion 215ff diode resonator 464ff discrete feedback control 528ff, 543ff double pendulum 158 double-scroll oscillator 357ff drift wave chaos and turbulence 55 Iff drive system 275 driven pendulum 617, 626, 637ff Duffing oscillators 230 dynamical zeta functions 208

dynamics of TSS lines 392ff Eckhaus stable range 125 Eckman-Kamphorst-Ruelle-Ciliberto (EKRC) algorithm 346 effect of noise lOf, 169f effective Lyapunov exponent 621, 618ff effects of noise 169ff El Nino's dynamics and chaos 366ff electroencephalograms (EEG) 411 embedded periodic orbits 314ff embedding dimensions 623 encoding generalized synchronization 294f epileptic activity 450 error signal 464 escape rate 184 extended state-space approach 565ff extensions of the OGY-control 649ff eyelet intermittency 317 Fabry-Perot resonator 500 feedback - gain matrix 90 - stabilization of turbulence 553f Feigenbaum 567 ferromagnetic - insulators 563 - resonance 563ff fiber laser 497ff fibrillation 428f filter gain matrix 105 first-order Suhl instability 564 Floquet - exponents 26ff, 581 - multiplier 496,506ff - theory 23,515 fractal basin boundaries 181 Fredholm - condition 37 - determinant 209,219 fundamental cycles 211,221 generalized dimensions 214 generalized - Lyapunov exponents 215 - metric entropies 214 - synchronization 255ff, 286ff, 331f,329ff - synchronization of chaos in experiment 329ff - transfer operator 224

Index

globally coupled chaotic oscillators 32Off glow discharge 513, 539ff Grassberger-Procaccia algorithm 409 Henon - attractor 170 - map 164ff, 183, 635ff Hopf - bifurcation 502 - instability 28 hyperchaos 76, 257ff, 319 hysteresis and nonlinear oscillations 53Iff identical - synchronization 23ff, 329 - systems 290f improved - control algorithm 199f - improved control of chaotic saddles 19Iff information dimension 165 interacting - oscillators 318f - of prediction and control 638ff intermittency type III 572 intermittent map 220 International-Sun-Earth-Explorer-3 159 intersections 167ff intracavity second harmonic generation 492 invasive vs noninvasive methods 503 ionization - instabilities 513 - threshold 532 - wave 542 - wave chaos 539ff, 544, 548 KAM surfaces 190,191 Kelvin waves 368, 374, 379 kicked double rotor 142 Kramers time 538 Kuramoto-Sivashinsky (KS) equations 284 lag synchronization 292,319 Lang-Kobayashi model 493 Langmuir-mode 533 laser - chaos 487ff - diode 497 - with modulated parameters 490f lattice - of chaotic oscillators 321

689

- partitioning 107f limits of the simple feedback method 27ff linear control 5ff linear-quadratic control (LQC) 87, 94 local - control of low-dimensional chaos 615ff, 630 - Lyapunov exponent 528 logistic map 2, 159 Lorenz - attractor 160ff, 162 - model 163, 164, 277, 616, 621, 627 Lyapunov function 287, 289 machineless TSS lines 395 Mackey-Glass equation 357ff, 406, 407 magneto - elastic buckled beam 659 - elastic ribbon 171 - plasma 534 - static modes 565 manifold corrections 506ff marginal fixed points 219ff mean prediction error 622 mechanism of delayed feedback control 22ff minimal density of pinning sites 112 modeling errors 169f mode-locked 542 modifications of the OGY-control 521 ff, 645, 658 motion detection 416f multi - armed spirals 604 - channel communication 72, 81 - channel feedback control 607 - magnon processes 565 - mode lasers 492 - parameter control 553 mutual correlation 73 natural - distribution 184 - measure 187 Nd-doped optical fiber laser 491, 498 nerve action potentials 448 network of oscillators 408ff Newton' s method 147 NINO3 index 369,370,371,381 noise amplification factor 100 non

690

Index

- attracting 181 - autonomous and filtered synchronization circuit 237,241,243 - delay coordinates 38 If - feedback control of chaos 503ff - hyperbolic component 189 - identical chaotic systems 286ff, 320, 329 - linear reaction-diffusion equations 591 - resonant parametric modulation 569ff Nozakki-Bekki hole 122, 126 observability 108 Occasional Proportional Feedback 460ff, 497,497,521,574ff one - dimensional Gaussian maps 279 - way coupled CML systems 57ff, 341 online time-delay 547 on-off intermittency 339ff open - flow systems 475f - loop control 277, 516ff, 555 optical feedback 493 optimal targeting 151 ff optimizing performance 401 orbit correction 635ff Oregonator model 595, 603 oscillating thermionic plasma diodes 555 Ott-Grebogi-Yorke (OGY) approach Iff, 119, 518, 565, 574, 629, 648, 652 output feedback 113f oxygen-inhibition 596ff pacemaker 413 pairs of synchronizing systems 275ff parameter modulation 293, 489, 564 partial differential equations (PDE) 44 pattern discrimination 415 pendulum 616ff, 638 period doubling 526ff period tripling s 567 periodic - array of pinnings 92ff - orbit of higher period 1 Iff - orbit theory 205ff - windows 53, 181 phase - control 503 - defect 410 - of a chaotic oscillator 309ff

- synchronization 292, 305ff, 312ff, 318ff, 330 photosensitive BZ reaction 610 Pierce-diode 524ff pinning - control 90f - gain 48 plasma diodes 523ff pole placement method 13ff Pomeau-Manneville route 567 predictability 615ff - contours 624ff - portraits of the pendulum 618 production lines 387ff pruning fronts 211 pulsed - control 497 - corrections 496f quasi - continuous control 649ff, 656, 673 - periodic 307,542 range of cooperation 506 reaction-diffusion systems 134ff recurrent points 663 recycling measures of chaos 212ff resonance attractors 608, 609 response system 275 restriction for delayed feedback 31 Riccati equation 96ff riddled basins 252ff, 255 Roessler - attractor 310,319,528 - circuits 246,249 routes to chaos 567ff Ruelle resonances 223 Ruelle-Takens-Newhouse scenario 551, 567, 568 saddle-node curve 126 saturable absorber 491 f secure communication 70, 71, 73 semiconductor lasers 493 self-organized order 399ff Shinriki oscillator 353 sigmoidal firing function 419 signal-to-noise ratio 538, 566 single channel feedback 607 single mode CO 2 laser 488ff

Index

singular value decomposition 651 skeleton for the attractor 316 space-time chaos 322 spatially extended chaotic systems 87ff, 284ff, spatiotemporal - chaos 43ff, 68ff, 473ff - dynamics of control 548ff - temporal model 365ff, 375ff spin - systems 563ff - waves 565ff spiral waves 592ff sporadic driving 28Iff stability of plane waves 125 stabilizability condition 90 stabilization of unstable steady states 500ff stabilizing - a fixed point 5ff - a metastable state 185 stable manifold 145,183 state - estimate 104 - feedback 108ff - reconstruction 103ff statistical approach 313f steady state control 94ff stochastic resonance 516ff, 555 stroboscopic surface of section 16 strong synchronization 330 subharmonic - entrainment 288 - modulation 504,505 subharmonics 489 Suhl threshold 566 suppressing spatiotemporal chaos in CML 46ff suppression - of chaos 533f - of chaos by resonant perturbations 516 - of spin-wave chaos 57Iff switching between periodic orbits 150f switching time 151 symbol recognition 415 synchronization 271 ff - circuit 239ff - in chaotic systems 229ff - in data 322 - manifold 332f, 357 - of periodic oscillations 306ff

колхоз 6/2/06

691

- region 312 synchronized arrays 25 If synchronizing spatiotemporal chaos 71 taming turbulence 550ff tangent space 667 targeting 17, 14 Iff Tauberian theorems 223 temporal chaos 119 thalamic pacemaker 412 thermionic diode 5 31 ff time - delay coordinates 645ff - delayed feedback control 2Iff, 578ff - series analysis 344ff - to achieve control 7ff - to reach target 165ff topological defects 119ff, 126ff topological entropy 169, 198 Toyota Sewn Products Management System 388 trace formulas 213 tracking 67 5 ff, 677 transient chaos 53 transverse manifold 232ff traveling waves 121,125ff tree-targeting algorithm 148ff TSS production lines 389ff turbulent magnetized plasmas 514 two-dimensional Ginzburg-Landau equation 132f two-way mutual coupling 233 two-worker production 395 type-I intermittency 308 uncontrollable modes 93 uni directional coupling 238, 273, 290 unstable manifold 183 unstable periodic orbits 1, 623ff ventricular fibrillation 428ff, 431 voltage sensitive dye staining 431 weak and strong synchronization weak phase turbulence 550 winding number 307, 308 yttrium iron garnet 565 YAG laser 489,498,506

330ff