SIAM Journal on Applied Dynamical Systems ~ 2009 Volume 8, Issue 2, pp. 527-675

c 2009 Society for Industrial and Applied Mathematics SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 8, No. 2, pp. 527–553 N...

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c 2009 Society for Industrial and Applied Mathematics

SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 8, No. 2, pp. 527–553

Nonlinear Drillstring Dynamics Analysis∗ Christophe Germay†, Nathan Van de Wouw‡, Henk Nijmeijer‡, and Rodolphe Sepulchre† Abstract. This paper studies the dynamical response of a rotary drilling system with a drag bit, using a lumped parameter model that takes into consideration the axial and torsional vibration modes of the bit. These vibrations are coupled through a bit-rock interaction law. At the bit-rock interface, the cutting process introduces a state-dependent delay, while the frictional process is responsible for discontinuous right-hand sides in the equations governing the motion of the bit. This complex system is characterized by a fast axial dynamics compared to the slow torsional dynamics. A dimensionless formulation exhibits a large parameter in the axial equation, enabling a two-time-scales analysis that uses a combination of averaging methods and a singular perturbation approach. An approximate model of the decoupled axial dynamics permits us to derive a pseudoanalytical expression of the solution of the axial equation. Its averaged behavior inﬂuences the slow torsional dynamics by generating an apparent velocity weakening friction law that has been proposed empirically in earlier work. The analytical expression of the solution of the axial dynamics is used to derive an approximate analytical expression of the velocity weakening friction law related to the physical parameters of the system. This expression can be used to provide recommendations on the operating parameters and the drillstring or the bit design in order to reduce the amplitude of the torsional vibrations. Moreover, it is an appropriate candidate model to replace empirical friction laws encountered in torsional models used for control. Key words. drillstring dynamics, discontinuous delay diﬀerential equations, stick-slip vibrations AMS subject classifications. 34C15, 34C29, 34C60, 34D15, 37M99 DOI. 10.1137/060675848

1. Introduction. Self-excited vibrations are phenomena commonly observed in rotary drilling systems used by oil industries. According to down-hole measurements [16], drilling systems permanently experience torsional vibrations, which often degenerate into stick-slip oscillations. These oscillations are characterized by stick phases, during which the rotation stops completely, and slip phases, during which the angular velocity of the tool increases up to two times the nominal angular velocity. Stick-slip oscillations are an important cause for drillstring failures and drag bit breakages. In order to reduce the costs of failures, considerable research eﬀort has been dedicated in recent years to suppressing the large torsional vibrations. Diverse strategies, both active and passive, have been proposed in the literature to compensate for stick-slip vibrations; see [11, 13, 14, 19]. Control strategies usually operate at the ground surface by regulating the torque delivered to the drilling system or by adapting ∗

Received by the editors November 24, 2006; accepted for publication (in revised form) by J. Keener November 5, 2008; published electronically April 10, 2009. This work was supported by the Walloon Region and FSE agency (European Social Funding). http://www.siam.org/journals/siads/8-2/67584.html † Department of Electrical Engineering and Computer Science, University of Liege, BAT. B28 Syst`emes et mod´elisation, Grande Traverse 10, B-4000 Li`ege, Belgium ([email protected], [email protected]). ‡ Mechanical Engineering, Dynamics and Control, University of Technology, P.O. Box 513, WH 0.131, 5600 MB Eindhoven, The Netherlands ([email protected], [email protected]). 527

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the weight-on-bit. Most of these studies rely on one- or two-degree-of-freedom (DOF) models that account for the torsional dynamics only. The oscillation mechanism arises from the friction model, which empirically captures the bit-rock interaction. The most common friction models include (i) velocity weakening laws as in [3, 4], (ii) stiction plus Coulomb friction (see [11, 19]), and (iii) models including the Stribeck eﬀect (characterized by a decreasing friction-velocity map localized around zero velocity), with diﬀerent degrees of complexity such as Karnop and LuGre models [5] that can be found in [12, 13, 14]. The diversity of these diﬀerent friction models raises the question of the physical origin of the torsional vibrations. It complicates the synthesis of control laws designed to eliminate the oscillations in torsion and excludes the inﬂuence of the bit design on such vibrations. In the present paper, we undertake the analysis of a new model, proposed in [17, 18], based on a physical and geometrical modeling of the bit-rock interaction. In this model, stickslip vibrations do not result from an empirical friction model but rather from the dynamic coupling between the axial and torsional DOF of the drilling system. In this approach, the axial vibrations are sustained by the regenerative eﬀect associated with cutting. Namely, since the motion of the bit is helical, the thickness of the rock (or depth of cut) removed by a cutter at time t is aﬀected both by its own axial position and by the path of the cutter ahead. As a consequence, the cutting force depends on the current axial position of the bit and a delayed axial position of the bit. This model is consistent with studies of chattering in metal machining [7, 10, 20, 22]. The regenerative eﬀect is ultimately responsible for the coupling of the two modes of oscillations and for the existence of self-excited vibrations. A discontinuous term is present in the equations of motion due to the frictional contact taking place at the wearﬂat-rock interface. Numerical simulations of this complicated system of equations exhibit stick-slip oscillations or bit bouncing phenomena for sets of parameters consistent with quantities measured in real ﬁeld operations [17, 18]. Furthermore, an apparent bit-rock velocity weakening is recovered in the numerical simulations under certain conditions even though all the model parameters are rate-independent, including the friction coeﬃcient. It has also been shown that a key parameter related to the bit shape has a dominant inﬂuence on the existence of stick-slip torsional vibrations. A numerical analysis of the model is presented in [18]. It identiﬁes the intermittent losses of the frictional contact at the wearﬂat-rock interface as the cause of the apparent decrease of the torque with the bit angular velocity. The losses of contact also contribute to a gain in drilling eﬃciency, as an energy transfer from the frictional contact to the pure cutting process occurs. The complex and diversiﬁed numerical simulations in [18] motivate the analysis in the present paper aiming to identify the oscillation mechanisms and their parametric dependence. The proposed analysis exploits the presence of a large parameter in the axial governing equation leading to a two-time-scales separation between the fast axial dynamics and the slow torsional dynamics. The analysis uses a combination of averaging methods and a singular perturbation approach [21]. The study of the decoupled axial and torsional dynamics provides an explanation for the emergence of most of the diﬀerent dynamic regimes observed in parameter space. We also derive an approximate analytical expression of the velocity weak-

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ening law related to the physical parameters of the system. This expression can be used to provide recommendations on the operating parameters and the drillstring or the bit design in order to reduce the amplitude of the torsional vibrations. Moreover, it is an appropriate candidate to replace empirical friction laws encountered in torsional models used for control. The paper is organized as follows. Section 2 is devoted to the derivation of the mathematical model of the drilling system and its main features. In section 3, we brieﬂy present the methodology of analysis based on singular perturbation theory and the averaging method. The fast axial dynamics are analyzed in section 4. In section 5, an approximated analytical expression of the averaged axial dynamics is used in the analysis of the torsional dynamics. Section 6 shows some limitations of the two-time-scales approach. Finally, we draw some conclusions in section 7. 2. Drilling model. 2.1. Derivation of the dynamical model. A rotary drilling structure consists essentially of a rig, a drillstring, and a bit. The essential components of the drillstring are the bottom hole assembly (BHA), composed mainly of heavy steel tubes to provide a large downward force on the bit, and a set of drill pipes made of thinner tubes. For the idealized drilling system under consideration, we assume that the borehole is vertical and that there are no lateral motions of the bit. The lumped parameter model of the drillstring presented in [17, 18], which is stripped to its essential elements, consists of an angular pendulum of stiﬀness C ended with a punctual inertia I and a punctual mass M free to move axially (see Figure 1) to represent the BHA and the bit as a unique rigid body. At the top of the drillstring, an upward force Ho and a constant angular velocity Ωo are imposed. It is assumed that the weight-on-bit provided by the drillstring to the bit Wo = Ws − Ho is constant, which implies that the hook load Ho is adjusted to compensate for the varying submerged weight of the drillstring Ws . The equations of motion of the drill bit and the BHA are then given by

Figure 1. Simpliﬁed model of a drilling system.

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GERMAY, VAN DE WOUW, NIJMEIJER, AND SEPULCHRE

Figure 2. (a) Sketch of forces acting on a single cutter; (b) section of the bottom-hole proﬁle located between two successive blades of a drill bit (after [18]).

(2.1) (2.2)

I

d2 Φ(t) + C (Φ(t) − Ωo t) = −T (t), dt2 d2 U (t) = Wo − W (t), M dt2

where U , Φ and t denote vertical, angular positions of the drag bit and time, respectively. The reacting torque-on-bit T (t) and the reacting weight-on-bit W (t) originate from the process of rock destruction occurring at the bit-rock interface. The formulation of the bit-rock interface laws derives from a phenomenological model [6] of the forces acting on a single cutter of width w when removing rock over a constant depth d and constant longitudinal velocity, as sketched in Figure 2(a). The rock cutting consists of two independent processes: (i) a pure cutting process taking place at the cutting face (subscript c ) and (ii) a frictional contact process (subscript f ) along the interface between the wearﬂat of length (horizontal ﬂat surface below the cutter) and the rock. The total force on the cutter is the sum of the cutting force Fc and the friction force Ff , exerted on the cutting face and on the wearﬂat, respectively. The vertical (subscript n ) and horizontal (subscript s ) components (see Figure 2(a)) of the cutting force and the friction force are expressed as Fcs = εwd,

Fcn = ζFcs ,

Ffs = μFfn ,

Ffn = σw,

where ε is the intrinsic speciﬁc energy (the minimum amount of energy required to destroy a unit volume of rock), ζ is a number characterizing the orientation of the cutting force, μ is the coeﬃcient of friction, and σ is the maximum contact pressure at the wearﬂat-rock interface. When the wearﬂat is in conforming contact with the rock, σ is a constant parameter. Based on single cutter experiments, the value of this parameter can reasonably be assumed to be in the same range as ε [1, 2]. The distinction between cutting and friction forces is also relevant to modeling the generalized forces acting on a drill bit. The reacting torque-on-bit T and the reacting weight-on-bit W due to the operation of rock destruction account for both cutting and frictional processes,

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NONLINEAR DRILLSTRING DYNAMICS ANALYSIS

531

(2.3)

T (t) = Tc (t) + Tf (t),

(2.4)

W (t) = Wc (t) + Wf (t).

The idealized drag bit of radius a consists of n identical radial blades regularly spaced by an angle of 2π/n; see Figure 2(b). Each blade is characterized by a vertical cutting surface facing a depth of cut dn and a wearﬂat of constant width n orthogonal to the bit axis. The cumulative depth of cut of the bit is d = ndn , and the equivalent wearﬂat width for the bit is = nn . By integrating the eﬀects of all the individual cutters along the bit proﬁle, the cutting components of W and T are given by [6] Wc (t) = naζεdn (t) ,

(2.5)

Tc (t) = n

a2 εdn (t) , 2

both proportional to the depth of cut dn (t) removed at time t. When the bit experiences vibrations, the rock ridge facing the blades varies. Because of the helicoidal motion (the bottom hole proﬁle is dictated by the passage of the previous blade), the variable dn (t) is expressed by dn (t) = U (t) − U (t − tn (t)),

(2.6)

where the delay tn is the time required for the bit to rotate by an angle of 2π/n. The delay tn is the solution of the implicit equation t 2π dΦ(s) ds = Φ(t) − Φ(t − tn (t)) = . (2.7) ds n t−tn (t) A conceptual sketch is depicted in Figure 2(b). The frictional components of W and T are given by (2.8)

Wf = nan σ

(1 + sign( dU dt )) , 2

Tf = n

(1 + sign( dU a2 dt )) γμn σ , 2 2

where the parameter γ depends on the spatial orientation and distribution of the wearﬂats along the bit proﬁle [1]. The forces acting at the wearﬂat/rock are assumed constant once the wearﬂat is in conforming contact with the rock [18], i.e., when the bit moves downward dU dU dt > 0. When the bit moves upward ( dt < 0), we assume a complete loss of contact between the wearﬂat and the rock, so that the frictional components Wf and Tf vanish. Note that referring to Wf as a frictional term is a slight abuse of language since it is a reaction force. Our terminology emphasizes that both Tf and Wf arise from the frictional process. In the absence of vibrations, the nominal drilling solution ( dU dt > 0) is given by Tf + a(Wo2ζ−Wf ) (Wo − Wf ) Ωo , Uo (t) = t, (2.9) Φo = Ωo t − C aζε 2π where Tf + cut.

a(Wo −Wf ) 2ζ

is the nominal torque To and (Wo − Wf ) /aζε is the nominal depth of

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GERMAY, VAN DE WOUW, NIJMEIJER, AND SEPULCHRE

The expression of the dimensionless equations that govern the bit motion, also referred to in the literature as the threshold-type delay equations, yields (2.10)

u ¨ (τ ) = nψ [−vo (τn − τno ) − u (τ ) + u (τ − τn ) + λn g(u(τ ˙ ))] ,

(2.11)

˙ ))] − ϕ (τ ) , ϕ¨ (τ ) = n [−vo (τn − τno ) − u (τ ) + u (τ − τn ) + βλn g(u(τ τ 2π (ωo + ϕ˙ (s)) ds = , n τ −τn (τ )

(2.12)

where u(τ ) = (U − Uo )/L∗ and ϕ(τ ) = Φ − Φo represent the dimensionless axial and angular deviation to the nominal solutions, respectively. The dot denotes diﬀerentiation with respect to the dimensionless time τ = t/ I/C. The characteristic length is given by L∗ = 2C/εa2 . In the absence of torsional vibrations, ϕ = 0 and τn = τno = 2π/nωo ; in the absence of axial ˙ in (2.10) is deﬁned as vibrations, u = u (τ − τn ) = 0. The function g(u) 1 (2.13) g(u) ˙ = (1 − Sign(u˙ + vo )) . 2 Physically, the dimensionless normalized term g(u) ˙ is the complement of the normalized re˙ + Wf /nan σ = 1. acting force Wf /nan σ, i.e., g(u) The dimensionless parameters of the model (2.10)–(2.12)are the following: (i) the control parameters Wo = aWo /2ζC and ωo = Ωo I/C; (ii) the nominal dimensionless reacting force λ = nλn = na2 n σ/2ζC is proportional to the length of the wears (it is an image of the bluntness of the bit); (iii) the nominal axial bit velocity vo = ωo (Wo − λ) /2π; (iv) the lumped parameter β = μγζ characterizes the geometry of the bit; (v) the lumped parameter ψ = ζεaI/M C characterizes the drill string design. The set of equations (2.10)–(2.12) is nonlinear, coupled, and contains a state-dependent delay. Furthermore, the frictional process causes a discontinuous term g(u) ˙ in (2.10) and (2.11). The solutions of the discontinuous diﬀerential equation are deﬁned in Filippov’s sense. Filippov’s convex method [8] treats the discontinuous function g(u) ˙ as a convex set-valued mapping on the hyperplane u˙ = −vo ; i.e., Sign(x) maps 0 to the set [−1, 1]. 2.2. Stick modeling. 2.2.1. Axial stick. • A stick phase may occur in the axial dynamics, only when the axial vibrations cause the axial velocity U˙ to become zero for a limited period of time although the bit is ˙ > 0 ≡ (ϕ(τ still rotating forward Φ ˙ ) > −ωo ). We refer to this situation as the axial stick phase during which the axial position of the bit is stationary (U = const). It corresponds to the situation where the applied weight-on-bit Wo can be compensated by the cutting force Wc and a portion of the reacting force Wf . In dimensionless form, the latter is upperbounded by λ, and the mathematical conditions for an axial stick phase are as follows: λn (2.14) (1 − Sign(0)) 0 ∈ nψ −vo (τn − τno ) − u (τ ) + u (τ − τn ) + 2 λn 2 (2.15) ∈ [−1, 1] . ⇒ −vo (τn − τno ) − u (τ ) + u (τ − τn ) + 2 λn

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533

The value of g(u) ˙ during the axial stick phase is given by (2.16)

g(u˙ = −vo ) =

vo (τn − τno ) + u (τ ) − u (τ − τn ) . λn

2.2.2. Torsional stick. The torsional model (2.11) is valid as long as ϕ˙ > −ωo , which corresponds to a slip phase. The torsional vibrations may become so severe that the sign of the velocity is reversed. In this case, the magnitude of the frictional torque is assumed to be suﬃcient to restrain the bit from rotating backward. The system then enters a stick phase during which the bit sticks to the rock. The torsional stick phase is modeled by ˙ = 0 ≡ (ϕ(τ (2.17) Φ ˙ ) = −ωo ) , (2.18) U˙ = 0 ≡ (u˙ = −vo ) . The stick equations (2.17)–(2.18) are substituted into (2.10) and (2.11) until the right-hand side of (2.11) becomes positive and the bit enters a new slip phase. Physically, since the rotation of the drill pipes continues at the surface, the torque applied by the drillstring onto the BHA builds up until its magnitude is suﬃcient to overcome the reacting torque, causing the bit to rotate. 2.3. Bit bouncing. Model (2.10)–(2.12) loses its validity when the dimensionless depth of cut (2.19)

δ = n [vo τn + (u (τ ) − u (τ − τn ))]

becomes negative. This event will be referred to as bit bouncing, which is detrimental for the bit. It occurs when the bit experiences suﬃciently large axial vibrations to disengage completely from the rock formation. The objective of the design is to avoid it. 3. Two-time-scales analysis. In view of the complexity of the model, its mathematical analysis is not straightforward. However, two clearly distinct time scales (see Figure 11) emerge due to the magnitude of the parameter ψ, which is typically of order 102 –103 . In the remainder of the paper, we consider the model (2.10)–(2.12) in the singularly perturbed form (3.1) (3.2) (3.3)

¨ (τ ) = − [vo (τn − τno ) + u(τ ) − u (τ − τn ) − λn g(u˙ (τ ))] , 2 u

¨ (τ ) , ϕ¨ (τ ) + ϕ (τ ) = −n (1 − β) λn g(u˙ (τ )) − 2 u τ 2π , (ωo + ϕ˙ (t)) dt = n τ −τn (τ )

√ where 1/ nψ = > 0 is a small parameter. In this conﬁguration, the axial dynamics will be considered as “fast” dynamics, and the torsional dynamics as “slow” dynamics. Indeed, the characteristic dimensionless time of the torsional oscillations √ is τt ≈ 2π, while the characteristic dimensionless time of the axial oscillations is τa ≈ 2π/ ψ. In the classical singular perturbation theory, the fast system can be studied independently by freezing the slow variables. Commonly called the boundary layer system, it consists of

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trajectories that converge exponentially to a slow manifold. Upon convergence of the boundary layer, solutions evolve on the so-called reduced model that provides a good estimate of the slow dynamics. In the particular system (3.1)–(3.3), solutions of the boundary layer converge towards a family of limit cycles that depends on the parameters of the system. Nonetheless, the authors of [21] provide a uniﬁcation framework that combines averaging methods and singular perturbation approach to handle systems with complex fast dynamics. Although the theory has not been fully developed for our particular type of equation (threshold type with discontinuous terms), we adopt a similar approach to analyze the dynamics of the complete system (3.1)–(3.3). In the next section, the fast axial dynamics of (3.1) are studied under the simplifying assumption that the delay τn is a ﬁxed parameter. Under certain conditions, numerical simulations show that stable oscillations in u˙ are observed. We will propose an analytical approximation of the fast axial solution. In section 5, this approximation will be used to study the slow torsional dynamics by means of averaging methods. 4. Axial dynamics. In order to observe the periodic oscillations in u˙ as a true limit cycle, a proper coordinate transformation is in place, where the new set of state variables must be of zero derivative mean over the limit cycle. Let us assume that periodic oscillations exist in u. ˙ Then, we may write that 1 τ +τa /2 u ¨ (s) ds = 0. (4.1) ¨ u(τ )a = τa τ −τa /2 In the simplest case where τn = τno , which is not contradictory with the existence of an axial limit cycle as will be shown below, the equality (4.2)

u(τ ) − u (τ − τn )a = λn g(u˙ (τ ))a

is obtained by averaging (3.1) with (4.1) over one axial limit cycle, where the mean value of the only nonlinear term g(u˙ (τ ))a in (3.1) is nonzero. Therefore, a drift of the solution in the u-direction with a velocity depending on λn g(u˙ (τ ))a exists. Note that g(u˙ (τ ))a is a priori unknown as it is a function of u˙ (τ ); i.e., it depends on the solution of (3.1). For this reason, we introduce a new set of state variables of zero derivative mean over an axial limit cycle τ ) = u (¯ τ − τ¯n ) − u (¯ τ ) − v¯o (¯ τn − τ¯no ) , w1 (¯ τ ) = u˙ (¯ τ ) / nψ, w2 (¯ √ √ √ that evolves √ in the fast time scale τ¯ = τ nψ and where v¯o = vo / nψ, τ¯n = τn nψ, and τ¯no = τno nψ. Physically, the new variable w1 represents the negative discrepancy between the dimensionless form of the depth of cut and the nominal one. In the new variables, the axial equation (3.1) admits the state-space representation (4.3)

τ ) = w2 (¯ τ − τ¯n ) − w2 (¯ τ) , w ˚1 (¯

(4.4)

τ ) = w1 (¯ τ ) + λn g¯(w2 (¯ τ )), w ˚2 (¯

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NONLINEAR DRILLSTRING DYNAMICS ANALYSIS

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with 1 (1 − sign (w2 + v¯o )) . 2 The limit cycle observed in the new state (w1 , w2 ) space will be understood as the axial limit cycle. The system (4.3)–(4.4) is referred to as the fast system. The round dot denotes diﬀerentiation with respect to the stretched time τ¯. The initial condition required to solve the inﬁnite dimensional axial dynamics (3.1) is the function u(·) on the time interval [−τn , 0]. In the new variables, this initial condition translates into the function w2 (·) on the time interval [−¯ τn , 0] and 0 vo (¯ τn − τ¯no ) − w2 (t)dt. (4.6) w1 (0) = −¯ (4.5)

g¯(w2 ) =

−¯ τn

vo (¯ τn − τ¯no ) /¯ τn ). PhysThe equilibrium solution of (4.3)–(4.4) with (4.6) is (w1 , w2 ) = (0, −¯ ically, it corresponds to the rigid translation u˙ = −vo (τn − τno ) /τn , u (τ − τn ) − u (τ ) = vo (τn − τno ). In a neighborhood of this equilibrium, the reacting force Wf at the wearﬂat-rock interface vo ), as its dimensionless complement g¯(w2 ) is 0. The dynamics w ˚1 (¯ τ) = is permanent (w2 > −¯ τ − τ¯n ) − w2 (¯ τ ) and w ˚2 (¯ τ ) = w1 (¯ τ ) are then linear. Stability of the equilibrium is thus w2 (¯ determined by the location of the roots of the characteristic function P (s) = s2 + 1 − e−s¯τn .

(4.7)

√ This function does not have roots in the right half of the complex plane when τ¯n < π/ 2. Two complex conjugated roots pass from the left half-plane to the right half-plane at the critical √ √ the value τ¯n = π/ 2. They remain in the open right half-plane for τ¯n > π/ 2. Consequently, √ vo (¯ τn − τ¯no ) /¯ τn ) is exponentially stable when τ¯n < π/ 2, equilibrium point (w1 , w2 ) = (0,√−¯ √ = π/ 2, and unstable when τ¯n > π/ 2. In typical ﬁeld operations, marginally stable when τ¯n √ the delay satisﬁes τ¯n > π/ 2, meaning that the equilibrium solution is unstable. The growth of the solutions of (4.3)–(4.4) away from this unstable equilibrium is limited by the nonlinear friction. Under certain conditions, this mechanism is responsible for the existence of an axial stick-slip limit cycle. The next section provides a qualitative description of this limit cycle in the phase plane (w1 , w2 ). 4.1. Analysis of the axial limit cycle. When the equilibrium of (4.3)–(4.4) is unstable √ (¯ τn > π/ 2), numerical simulations indicate that the solutions of (4.3)–(4.4) either grow unbounded (ultimately leading to bit bouncing as described in section 2.3) or converge to a limit cycle that ﬁts the qualitative description of Figure 3. In the state space (w1 , w2 ), the axial limit cycle illustrated in Figure 3 can be decomposed into three diﬀerent phases: a slip phase, a stick phase, and a sliding phase. By choosing ˚1 (0) > 0, the temporal arbitrarily the origin of time as w1 (0) = 0, w2 (0) = −vo , and w sequence of these three phases during one period [0, τ¯a ] of the cycle is as follows: τ ) > −vo . As a (i) The slip phase (¯ τ ∈ [0, τ¯k ]) is characterized by the condition w2 (¯ consequence, g¯(w2 ) = 0, and the solution obeys the unstable linear dynamics (4.8)

τ ) = w2 (¯ τ − τ¯n ) − w2 (¯ τ) , w ˚1 (¯

(4.9)

τ ) = w1 (¯ τ) . w ˚2 (¯

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GERMAY, VAN DE WOUW, NIJMEIJER, AND SEPULCHRE

Figure 3. Axial limit cycle in the space (w1 , w2 ) and the corresponding evolution of w1 , w2 , and λn g¯(w2 ) in time.

During this phase, w2 (¯ τ − τ¯n ) is initially greater than −vo (due to the previous slip τ ); i.e., w1 (¯ τ ) and w2 (¯ τ ) increase according to (4.8) and phase) and larger than w2 (¯ τ ) becomes larger than w2 (¯ τ − τ¯n ), w1 (¯ τ ) decreases and (4.9), respectively. When w2 (¯ τ ) to decrease. During that lapse of time, w2 (¯ τ − τ¯n ) becomes negative, causing w2 (¯ τ − τ¯n ) = −vo ). When w2 (¯ τ ) becomes equal has reached the previous stick phase (w2 (¯ to −vo , the system enters the stick phase. vo (ii) The stick phase (¯ τ ∈ [¯ τk , τ¯n ]) is characterized by constant axial velocity w2 = −¯ and constant friction λn g¯(w2 ) = −w1 (¯ τk ). The projection of the solution in the phase plane (w1 , w2 ) has shrunk to one point. This phase will last until the delayed solution τ − τ¯n ) enters a slip phase described in (i). It happens when τ¯ = τ¯n . Note that w2 (¯ the existence of the stick phase, necessary to observe the axial stick-slip limit cycle, τk ) ≤ λn and τ¯k ≤ τ¯n . relies on the conditions 0 ≤ −w1 (¯ τ ) = −¯ vo ), (iii) During the sliding phase (¯ τ ∈ [¯ τn , τ¯a ]), the axial velocity is still at rest (w2 (¯ τ − τ¯n ) > −¯ vo when τ¯ > τ¯n , causing w1 (¯ τ ) to but the delayed axial velocity w2 (¯ τ ) = −¯ vo in the state space (w1 , w2 ). The term λn g¯(w2 (¯ τ )) slide along the line w2 (¯ τ )) can decreases accordingly until it reaches 0, i.e., the minimum value that g¯(w2 (¯

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NONLINEAR DRILLSTRING DYNAMICS ANALYSIS

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attain. Then a new cycle starts. The reader should note that the stick phase deﬁned in section 2.2.1 (U and U˙ constant) vo and w˙ 1 > 0) consists of the stick phase (w1 and w2 constant) and the sliding phase (w2 = −¯ in the new variables. Numerical simulations show also the existence of the axial limit cycle when −w1 (¯ τk ) is slightly greater than λn . The trajectories pass below the line w2 = −¯ vo and vo somewhat later. We do not consider this case here since it arises in a stick onto w2 = −¯ very small region of the set of parameters. In the next section, we derive an approximate solution for this limit cycle that allows for an analytic prediction of the inﬂuence of key system parameters on this limit cycle. These analytical predictions are compared to numerical solutions obtained from solving the fast dynamics by a shooting method [15]. 4.2. Analytical approximation of the axial limit cycle. Slip phase: To approximate the time evolution of the limit-cycle solution shown in Figure 3, τ − τ¯n ) = −¯ vo ) during the slip we assume that the delayed axial velocity is zero (w2 (¯ phase τ¯ ∈ [0, τ¯k ]. The resulting linear system (4.10)

vo − w2 , w ˚1 = −¯

(4.11)

w ˚2 = w1 ,

vo , which yields is solved for the initial condition w2 (0) = −¯ (4.12)

τ ) = C1 cos τ¯, w1 (¯

(4.13)

τ ) = C1 sin τ¯ − v¯o , w2 (¯

τ¯ ∈ [0, τ¯k ] .

Stick phase: The approximated solution (4.12)–(4.13) enters the stick phase at τ¯k = π when τk ) becomes equal to −¯ vo . The dimensionless term λn g¯(w2 ) associated with the w2 (¯ τk ) = C1 according to (4.12)) frictional process increases suddenly (λn g¯(w2 ) = −w1 (¯ τ ) = 0 during the stick phase. The necessary condition for observing a such that w ˚2 (¯ τ − τ¯n ) = w2 (¯ τ ) = −¯ vo ⇒ stick phase is w2 (¯ √ τ¯n > π, which satisﬁes the condition of τk ) ≤ λn . instability of the axial equilibrium (¯ τn > π/ 2), and −w1 (¯ Sliding phase: The solution in the sliding phase is given by (4.14)

τ ) = −C1 cos (¯ τ − τ¯n ) , w1 (¯

(4.15)

τ ) = −¯ vo . w2 (¯

τ ) = C1 , i.e., at time The (approximate) solution returns to the initial state when w1 (¯ τ¯a = π + τ¯n . The free constant C1 in (4.12)–(4.13) is determined from the initial condition 0 w2 (t) + v¯o dt. w1 (0) = C1 = v¯o τ¯no − −¯ τn

τ ) = −¯ vo over [−¯ τn , 0], we obtain Since w2 (¯ (4.16)

C1 = v¯o τ¯no ,

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538

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Figure 4. Top: Limit cycle of w1 and w2 with τ¯. Bottom: Approximated limit cycle of w1 and w2 in terms of τ¯.

which corresponds to the nominal depth of cut per blade. The time evolution of the axial limit cycle solution and its approximation are illustrated in Figure 4. The approximate solution of the limit cycle provides the following predictions: • The period of the limit cycle is estimated as τ¯a = π + τ¯n . It grows linearly with the delay and is independent of the parameters v¯o and τ¯no . • The amplitude of the limit cycle is estimated as C1 = v¯o τ¯no , regardless of the delay τ¯n . It must be less than λn to observe a stick phase, which is essential to the existence of the axial limit cycle, as otherwise bit bouncing will eventually take place. • In the next section, we will see that the axial dynamics inﬂuence the torsional dynamics τ )) over one axial limit cycle. By using (2.16) through the average value of g¯(w2 (¯ τ )) is diﬀerent from zero only during the stick and the sliding with (4.5), λn g¯(w2 (¯ τ ). Figure 4 illustrates that −w1 (¯ τ ) takes both phase where it takes the value −w1 (¯ positive and negative values over one period of the approximate limit cycle, which is in contrast to the (physical meaningful) property that g¯ should take only positive values in the exact model. To correct for this artifact, we compute the averaged g (w2 (¯ τ ))a using the approximate limit cycle, but we restrict the frictional term λn ¯ τ )) only, i.e., on the time-interval interval of integration to the positive values of g¯(w2 (¯ [0, τ¯n + π/2], yielding 1 τ¯a (4.17) g(w2 (¯ τ ))a = λn g¯(w2 (s))ds λn ¯ τ¯a 0 τ¯n +π/2 v¯o τ¯no (¯ τn − π + 1) 1 (4.18) . w1 (s) ds = ≈− τ¯n + π π (¯ τn + π)

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Figure 5. Top: Period of the axial limit cycle for diﬀerent values of τ¯n , v¯o , and τ¯no . Middle: Maximum vo = 0.063, τ¯no = 6.8), v¯o (¯ τn = 6.8, τ¯no = 5.8), and τ¯no (¯ vo = 0.043, values of g¯(w2 ) with respect to τ¯n (¯ τ¯no = 10.18), respectively (these results are obtained by solving the fast system with the shooting method). Bottom: Comparison between the results obtained with the approximative and the fast systems.

It is linear in the parameters v¯o and τ¯no . It is a monotonic function of the delay, τ ))a becomes but its dependency saturates as the delay increases. Note that ¯ g (w2 (¯ zero when τ¯n =√π − 1, which almost agrees with the condition of stability of the axial equilibrium π/ 2. Figure 5 illustrates the excellent match between these analytical predictions and the numerical results obtained from a shooting method applied to the exact model. We see that τ¯n does not aﬀect the amplitude of the periodic orbit but varies linearly with v¯o and τ¯no as predicted by the approximated system. The period of the orbit is inﬂuenced only by τ¯n , and the average value is an excellent approximation of the solutions obtained from the full model.

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540

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When the bit angular velocity is small, the delay is large, and the system spends most of the time in the stick phase where the frictional contact force between the wearﬂat and the g (w2 (¯ τ ))a with τ¯n . rock is constant. This corresponds in Figure 5 to the increase of λn ¯ 4.3. Bit bouncing. Bit bouncing occurs when both the equilibrium solution and the limit cycle solution of the axial dynamics are unstable. The amplitude of the vibrations grows exponentially, and the bit ultimately loses contact with the rock completely. The existence of the axial limit cycle discussed in the previous section requires the condition (4.19)

0 ≤ −w1 (¯ τk ) ≤ λn .

(4.20)

Wo /λ < 2,

√ We use condition (4.19) together with the parametric condition τ¯n > π/ 2 (which guarantees that the equilibrium is unstable) as a prediction of the parametric range in which a stable axial limit cycle exists. τk ) = v¯o τ¯no in the approximation (4.12)–(4.16), equation (4.19) translates Because −w1 (¯ into the parametric condition v¯o τ¯no < λn , which, rewritten in the original parameters of the model, is equivalent to the condition

where we recall that Wo is related to the applied weight-on-bit and λ is proportional to the length of the wears (bluntness of the bit). This prediction is in good agreement with the stability map in Figure 6, numerically computed from the full model (2.10)–(2.12), as shown in [9]. To draw the map, we simulated 300 bit revolutions for each pair of values (Wo , λ). If the depth of cut becomes negative, the computation is stopped and the corresponding value (Wo , λ) is given the dark grey color; otherwise a light grey color is chosen. The black region indicates parameter values for which the bit is not drilling but is only in frictional contact. In that case, the dimensionless applied weight-on-bit Wo does not overcome the nominal dimensionless frictional term λ (i.e., the weight-on-bit transmitted by the wearﬂats when the bit is drilling). In the absence of torsional vibrations, i.e., when ϕ˙ = 0, the theoretical analysis predicts no bit bouncing when √ the axial equilibrium is stable (¯ τno = τ¯n < π/ 2 ⇔ ωo > ωos = 2 2ψ/n) or when Wo /λ < 2. These predictions are mainly illustrated in Figure 7, although the numerical results are obtained from the complete system (2.10)–(2.12) where the bit experiences torsional vibrations. The three diﬀerent axial regimes (stable equilibrium, stable limit cycle, bit bouncing) are represented in the parametric plane (ωo , Wo ). We conclude that the predictions of the analytical approximation of the axial limit cycle are accurate in detecting the transitions, such as stability of the axial equilibrium, stability of the axial limit cycle, and bit bouncing. 5. Torsional dynamics. The reduced (slow) dynamics governing the torsional motion is obtained by assuming the following: 1. The slow variables are constant over the period of oscillations of the axial vibrations: 1 τ +τa /2 f (τ + s) ds ≈ f (τ ) , (5.1) f (τ )a = τa τ −τa /2 √ where f can be either ϕ, ϕ, ˙ ϕ¨ or τn and τa = τ¯a / nψ.

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Figure 6. Map of stability for values of (Wo , λ) (n = 6, β = 0.276, ωo = 4, ψ = 63.1); after [9].

Figure 7. Numerical map of stability for diﬀerent rotational speeds ωo and ψ when n = 6, β = 0.43, and λ = 4.2; after [18].

2. The mean axial acceleration along a periodic solution of the axial limit cycle is zero: (5.2)

¨ u (τ, τn )a = 0.

By averaging (3.2) over a period of the axial limit cycle τa (τn ), the reduced model yields

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542

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Figure 8. Comparison between the real delay τn and the approximation

(5.3)

2π n(ωo +ϕ) ˙

during three stick-slip cycles.

ϕ¨ (τ ) + ϕ (τ ) = −n (1 − β) λn g(u˙ (τ, τn ))a .

Together with (3.3), this equation forms the reduced system. 5.1. Analytical approximation of the torsional dynamics. To facilitate the analysis of the slow dynamics, we derive an explicit relationship linking τn (τ ) and the bit angular velocity ϕ˙ (τ ). Most of the time, the delay τn is of the order of 10−1 , which is small compared to the characteristic time 2π of the torsional oscillations. For this reason, we treat the torsional variable ϕ˙ (τ ) as a constant over the delay τn (τ ), and its expression yields τ 2π (5.4) (ωo + ϕ˙ (t)) dt ≈ τn (τ ) (ϕ˙ (τ ) + ωo ) = n τ −τn (τ ) 2π (5.5) . ⇔ τn (τ ) ≈ n (ωo + ϕ˙ (τ )) It should be noted that this approximation is no longer valid when the bit is in the torsional stick phase (i.e., when ωo + ϕ˙ (τ ) ≈ 0). See Figure 8 for a comparison between the real delay τn and the approximation in (5.5). By combining the results obtained in section 4.2 and the approximation (5.5), we can construct an analytical approximation Ga (ωo + ϕ˙ (τ )) of g(u˙ (τ, τn ))a that is valid when the axial limit cycle exists and is stable (¯ τn > π − 1 ⇔ ωo + ϕ˙ (τ ) < ωos and Wo /λ < 2): √ √ (Wo − λ) 2π ψ − (π − 1) n (ωo + ϕ˙ (τ )) √ . (5.6) Ga (ωo + ϕ˙ (τ )) = √ λ 2π ψ + π n (ωo + ϕ˙ (τ )) By substituting Ga (ωo + ϕ˙ (τ )) into (5.3), we obtain an approximate equation governing the slow torsional vibrations (5.7)

ϕ¨ (τ ) + ϕ (τ ) = −n (1 − β) λn Ga (ωo + ϕ˙ (τ ))

that becomes autonomous and nonlinear because of the term Ga (ωo + ϕ˙ (τ )). This equation (or reduced model) will be helpful to characterize the origin and the nature of torsional vibrations. The following observations are drawn from (5.6):

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543

Figure 9. Left: Ga with respect to ψ when n = 6, ωo + ϕ(τ ˙ ) = 4, Wo = 8, and λ = 5.4. Right: Ga (ωo + ϕ(τ ˙ )) ˙ ) for three diﬀerent values of ω (1, 3, 6). The bifurcation between the stable axial limit with respect to ωo + ϕ(τ o cycle and the stable axial equilibrium occurs at 2 2ψ/n = 9.17 in this example.

Figure 10. Torque averaged over several bit revolutions for diﬀerent values of the rotational speed obtained from numerical simulations of the complete system.

• Ga (ωo + ϕ˙ (τ )) is monotonically increasing with the parameter ψ (see left panel in Figure 9) for all other parameters ﬁxed. • In Figure 9, Ga (ωo + ϕ˙ (τ )) is plotted for diﬀerent values of ωo + ϕ˙ (τ ). When ωo + ϕ˙ (τ ) > ωos , then Ga (ωo + ϕ˙ (τ )) = 0, which corresponds to the exponential local stability of the axial equilibrium point. The monotonic decrease of Ga (ωo + ϕ˙ (τ )) with ωo + ϕ˙ (τ ) recovers the so-called velocity weakening law, often empirically assumed to be an intrinsic property of the bit-rock interaction (see [3, 4]) and the essential cause of the torsional vibrations. In the present model, the velocity weakening law is a consequence of the axial vibrations and more precisely of the decreases of the contact forces occurring at the wearﬂat-rock interface. The velocity weakening eﬀect in the torque is further illustrated in Figure 10. vo τ¯no /λ, which is • The function Ga (ωo + ϕ˙ (τ )) is directly proportional to Wo /λ − 1 = n¯ the cumulative nominal dimensional depth of cut scaled by the dimensionless frictional contact. 5.2. Local stability analysis. By using the approximate analytical expression of Ga in (5.6), we can perform a local stability analysis of the equilibrium point of the reduced model.

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Figure 11. Stick-slip torsional and axial vibrations. The frequency of the torsional dynamics ϕ˙ diﬀers strongly from the axial dynamics u˙ (n = 6, ψ = 63.1, β = 0.276, ωo = 4, Wo = 3.44).

Jacobian linearization of the model yields ϕ¨ (τ ) − (1 − β) λGa (ωo ) ϕ˙ (τ ) + ϕ (τ ) = − (1 − β) λGa (ωo ) ,

(5.8) with

Ga

(5.9)

√ dGa (ω) (Wo /λ − 1) 2 nψ (ωo ) = = − (2π − 1) √ 2 . dω ω=ωo π 2 nψ + nωo

The bits commonly used in the petroleum industry are characterized by β < 1. Since Wo > λ (a necessary condition for drilling), we can conclude the following: 1. The equilibrium point of (5.7) is given by (5.10)

ϕ (τ ) = − (1 − β) λGa (ωo )

and ϕ˙ (τ ) = 0.

It thus depends on the fast axial dynamics (Ga (ωo )). √ 2. The derivative Ga (ωo ) is always negative when τ¯n ≈ 2π nψ/n (ωo + ϕ˙ (τ )) > π − 1, meaning√that equilibrium point in torsion is unstable. 3. τ¯n ≈ 2π nψ/n (ωo + ϕ˙ (τ )) < π − 1, Ga (ωo ) = 0, and the reduced model reduces to a harmonic oscillator. Then, the equilibrium point of the reduced model is marginally stable. Marginal stability of the reduced model gives rise to the quasi-limit cycle discussed in section 5.4. 5.3. Large torsional vibrations. Large torsional vibrations are observed when the axial dynamics exhibits a stable limit cycle with stick and slip phases. These torsional oscillations are characterized by a fast growth of the amplitude of the torsional vibrations and under certain conditions a large torsional limit cycle that exhibits alternating stick (ϕ˙ = −ωo ) and slip phases (ϕ˙ > −ωo ). The dominant frequencies occurring in the axial and torsional modes diﬀer strongly (see Figure 11). The local stability analysis of the analytical approximation of the reduced model predicts √ that the torsional oscillations will appear when 2π nψ/n (ωo + ϕ˙ (τ )) > π − 1. The numerical

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Figure 12. Comparison between the standard deviation of the solution ϕ˙ of the full model and the approximated reduced model with n = 6, β = 0.3, ψ = 63.1, Wo − λ = 2.6, ωo = 3 obtained through numerical simulation with a simulation time T = 5.

solutions of the analytical approximation of the torsional system and the numerical solutions of the full model are compared by measuring the standard deviation of the bit angular velocity around its nominal value, deﬁned as std(ϕ) ˙ =

1 T

T

ϕ˙ 2 (t)dt.

0

In Figure 12, we have arbitrarily chosen initial conditions at rest for both systems (ϕ(0) = ϕ(0) ˙ = 0). The deviation of the initial condition from the equilibrium is thus given by − (1 − β) λGa (ωo ) (see (5.10)). The eﬀect of the magnitude of the initial deviation is mainly observed when the simulation time T is relatively short (T = 5). The full model and the reduced model are in good agreement except for the so-called antiresonance zone, which will be brieﬂy discussed in section 6.1. We see, for instance, that the results in Figure 12(b) and 12(c) are consistent with the results presented in Figure 9. In Figure 13, the simulations are initialized near the equilibrium value of the reduced model (see (5.10)). The initial condition on the bit rotational velocity ϕ˙ (0) is set to 10−3 ωo in order to trigger the oscillations in the analytical approximation of the torsional dynamics. There again, we observe coherent numerical results between the two models.

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Figure 13. Comparison between the standard deviation of the solution ϕ˙ of the original and slow systems with n = 6, β = 0.3, ψ = 63.1, Wo − λ = 2.6, ωo = 3 obtained through numerical simulation with a simulation time T = 130.

Figure 14. Diﬀerent types of evolution of the torsional vibrations when n = 6, ψ = 50, Wo − λ = 2, ωo = 5; (a) β = 0.3, (b) β = 1.3; after [18].

It should be emphasized that the amplitude of the torsional limit cycle in the full model may depend on the initial condition. Although the numerical simulations usually match the rate of growth of the oscillations predicted with the reduced equation (5.7), the torsional limit cycles may diﬀer slightly. Nevertheless, the vibrations remain large and can be considered as detrimental for the drillstring. Our analysis identiﬁes the mean eﬀect of axial vibrations Ga (ωo + ϕ˙ (τ )) as a critical damping term in the torsional dynamics. Inspection of Ga (ωo + ϕ˙ (τ )) provides simple recommendations for the drillers: a parameter β > 1 guarantees positive damping, i.e., the absence of stick-slip vibrations, as seen in Figure 14. Furthermore, geometric parameters should be designed to minimize the term (1 − β) λGa (ωo ).

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Figure 15. Quasi-limit cycle in the phase plane (ϕ, ϕ). ˙ The parameters are n = 6, ψ = 63.1, β = 0.276, ωo = 10, Wo − λ = 3.44.

5.4. Quasi-limit cycle. A quasi-limit cycle is observed when the axial equilibrium is stable and when the function Ga (ωo + ϕ˙ (τ )) in (5.6) vanishes, resulting in the reduced dynamics ϕ¨ (τ ) + ϕ˙ (τ ) = 0. Figure 15 depicts a cross section of the phase diagram (ϕ, ϕ) ˙ in this regime. Each loop takes about 2π units of dimensionless time. The amplitude of the torsional limit cycle strongly depends on the initial conditions. Although the right-hand side of (5.7) is zero, we observe a slow ampliﬁcation of the torsional vibrations in the full model (see Figure 15). The marginal stability of the reduced model is inconclusive for the stability of the global system. It should be emphasized that the damping term in (5.8) is destabilizing when β < 1, unless Ga (ωo ) = 0, which characterizes the just described quasi-limit cycle regime. It is seen in Figure 10 that this parametric condition will occur for ωo suﬃciently large. The consequence is that increasing the rotational speed is a way to avoid the exponential instability of the torsional equilibrium when β < 1. This is consistent with ﬁeld practice where drilling structures are often equipped with a down-hole motor. 6. Limitations of the two-time-scales approach. The two-time-scales approach in the previous section provides an accurate prediction of the diﬀerent behaviors of the model in parametric regions when there is a clear separation between the fast time scale of axial dynamics and the slow time scale of torsional dynamics. In this section, we brieﬂy describe additional phenomena that are observed when this time scale separation no longer holds. 6.1. Antiresonance. The antiresonance regime occurs when the axial dynamics exhibit stick-slip oscillations that eventually damp the torsional vibrations. It occurs mainly at very low nominal rotational speeds ωo , as seen in Figure 16. In order to understand the source of this destabilizing mechanism, we simulated the axial dynamics while imposing ϕ˙ as a harmonic signal of amplitude ωo /2. Figure 17 illustrates the ˙ at two diﬀerent speeds: evolution of λn g(u) • on the left, ωo = 1, and the antiresonance phenomenon occurs; • on the right, ωo = 3, and the torsional stick-slip oscillations are fully developed. The antiresonance process is clearly identiﬁed as a synchronization of the amplitude of

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Figure 16. Antiresonance regime: Torsional vibrations are stabilized by the axial vibrations (n = 6, ψ = 63.1, β = 0.276, ωo = 1, Wo − λ = 3.44).

Figure 17. Evolution of vo (τn − τno ) and n(1 − β)λn g(u) ˙ when the angular bit velocity ϕ˙ + ωo is imposed to be harmonic and with a bias of ωo /2. On the left, the rotational speed ωo is such that we observe numerically the antiresonance mode, while on the right, the steady state corresponding motion is stick-slip vibrations in torsion.

the plateaus of g(u) ˙ with the angular velocity, stabilizing the torsional equilibrium at hand. This particular stabilization mechanism of the torsional equilibrium occurring at small ωo is not predicted by the two-time-scales approach. Furthermore, the two-time-scales approach predicts that the amplitude of the axial limit cycle is inﬂuenced only by vo τno and not by ˙ with the τn (see Figure 18). However, the variation of the amplitude of the plateaus of g(u) angular velocity or equivalently the delay τn is clearly noticeable in Figure 17. Figure 19 suggests that the variation of the delay during the axial slip phase, i.e., when g(u) ˙ = 0, is a passive source of the stability mechanism. In Figure 19, the delay is ﬁrst constant and then increases linearly. Therefore, the variation of the magnitude of the plateaus of g(u) ˙ depends on the slope of τn during the axial slip phase. As a matter of fact, the axial dynamics act as a sampler of the derivative of the delay at each slip phase. The value of the derivative of the delay at these particular instants aﬀects the height of the plateaus of g(u). ˙ The maximum ˙ are plotted in Figure 20 for diﬀerent values of the slope of τn . values of n(1 − β)λn g(u) The antiresonance regime is advantageous because it stabilizes the torsional equilibrium.

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Figure 18. Evolution of the maximum value of n (1 − β) λn g(u) ˙ over one axial limit cycle for diﬀerent 2π − ωo ). values of ϕ˙ obtained with the shooting method (we use the approximation ϕ˙ ≈ nτ n

Figure 19. Evolution of τn and n (1 − β) λn g(u) ˙ when the delay changes as an increasing ramp.

Figure 20. Discrepancy of max(n (1 − β) λn g(u)) ˙ for diﬀerent slopes of ramp of τn .

Unfortunately, it occurs only at extremely low rotational speed, which makes it an impractical solution in drilling applications. 6.2. Delayed bifurcations. 6.2.1. Bit bouncing. The analysis in section 4.3 predicted bit bouncing √ when the axial equilibrium is unstable and the axial limit cycle does not exist, i.e., τ¯n > π/ 2 and Wo > 2λ (see Figure 7). When the bit experiences torsional √ and it may √ √vibrations, ϕ˙ (τ ) oscillates around zero, happen that the delay τ¯n (τ ) ≈ 2π ψ/ n (ωo + ϕ˙ (τ )) oscillates around π/ 2. This occurs

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Figure 21. Map of stability for diﬀerent rotational speeds ωo (n = 6, β = 0.276, ψ = 63.1); after [9].

when ωo approaches the bifurcation value ωos = 2 2ψ/n. The asymptotic solutions of the (delay-frozen) fast axial dynamics then alternate between stable and unstable axial equilibrium. In the full model, this phenomenon delays the bit bouncing bifurcation predicted at Wo = 2λ. Figure 21 illustrates that the transition from stable drilling to bit bouncing in the full model moves away from the theoretical prediction s Wo = 2λ as ωo approaches the bifurcation value ωo = 2 2ψ/n = 9.17. This eﬀect is also visible in Figure 7. 6.2.2. Quasi-limit cycle. For the same reason, the results obtained for the reduced model and for the full model may diﬀer when the parameters are in the vicinity of the bifurcation ωos = 2 2ψ/n, as displayed in Figure 22. The parametric region where a quasi-limit cycle is observed in numerical simulations is much larger than the one predicted from the reduced model. This is because the transient time to pass from the axial equilibrium point to the stable axial limit cycle is not negligible. The fast axial solutions do not reach steady state over this time frame, which reduces the averaged frictional term λn g(u˙ (τ ))a and therefore delays the instability of the torsional dynamics. 7. Conclusions. A novel approach to modeling stick-slip vibrations of drag bits in drilling structures accounts for the coupling between the axial and the torsional modes of vibrations via the bit-rock interface laws. This coupling introduces a state-dependent delay and a discontinuous friction term in the governing equations. Numerical simulations (see [18]) show the

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551

T Figure 22. Mean value of the friction term over one limit cycle in torsion (λn g(u) ˙ t = λTn 0 g(u)dt) ˙ in terms of ωo . The black dots are the solutions of the full model, while the white dots are the solutions of the fast subsystem.

existence of diﬀerent steady-state behaviors, such as axial and torsional stick-slip oscillations, antiresonance regime of the torsional dynamics, and a quasi-limit cycle or bit bouncing, in the torsional or axial direction, respectively. Furthermore, a parametric analysis reveals that the apparent decrease of the mean torque with the angular velocity responsible for the growth of the amplitude of the torsional vibrations is a consequence of the axial vibrations and more precisely of the intermittent decreases of the frictional contact forces at the wearﬂat-rock interface. The dimensionless formulation exhibits a large parameter ψ in the model, which enables a two-time-scales analysis of the axial and torsional dynamics. The axial mode oscillates much faster than the torsional mode of vibration. In this paper, we present an asymptotic analysis that decouples fast axial dynamics (with a frozen constant delay) from the slow torsional dynamics, inﬂuenced only by the √averaged behavior of the fast dynamics. When the delay is larger than a critical value π/ 2nψ, where n is the number of blades mounted on the bit, a stable limit cycle in the axial direction is observed over a certain parametric range. An approximate model of the axial dynamics is proposed to provide an analytical characterization of the limit cycle. The resulting analytical predictions match the numerical observations well. They are useful for characterizing the phenomenon of bit bouncing, which originates from the instability of the axial solutions. The approximate model also provides an analytical expression of the averaged reacting torque-on-bit that inﬂuences the torsional dynamics. Its variation in terms of the bit angular velocity recovers the empirical velocity weakening law observed in experiments. The analysis of the slow torsional dynamics predicts the emergence of the diﬀerent regimes of torsional vibrations (stick-slip vibrations or a quasi-limit cycle) in parametric ranges that agree with the numerical simulations. The analytical predictions provide useful recommendations for the design of drilling structure, the selection of the operating parameters, or the control synthesis. We also discuss some limitations of the two-time-scales approach to capturing phenomena such as antiresonance or delayed bifurcations. The antiresonance regime is characterized by small vibrations of the bit angular velocity around its nominal value, although the bit experiences intermittent losses of frictional contact. This regime occurs at low rotational speed, or equivalently at large delay. It is only observable when the axial stick time is large

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enough to generate phase locking with the bit angular velocity. This work possibly opens new perspectives for the synthesis of passive control laws to reduce the amplitude of the torsional vibrations. Most notably, it was shown that the equilibrium of the torsional dynamics could become exponentially stable by changing the bit design through the parameter β or the number of active blades n. Acknowledgments. This paper presents research partially supported by the Belgian Programme on Inter-university Poles of Attraction, initiated by the Belgian State, Prime Minister’s Oﬃce for Science, Technology and Culture. This research was initiated at the University of Minnesota under the supervision of Pr. Emmanuel Detournay through the Ph.D. thesis of Dr. Thomas Richard and the M.Sc. thesis of Christophe Germay. The authors would also like to thank Pr. Emmanuel Detournay for his careful review of the manuscript. REFERENCES [1] J. I. Adachi, E. Detournay, and A. Drescher, Determination of rock strength parameters from cutting tests, in Proceedings of the 2nd North American Rock Mechanics Symposium (NARMS 1996), Rotterdam, 1996, Balkema, Leiden, The Netherlands, 1996, pp. 1517–1523. [2] R. Almenara and E. Detournay, Cutting experiments in sandstones with blunt PDC cutters, in Proceedings of the International Society of Rock Mechanics Symposium EuRock ’92, Thomas Telford, London, 1992, pp. 215–220. [3] J. F. Brett, The genesis of torsional drillstring vibrations, SPE Drill. Eng., September (1992), pp. 168–174. [4] N. Challamel, Rock destruction eﬀect on the stability of a drilling structure, J. Sound Vibration, 233 (2000), pp. 235–254. [5] C. Canudas de Wit, H. Olson, K. J. Astrom, and P. Lischinsky, A new model for control of systems with friction, IEEE Trans. Automat. Control, 40 (1995), pp. 419–425. [6] E. Detournay and P. Defourny, A phenomenological model of the drilling action of drag bits, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 29 (1992), pp. 13–23. [7] R. P. H. Faassen, N. Van de Wouw, J. A. J. Oosterling, and H. Nijmeijer, Prediction of regenerative chatter by modeling and analysis of high-speed milling, Internat. J. Machine Tool and Manufacture, 43 (2003), pp. 1437–1445. [8] A. F. Filippov, Diﬀerential equations with discontinuous right-hand side, Amer. Math. Soc. Transl., 42 (1964), pp. 199–231. [9] C. Germay, Self-Excited Vibrations of Drag Bits, Master’s thesis, University of Minnesota, 2002. [10] T. Insperger, G. Stepan, and J. Turi, State dependent delay model for regenerative cutting processes, in Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference (ENOC 2005), Eindhoven, The Netherlands, 2005, pp. 1124–1129. [11] J. Jansen, L. van den Steen, and E. Zachariasen, Active damping of torsional drillstring vibrations with a hydraulic top drive, SPE Drilling Completion, 10 (1995), pp. 250–254. [12] R. I. Leine, D. H. van Campen, and W. J. G. Keultjes, Stick-slip whirl interaction in drillstring dynamics, J. Vibration Acoustics, 124 (2002), pp. 209–220. [13] N. Mihajlovic, A. A. van Veggel, N. van de Wouw, and H. Nijmeijer, Analysis of friction induced limit cycling in an experimental drill-string system, J. Dynam. Systems Measurement and Control, 126 (2004), pp. 709–720. [14] E. M. Navarro-Lopez and R. Suarez-Cortez, Practical approach to modelling and controlling stickslip oscillations in oilwell drillstrings, in Proceedings of the IEEE International Conference on Control Applications, 2004, IEEE Press, Piscataway, NJ, pp. 1454–1460. [15] T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York, 1989.

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[16] D. R. Pavone and J. P. Desplans, Application of high sampling rate downhole measurements for analysis and cure of stick-slip in drilling, in Proceedings of the SPE Annual Technical Conference and Exhibition, 1994, SPE 28324, pp. 335–345. [17] T. Richard, C. Germay, and E. Detournay, Self-excited stick-slip oscillations of drill bits, C. R. M´ecanique, 332 (2004), pp. 619–626. [18] T. Richard, C. Germay, and E. Detournay, Self-excited vibrations of drilling systems with drag bits, J. Sound Vibration, 305 (2007), pp. 432–456. [19] A. F. A. Serrarens, M. J. G. van de Molengraft, J. J. Kok, and L. van den Steen, H 1 control for suppressing stick-slip in oil well drillstrings, IEEE Control Systems Mag., 18 (1998), pp. 19–30. [20] G. Stepan, Delay-diﬀerential equation models for machine tool chatter, in Dynamics and Chaos in Manufacturing Processes, F. C. Moon, ed., Wiley, New York, 1998, pp. 165–191. [21] A. R. Teel, L. Moreau, and D. Nesic, A uniﬁed framework for input-to-state stability in systems with two time scales, IEEE Trans. Automat. Control, 48 (2003), pp. 1526–1544. [22] J. Tlusty and M. Polacek, The stability of machine tool against self-excited-vibrations in machining, in Proceedings of the ASME Production Engineering Research Conference, American Society of Mechanical Engineers, New York, 1963, pp. 465–474.

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c 2009 Society for Industrial and Applied Mathematics

SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 8, No. 2, pp. 554–575

Singular Continuations of Attractors∗ A. Giraldo† and J. M. R. Sanjurjo‡ Dedicated with aﬀection to Professor Jos´e M. Montesinos on the occasion of his 65th birthday Abstract. We study dynamical and topological properties of the singularities of continuations of attractors of ﬂows on manifolds. Despite the fact that these singularities are not isolated invariant sets, they ˇ share many of the properties of attractors; in particular, they have ﬁnitely generated Cech homology ˇ and cohomology, and they have the Cech homotopy type of attractors. This means that, from a global point of view, the singularities of continuations are topological objects closely related to ﬁnite polyhedra. The global structure is preserved even for weaker forms of continuation. An interesting case occurs with the Lorenz system for parameter values close to the situation of preturbulence. A general result, motivated by this particular case, is presented. Key words. dynamical system, continuation of isolated invariant sets, singular continuation, attractor, quasi attractor, shape, Lorenz system AMS subject classifications. 37B25, 37D45, 37C70, 37B30, 54C56, 55P55 DOI. 10.1137/080737356

1. Preliminaries. In this paper we study some dynamical and topological features of the continuations of attractors of ﬂows. Given a compact invariant set of a ﬂow ϕ : X ×R −→ X on a locally compact metric space X, K is said to be an isolated invariant set if there exists a compact neighborhood N of K in X such that K is the maximum invariant subset of N . In this case, N is then an isolating neighborhood for K in X. A compact set K ⊂ X is an attractor if there exists a neighborhood U of K in X such that every x ∈ U satisﬁes that for every neighborhood V of K in X there exists t ∈ R+ such that ϕ({x} × [t, ∞)) ⊂ V . Moreover, K is an asymptotically stable attractor if it is an attractor such that every neighborhood of K contains a positively invariant neighborhood of K. In this paper, all attractors will be supposed to be asymptotically stable attractors. Let X be a locally compact metric space, and let ϕλ : X × R −→ X be a parametrized family of ﬂows (parametrized by λ ∈ I, the unit interval) such that Φ : X × I × R −→ X × I, given by Φ(x, λ, t) = (ϕλ (x, t), λ), is a ﬂow in X × I. Let K0 ⊂ X be an isolated invariant set for ϕ0 , and K1 ⊂ X an isolated invariant set for ϕ1 . We say that K0 and K1 are related by continuation if there is an isolated invariant set K ⊂ X ×I for Φ such that K0 = K ∩(X ×{0}) and K1 = K ∩ (X × {1}). This is equivalent to the following: K0 and K1 are related by ∗ Received by the editors October 6, 2008; accepted for publication (in revised form) by T. Sauer January 14, 2009; published electronically April 10, 2009. This work was supported by the Direccion General de Investigaci´ on. http://www.siam.org/journals/siads/8-2/73735.html † Departamento de Matem´ atica Aplicada, Facultad de Inform´ atica, Universidad Polit´ecnica de Madrid, Madrid 28040, Spain ([email protected]). ‡ Departamento de Geometr´ıa y Topolog´ıa, Facultad de CC.Matem´ aticas, Universidad Complutense de Madrid, Madrid 28040, Spain (Jose [email protected]).

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continuation if there is a family Kλ , λ ∈ [0, 1], with Kλ an isolated invariant set for ϕλ , such that if Nλ0 is an isolating neighborhood for Kλ0 , then there exists ε > 0 such that Nλ0 is an isolating neighborhood for Kλ for every λ ∈ (λ0 − ε, λ0 + ε) ∩ [0, 1]. If we have a continuous parametrized family of ﬂows ϕλ : X × R −→ X with λ ∈ [0, 1], and {Kλ | λ ∈ I} is a continuation relating two attractors, K0 and K1 , then on some occasions it is possible to replace at the parameter value λ = 1 the attractor K1 by another one, ˆ 1 , in such a way that the same continuation (with λ < 1) also relates K0 and the new K ˆ 1 . In some particular cases we may even have a nested sequence of attractors of attractor K n+1 ⊂ K1n . . . , all of which are related to K0 through the same continuation. This ϕ1 , . . . K1 is illustrated in this paper by an example. In this situation the “natural” continuation of K0 through {Kλ } seems to be K1 = K1n despite the fact that K1 may be nonisolated and hence, possibly, a nonattractor. We call K1 a singularity of the continuation {Kλ }. K1 is, in fact, a quasi attractor, often with a complicated topological structure. This situation is far from being exceptional in dynamical systems. As was remarked by Kennedy and Yorke in [24], “bizarre topology is natural in dynamical systems.” Moreover, Kennedy, dealing with discrete dynamical systems (generated by a homeomorphism) in a large class of compact metric spaces, including manifolds of dimension at least two, proved in [23] that the property of admitting an inﬁnite collection of attractors, each of which has nonempty interior and cannot be reduced to a “smallest” attractor, is generic. Hurley generalized this result in [20] by showing that this property holds for all attractors of a generic homeomorphism (see [1] for many related matters). In this paper we study properties of the quasi attractors obtained in situations similar to the one described above. We focus, in particular, on properties of the singularities of continuations. We introduce the continuation skeleton of an attractor K0 , which gathers information from all the continuations of K0 , and the related spectrum of K0 , which is the quasi attractor of the terminal ﬂow, ϕ1 , and which “survives” all possible continuations of K0 . In spite of their weird local topological structure, singularities of continuations and spectra of attractors have rather regular global topological properties, which agree with those of K0 . The proof of this fact is one of the aims of this paper. The examples that we ﬁnd in our study motivate the introduction of the class of tame quasi attractors, which share many of the topological properties of attractors. Tameness turns out to be a persistent property for most of the ﬂows deﬁned on a manifold. In order to formulate our results we use those parts of topology designed to study spaces ˇ ˇ from a global point of view, in particular Cech homology and cohomology [13] and Cech homotopy [28, 29]. The last one, also known under the name of shape theory, was introduced by Borsuk [5] in order to study geometric properties of compact metric spaces with not necessarily good local behavior. In fact, homotopy theory appears as a strong tool for such study in good spaces like polyhedra, manifolds, and, more generally, ANR (absolute neighborhood retract) spaces. However, homotopy theory is not eﬃcient in the absence of good local properties like local connectedness. On the other hand, spaces with complicated local behavior appear naturally, not only in pure mathematics but also in the mathematical formulation of natural phenomena, e.g., solenoids, Lorenz attractors, etc. Hence, it is natural to look for another adequate tool for handling these problems. This seems to be the role of shape theory in this context. Further,

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shape theory does not modify homotopy theory in the good framework; in particular, two polyhedra have the same shape if and only if they are homotopically equivalent. The Lorenz system is a source of attractors and, in general, invariant sets with a very complicated structure (this is the reason why they are referred to as strange sets in the literature). On the other hand, the remarkable chaotic ﬂow exhibited by the Lorenz equations has served as inspiration to important results on dynamical systems. A recent example can be found in [26], where it is shown that any C 1 robust attractor with singularities for a ﬂow on a closed 3-manifold resembles a geometric Lorenz attractor (see [27] for recent related results). In the last section of the paper we consider the Lorenz system for the classical values of σ = 10, b = 83 , and 24.06 ≤ r ≤ 24.74, which correspond to an interval of values for the parameter r just after the preturbulence phenomenon, between the appearance of the Lorenz attractor and the loss of stability of the ﬁxed points at the Hopf bifurcation. This is an interesting situation, since when we observe the evolution of the system as r decreases, there is a continuation of attractors in (24.06, 24.74] which cannot be continued for r = 24.06. In fact, at r = 24.06 the Lorenz attractor has not yet appeared, although there is an unstable strange set which is the upper limit of the sequence of attractors and which has the same shape as the attractors of the continuation. This particular situation serves as a motivation for a shape-theoretical theorem which holds for continuations of attractors in [0, 1) which cannot be extended to [0, 1] but which experience a transition from attractor to repeller when the parameter takes the value 1. We show that, under quite general hypotheses, the repeller has the same shape as the attractors of the continuation. For complicated spaces it is often useful to use a special family of results in shape theory called complement theorems. The ﬁrst of these theorems is due to Borsuk, who proved that two continua in R2 have the same shape if and only if they divide the plane into the same number of components. Later, Chapman [6] proved that two (properly embedded) compacta in the Hilbert cube have the same shape if and only if their complements are homeomorphic. There exist other results in ﬁnite dimension [7, 21]. Some of these results are interpreted in terms of attractor-repeller duality in dynamical systems [16]. On the other hand, Robinson [31] uses results of Geoghegan and Summerhill [14] combined with several other results to show that there is a diﬀerential equation on a ﬁnite-dimensional Euclidean space with some special dynamical properties. An important property of attractors has been formulated in [4, 18, 36] at various levels of generality using the following kind of topological idea (see also [15, 17, 33, 38] for other related results). Theorem 1.1. Every attractor of a ﬂow on a locally compact ANR (particularly in a manifold) has the shape of a (ﬁnite) polyhedron. For information about the basic results of shape theory we recommend the books [5, 11, 25, 10] and the paper [35]. We also use notions and results from Conley index theory [8, 12]. Roughly speaking, the Conley index of an isolated invariant set K is the homotopy type of the pointed space obtained from an isolating neighborhood of K (with some special properties) on collapsing the exit set to one point. The Conley index has the important property of being invariant under continuation. In the case of a continuation of attractors, this fact implies that the corresponding isolating neighborhoods are homotopically equivalent, and from this it is possible to prove that the

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attractors must have the same shape. On the other hand, under certain conditions of regularity and triviality of the shape of the exit set, the Conley index of an isolated nonsaddle set is uniquely determined by its shape [16]. For the general theory of dynamical systems we recommend the books [3, 41, 2, 32]. Finally, we recommend the book [40] and the survey article [42] for general information about the Lorenz attractor. 2. Decomposition and shape properties of the continuations of attractors. If K0 is an attractor and K1 is related to K0 by continuation, then K1 does not necessarily have to be an attractor, as is shown in the following example. Example 1. Consider -

ar

-

ar Kλ ar K0

-

br

where all the ﬂows are identical on a neighborhood of a, while away from a the ﬂow slows down until ﬁnally a new ﬁxed point b appears when λ = 1. Then [a, b], {a, b}, and {a} are all related by continuation to K0 = {a}. Now, K0 is an attractor and, for every 0 < λ < 1, Kλ = {a} is also an attractor. However, if we consider the three possible continuations of K0 , then [a, b] and {a} are attractors, while {a, b} is not an attractor. This example motivates the following theorem. Theorem 2.1. Suppose that ϕλ : M × R −→ M is a parametrized family of ﬂows (parametrized by λ ∈ I, the unit interval) on a locally compact ANR, M , and suppose that K0 ⊂ M is a connected attractor for ϕ0 . Let K1 be related by continuation to K0 . Then, the following hold. (i) If K1 is connected, then K1 is an attractor and Sh(K0 ) = Sh(K1 ). (ii) If K1 is not connected, then K1 = K11 ∪ K12 (with K11 ∩ K12 = ∅), where K11 is a connected attractor that continues K0 , and K12 is an invariant compactum with trivial Conley index. Moreover, Sh(K0 ) = Sh(K11 ). Proof. (i) If K0 is a connected attractor, then there is an index pair (N0 , L0 ) for K0 , with N0 a connected set, such that L0 = ∅. Hence its Conley index is the homotopy type of the space (N0 ∪ {∗}, ∗), i.e., a 2-component space. Hence, the Conley index of K1 is also the pointed homotopy type of a 2-component space. Since K1 is connected, this is possible only if there is an index pair (N1 , L1 ) for K1 , with N1 a connected set, such that L1 = ∅, but this implies that K1 is an attractor. We now prove that Sh(K1 ) = Sh(K0 ). This is a consequence of the fact that if N is a positively invariant connected isolating neighborhood of a connected attractor K, contained in its region of attraction, then Sh(K) = Sh(N ). This follows from a standard argument (see, for example, [16, Theorem 4]), which we reproduce here for the sake of completeness. By [30, Theorem 5.2], there exists a map f : N −→ R such that f (x) = 0 if x ∈ K, and f (xt) < f (x) if x ∈ N \ K and t > 0. Now, ω + (x) ⊂ K for every x ∈ N , and hence f (x) > 0

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if x ∈ N \ K. If W is an arbitrary open neighborhood of K with its closure contained in the interior of N , then there is a t0 > 0 such that f −1 ([0, t0 ]) ⊂ W . Otherwise, there would exist a sequence of points (xn ) ⊂ N \ W and a null sequence (tn ) of positive numbers such that f (xn ) < tn . From this we could deduce the existence of a point x0 ∈ N \ K with f (x0 ) = 0, in contradiction with the previous remark. This implies that, for any null sequence (tn )n≥0 starting in t0 , the sets f −1 ([0, tn ]) form a neighborhood basis of K in M . Consider any such null sequence (tn )n≥0 . We ﬁrst construct a retraction r : f −1 ([0, t0 ]) −→ f −1 ([0, t1 ]) in the following way: If x ∈ f −1 ([0, t1 ]), we deﬁne r(x) = x. On the other hand, if x ∈ f −1 ([0, t0 ]) \ f −1 ([0, t1 ]) and f (x) > t1 , then there exists a unique tx > 0 such that f (xtx ) = t1 . We then deﬁne r(x) = xtx . A strong deformation retraction from f −1 ([0, t0 ]) to f −1 ([0, t1 ]) is given by the homotopy θ : f −1 ([0, t0 ]) × [0, 1] −→ f −1 ([0, t0 ]), deﬁned as θ(x, s) = x(tx s) if f (x) ∈ [0, t1 ] and θ(x, s) = x if f (x) ∈ [0, t1 ]. In an analogous way we may construct a strong deformation retraction from f −1 ([0, t0 ]) to f −1 ([0, tn ]) for every n ∈ N. Since the sets f −1 ([0, tn ]) form a neighborhood basis of K in M we can deﬁne a strong shape deformation retraction from f −1 ([0, t0 ]) to K. Therefore K has the shape of f −1 ([0, t0 ]), where this is an isolating neighborhood of K. Finally, since all positively invariant connected isolating neighborhoods of K are homotopically equivalent, Sh(K) = Sh(N ). Now, the invariance of the Conley index by continuation in the case of attractors implies that the corresponding isolating neighborhoods are homotopically equivalent and hence have the same shape. Therefore K1 and K0 have the same shape. (ii) We have just seen that, if K1 is related by continuation to K0 , then its Conley index is the pointed homotopy type of a 2-component space. Now, if K1 is not connected, then it cannot be an attractor (since, in this case its Conley index would be the pointed homotopy type of a space with at least 3 components). On the other hand, since K1 is not an attractor, there is an index pair (N1 , L1 ) for K1 , with L1 = ∅. Then N1 cannot be a connected set (if it were, the homotopy type of (N1 /L1 , [L1 ]) wouldbe that of a connected set). Now, N1 has a ﬁnite number of components, such as N1 = ki=1 N1i , and we consider, for each i ∈ {1, 2, . . . , n}, Li1 = L1 ∩ N1i and K1i = K1 ∩ N1i . Then, there exists a component, and we may suppose that it is N11 , such that L11 = ∅. But this implies that K11 is an attractor. We see now that K1 \ K11 = ni=2 K1i has trivial Conley index. To see this, observe that the Conley index of K1 is the homotopy type of n N1i /L1 , [L1 ] , N11 ∪ i=2

and this agrees with the Conley n index of K0 , which is the homotopy type of the space 1 and i /L are disjoint connected sets, the homotopy type of ∪ {∗}, ∗). Since N N (N 0 1 i=2 1 1 n i 1 i=2 N1 /L1 , [L1 ] has to be the same as the homotopy type of ({∗}, ∗). Therefore K1 \ K1 has trivial Conley index. We see now that K11 continues K0 . Observe ﬁrst that, reasoning as above, we can deduce that for every λ ∈ [0, 1] we have the same situation as for λ = 1; i.e., Kλ decomposes as Kλ = Kλ1 ∪ Kλ2 , where Kλ1 is a connected attractor, and Kλ2 has trivial Conley index. To see that K11 continues K0 we consider Λ = {λ ∈ [0, 1] | Kλ1 continues K0 } such that 0 ∈ Λ, and we will prove that Λ = [0, 1].

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Consider N0 , a positively invariant connected isolating neighborhood of K0 contained in its region of attraction. Then, it was shown in [37] that there exists ε > 0 such that, for every λ ∈ [0, ε), N0 is an isolating neighborhood for ϕλ and the maximal invariant set in N0 is an attractor. On the other hand, we can take ε > 0 in such a way that, for every λ ∈ [0, ε), N0 is an isolating neighborhood of Kλ . This implies that Kλ is a (connected) attractor; hence Kλ1 = Kλ and continues K0 . This same argument can be adapted to prove that whenever Kλ10 continues K0 , there exists ε > 0 such that Kλ1 continues K0 for every λ ∈ [λ0 , λ0 + ε). On the other hand, suppose that Kλ1 continues K0 for every λ ∈ [0, λ0 ). Consider Nλ10 , a positively invariant connected isolating neighborhood of Kλ10 contained in its region of attraction and not meeting Kλ20 . Consider an isolating neighborhood Nλ0 = Nλ10 ∪ Nλ20 of Kλ0 such that Nλ20 is an isolating neighborhood of Kλ20 . Then there exists ε > 0 such that Nλ0 is an isolating neighborhood of Kλ for every λ ∈ (λ0 − ε, λ0 ]. On the other hand, we can take ε > 0 in such a way that, for every λ ∈ [λ0 − ε, λ0 ), Nλ10 is an isolating neighborhood for ϕλ and the maximal invariant set in Nλ10 is an attractor. This implies that this maximal invariant set has to be Kλ1 , and hence Nλ10 is an isolating neighborhood of Kλ1 . Therefore, Kλ1 continues K0 for every λ ∈ [0, 1], and, in particular, K11 continues K0 . Remark 1. In the conditions of the theorem we have the following: (a) Despite having trivial Conley index, K12 can be topologically very complicated. It can be proved, by an argument similar to that in the proof of Theorem 3 in [16], that any ﬁnite-dimensional compactum K can be embedded in Rn , for suitable n, in such a way that there is a parametrized family of ﬂows ϕλ : Rn × R −→ Rn (parametrized by λ ∈ I, the unit interval) such that K0 = {0} is a connected attractor for ϕ0 , and K1 = K0 ∪ K (disjoint union) is related by continuation to K0 . (b) The case of nonconnected attractors is only slightly more complicated, since attractors have a ﬁnite number of components. (c) The conclusion in (i) about K and K1 (K11 in (ii)) having the same shape is a consequence of the following more general result. Theorem 2.2 (Theorem 8 in [16]). Let K and K be isolated nonsaddle sets (K, K = ∅) of ﬂows ϕ and ϕ deﬁned on locally compact ANRs M and M , respectively. Suppose that K and K admit index pairs (N, L) and (N , L ) such that N and N are connected regular isolating neighborhoods of K and K and such that L and L (the exit sets) are both empty or both contractible (or, more generally, with trivial shape). Suppose, in addition, that N and N are ANRs. Then Sh(K) = Sh(K ) if and only if they have the same (unpointed) Conley index. 3. Quasi attractors. A quasi attractor of a ﬂow ϕ is a compact, nonempty, invariant set K that is an intersection of attractors of ϕ. In the present framework, any quasi attractor can be expressed in the form K = ∞ n=1 An , with each An an attractor of ϕ. It is known that, for a ﬂow in the plane, the only possible attractors are compacta with a ﬁnite number of components decomposing the plane into a ﬁnite number of components (i.e., compacta with the shape of a ﬁnite polyhedron). The notion of a quasi attractor is much more general, as shown in the following theorem. Theorem 3.1. Every continuum of the plane can be a quasi attractor of a dynamical system on R2 . Proof. Let K be a continuum of the plane, and suppose that R2 \K has an inﬁnite number

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2 of components (the ∞case of R \ K having a ﬁnite number of components is similar but easier). 2 Hence R \ K = i=0 Ai , where A0 is unbounded while Ai is bounded for every i ≥ 1. ¯k+1 ⊂ Uk , for Consider a family {Uk }k∈N of connected neighborhoods of K such that U every k ∈ N, and such that • the boundary of Uk is the disjoint union of a family {Cki }i=0,1,2,...,k of simple closed curves such that Cki ⊂ Ai for every i = 0, 1, 2, . . . , k, • there exists a null sequence {εk } such that Ai ∩Uk ⊂ Bεk (K) for every i = 0, 1, 2, . . . , k. In these conditions, it is easy to construct a ﬂow in R2 such that • all the points in K are stationary, • the ﬂow in all the curves {Cki } is stationary, • the ﬂow restricted to the annulus bounded by two (stationary) curves Cki and C(k+1)i (k ≥ i ≥ 0) will have Cki as a repeller and C(k+1)i as an attractor. This completes the proof of the theorem. Modifying the construction in the proof of the theorem, we can prove that more complicated sets are quasi attractors. We can even obtain those quasi attractors as an alternated intersection of attractors and repellers. The following example illustrates in the context of ﬂows some of the generic situations described by Kennedy [23] and Hurley [20]. Example 2. The solenoid is a quasi attractor of a ﬂow in R3 . Observe that the solenoid cannot be an attractor of a ﬂow, since it does not have polyhedral shape. The solenoid can be obtained in the following manner. We start with a solid torus T0 and a second solid torus T1 winding twice inside the ﬁrst one:

The ﬂow will have T0 as an attractor and will be stationary between T0 and T1 . We consider now another solid torus T2 ⊂ T1 isotopic to T1 :

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The ﬂow will have T2 as a repeller. We consider now two isotopic solid tori T4 ⊂ T3 winding twice into T2 :

The ﬂow will be stationary between T2 and T3 and will have T4 as an attractor. We consider now two isotopic solid tori T6 ⊂ T5 winding twice into T4 :

The ﬂow will be stationary between T4 and T5 and will have T6 as a repeller. Following this process, we obtain a ﬂow having the solenoid

as a quasi attractor in such a way that there exists a decreasing sequence {Ti } of tori such that • T0 , T4 , T8 , . . . are attractors,

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• T , T , T , . . . are repellers, ∞ 2 6 10 and i=0 T4i is the solenoid. The following result relates the property of a compactum being a quasi attractor to the fact that this property holds for its connected components, whenever the phase space of the ﬂow is a locally compact and locally connected metric space. Proposition 3.2. Let ϕ : X × R → X be a ﬂow on a locally compact and locally connected metric space, and let K be a compactum in X. Then K is a quasi attractor if and only if all the connected components of K are quasi attractors. Proof. It has been proved by Conley in [9] that if K is a quasi attractor, then all its components are quasi attractors. Conversely, if every component of K is a quasi attractor of ϕ and U is an arbitrary neighborhood of K in X, then for every component Kα of K there is an attractor Cα such that Kα ⊂ Cα ⊂ U . Since the basin of attraction Aα of Kα is an open set in X and the family {Aα } covers K, there exists, by the compactness of K, a ﬁnite subfamily, {Aα1 , . . . , Aαn }, such that K ⊂ Aα1 ∪ · · · ∪ Aαn . Moreover, Cα1 ∪ · · · ∪ Cαn is an attractor and attracts Aα1 ∪ · · · ∪ Aαn . Then, since K is invariant and is contained in the basin of attraction of Cα1 ∪ · · · ∪ Cαn , necessarily K ⊂ Cα1 ∪ · · · ∪ Cαn ⊂ U . Hence K is a quasi attractor. We now turn our attention to the topological structure of quasi attractors. Definition 3.3. Suppose that ϕ : M × R → M is a ﬂow on a locally compact ANR, M . A quasi attractor K of ϕ is said to be “tame” if for every neighborhood U of K in M there is an attractor C ⊂ U such that the inclusion i : K → C is a shape equivalence. It is easy to see that a quasi attractor K is tame if and only ∞if there exists a nested sequence of attractors · · · ⊂ Cn+1 ⊂ Cn ⊂ · · · ⊂ C1 such that K = n=1 Cn and the inclusion i : K → Cn is a shape equivalence for every n. In contrast with general quasi attractors, tame quasi attractors can display only a limited amount of “bizarre” topological properties. This is a consequence of the fact that attractors of ﬂows in locally compact ANRs have polyhedral shape [36], which compels tame quasi ˇ attractors to also have polyhedral shape and hence ﬁnitely generated Cech homology and cohomology. The solenoid, for instance, has nonpolyhedral shape; hence it cannot be realized as a tame quasi attractor of a ﬂow in an ANR. Compacta with inﬁnite connected components are also excluded. Tame quasi attractors appear in a natural way when dealing with singularities of continuations and spectra of attractors. Moreover, they enjoy an interesting dynamical property related to the notion of persistence introduced by Hurley [19]. We adapt here Hurley’s deﬁnition to our setting. Definition 3.4. A tame quasi attractor K for a ﬂow ϕ in a C r -manifold, M , is C r -persistent if, whenever ϕn is a sequence of ﬂows converging to ϕ in the C r -topology, then each ϕn has a tame quasi attractor Kn and Kn approaches K in the Hausdorﬀ topology. We shall see that persistence holds for all tame quasi attractors of each ﬂow in a C r -residual subset in the set of all C r -ﬂows on M for every r ≥ 0. Proposition 3.5. There is a residual subset C of the space F r (M ) of all C r -ﬂows on the manifold M such that all tame quasi attractors of ﬂows in C are C r -persistent. Proof. The proof is also an adaptation of Hurley’s arguments to our setting. In [19] it is proved that there exists a residual subset C ofthe space F r (M ) of all C r -ﬂows on M such that, if ϕ ∈ C, ϕn −→ ϕ ∈ F r (M ), and K = ∞ k=1 Ck with Ck an attractor for ϕ for every

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SINGULAR CONTINUATIONS OF ATTRACTORS

563

k ∈ N, then for each n ∈ N we have an attractor Cnn for ϕn close to some Ckn in the Hausdorﬀ metric in such a way that Cnn −→ K when n → ∞. If we take Kn = Cnn , then Kn is an attractor and hence a tame quasi attractor, which readily proves the proposition. 4. Singularities of continuations. From now on we shall concentrate on the case in which K1 is connected and hence an attractor. As a motivation for the notions introduced in this section, we present the following example based on the Warsaw circle:

which is deﬁned as the union of the closure of the graph of y = sin x1 , 0 < x ≤ π1 , and an arc from (0, −1) to π1 , 0 which is disjoint from the graph except at its end points. Example 3. Consider the following parametrized ﬂow ϕλ in R2 , λ ∈ [0, 1]. ϕ0 is as follows: ?

?

-

6

K0

q-

? 6

ϕ0

i.e., ϕ0 consists of a simple closed curve K0 , which is an attractor, and a repelling point in the bounded component of the complement of K0 . This ﬂow can be continuously deformed as follows: ?

?

?

? ?

6

K 12

q-

6

?

? ϕ 12

?

?

? ?

6

K 23

q-

?

? 6

?

ϕ 23

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?

6

K 34

q-

? 6

ϕ 34

564

A. GIRALDO AND J. M. R. SANJURJO

i.e., ϕλ consists of a simple closed curve Kλ , which is an attractor, and the same repelling point as in the case λ = 0, which is always contained in the bounded component of the complement of Kλ . Finally, ϕ1 will be as follows: ? ?

? ? 6 6

6 --

q -- ?

? ? 6 6

0 1 2 where there ∞ is ai sequence K1 ⊃ K1 ⊃ K1 ⊃ · · · of annuli such that the following hold: • i=0 K1 is a Warsaw circle, • the points of the Warsaw circle and those in the boundaries of the K1i are stationary, • K1i is an attractor, for every i ≥ 0, i ≤ λ < 1, • K1i is a neighborhood of Kλ , the attractor of ϕλ , for every λ, with i+1 and there exists also a repelling point in the bounded component of the complement of the ﬁrst annulus K10 . Then any of the K1i ’s is a continuation of K0 , while the Warsaw circle “should” be the natural continuation, although it is not isolated. In Example 3 we have obtained a continuation {Kλ } of an attractor K0 such that the terminal level is an attractor K1 = K10 which can be replaced by other attractors K1i (the family of annuli in the example), keeping the continuation unchanged for λ < 1 and still getting a continuation from K0 to K1i . Our following deﬁnition is motivated by this example. Suppose that we have a situation similar to the one described above, i.e., a continuation {Kλ } of connected attractors, and consider the family of all attractors K1i such that {Kλ }, with 0 ≤ λ < 1, deﬁnes a continuation relating K0 to K1i . If we take two of these attractors K11 and K12 , their intersection K11 ∩ K12 satisﬁes the same property. This is also true for ﬁnite intersections K11 ∩ K12 ∩ · · · ∩ K1n . i However, if the family is inﬁnite and we take the intersection K1 of all of them, then two diﬀerent situations can arise: i i1 i2 in 1. The intersection i K1 can be reduced to a iﬁnite intersection K1 ∩ K1 ∩ · · · ∩ K1 , in is one of the attractors K1 related to K0 by {Kλ }, with 0 ≤ λ < 1. which case K1 ﬁnite intersection. In this case K1i is 2. The intersection K1i cannot be reduced to a a quasi attractor but not anattractor. Then K1i is nonisolated, and the notion of continuation relating K0 to K1i is meaningless in the Conley index theory. Definition 4.1. Let ϕλ : X × R −→ X be a parametrized family of ﬂows (parametrized by λ ∈ I, the unit interval) on a locally compact metric space X, let {Kλ } be a continuation of connected attractors, and consider the family of all attractors K1i such that {Kλ }, with

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SINGULAR CONTINUATIONS OF ATTRACTORS

565

0 ≤ λ < 1, deﬁnes a continuation relating K0 to K1i . If the intersection K1 = K1i cannot be reduced to a ﬁnite intersection, we say that the quasi attractor K1 is a singularity of the continuation. The following theorem can be easily proved. Theorem 4.2. Let ϕλ : X × R −→ X be a parametrized family of ﬂows (parametrized by λ ∈ I, the unit interval) on a locally compact metric space X, let {Kλ } be a continuation of connected attractors, and let K1 be a singularity of the continuation. Then the set-valued map F : I −→ X given by Kλ if λ ∈ [0, 1), F (λ) = K1 if λ = 1 is upper-semicontinuous. In the following deﬁnition we introduce the spectrum of an attractor K0 , which is the quasi attractor of the terminal ﬂow, ϕ1 , which “survives” all possible continuations of K0 (and hence is independent of any particular continuation), and the continuation skeleton of K0 , which gathers information from all the continuations of K0 , at all levels of the parametrized ﬂow. Definition 4.3. Suppose that ϕλ : X × R −→ X is a parametrized family of ﬂows (parametrized by λ ∈ I, the unit interval) on a locally compact metric space X, and suppose that K0 ⊂ X is a connected attractor for ϕ0 . Consider

there exists a continuation from K0 to K1

, K1 = K1 ⊂ X

through connected (hence attractors) sets and, for every λ0 ∈ [0, 1],

there exists {Kλ } continuation of K0

. Kλ0 = K ⊂ X

through connected sets with Kλ0 = K ˜λ = If K1 = ∅, then Kλ = ∅ for every λ ∈ [0, 1], and we say that K Kλ ∈Kλ Kλ is the ˜ 1 , we shall λ-spectrum of K0 . When considering the most important one, the 1-spectrum K ˜ λ }λ∈[0,1] the continuation call it, simply, the spectrum of K0 . Finally, we call the family {K skeleton of K0 . Remark 2. (a) As was the case for singularities, the spectrum of an attractor may not be an isolated compactum, although it is a quasi attractor. (b) The spectrum of K0 need not be a singularity induced by a continuation {Kλ }. Consider the following ﬂow: -

ar1- a-br3br2 br1 r2 ar3r ar r

-

ar1- a-br3br2 br1 r2 ar3r ar r

-

br3br2 br1 ar1- a-r2 ar3r ar r

-

ar

-

ar K0

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566

A. GIRALDO AND J. M. R. SANJURJO

where all the ﬂows are identical for λ ≥ 12 , while for λ < 12 they are identical far enough from a although slowing down to allow the appearance of inﬁnitely many new 1 degenerate ﬁxed points at λ1 = 2 . Then, any continuation {Kλ } of attractors

must be ﬁxed as {a} for λ ∈ 0, 2 and as any of the intervals [ai , bj ] for λ ∈ 12 , 1 . As a consequence, the ﬂow has no singularity at λ = 1. Therefore, the spectrum of K0 , which can be easily seen to be {a}, cannot be a singularity of any continuation. ˜ λ = ∅ for every λ ∈ [0, 1], and hence the continuation skeleton and (c) If K1 = ∅, then K the spectrum of K0 are well deﬁned. ˜ λ = ∅ for ˜ 1 = ∅ (the same argument applies to prove that K Proof of (c). We prove that K every λ ∈ [0, 1]). Consider {Kλ } and {Kλ } two continuations of K0 to compacta K1 and K1 . Then there are isolated invariant sets K, K ⊂ X × I for Φ such that K0 = K ∩ (X × {0}) = K ∩(X×{0}), K1 = K∩(X×{1}), and K1 = K ∩(X×{1}). Then K∩K ⊂ X×I is an isolated invariant set for Φ such that (K ∩ K ) ∩ (X × {0}) = K0 and (K ∩ K ) ∩ (X × {1}) = K1 ∩ K1 . Hence K1 ∩ K1 is related by continuation to K0 and is nonempty, since an attractor cannot continue to the empty set. In a similar way, the intersection of a ﬁnite number of compacta related by continuation to K0 is still a nonempty compactum related by continuation to K0 . Finally, it is easy to see that the intersection of all compacta related by continuation to K0 is still a nonempty compact subset of X. The continuation skeleton of K0 induces in a natural way an application from the parameter space to the phase space. In the following theorem we show that this application is upper-semicontinuous. Theorem 4.4. Let ϕλ : X × R −→ X be a parametrized family of ﬂows (parametrized by λ ∈ I, the unit interval) on a locally compact metric space X. Let K0 ⊂ X be an attractor for ϕ0 . Then the set-valued map F : I −→ X, ˜λ λ −→ K is upper-semicontinuous. ˜λ = Proof. Consider λ0 ∈ I, and let U be an open neighborhood of F (λ0 ) = K 0 K∈Kλ0 K. Then there exists K ∈ Kλ0 such that K ⊂ U . To see this, suppose that K ∩ (X \ U ) = ∅, this implies that for every K ∈ Kλ0 . Since Kλ0 is a family closed by ﬁnite intersections, n K ∩ (X \ U ) = ∅ for every K , K , . . . , K ∈ K , and, hence, K ∩ (X \ U ) = ∅. i 1 2 n λ 0 i=1 K∈Kλ0 Therefore F (λ0 ) ∩ (X \ U ) = ∅, a contradiction. Consider {Kλ } related by continuation to K0 through connected sets such that Kλ0 = K, and consider V ⊂ U an isolating neighborhood for K = Kλ0 . Then there exists ε > 0 such that V is an isolating neighborhood for Kλ , for every λ ∈ (λ0 − ε, λ0 + ε). But this implies that F (λ) ⊂ V ⊂ U , for every λ ∈ (λ0 − ε, λ0 + ε). We end this section with one of the main results of the paper, in which we prove that, despite their weird local topological structure, singularities of continuations and the spectrum of an attractor K0 have rather regular global topological properties, which agree with those of K0 .

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SINGULAR CONTINUATIONS OF ATTRACTORS

567

Theorem 4.5. Let ϕλ : M × R −→ M be a parametrized family of ﬂows (parametrized by λ ∈ I, the unit interval) deﬁned on a locally compact ANR, M (in particular in a manifold), ˜ 1 = ∅ is the spectrum of K0 , then K ˜ 1 is a and let K0 ⊂ M be a connected attractor for ϕ0 . If K ˜ 1 ) = Sh(K0 ). In particular, K ˜ 1 is a continuum with polyhedral tame quasi attractor and Sh(K ˇ shape and ﬁnitely generated Cech homology and cohomology groups, and only ﬁnitely many of them are nontrivial. Moreover, if a continuation {Kλ } of K0 by connected attractors has a singularity, K1 , ˜ 1 also hold for K . then the properties stated for K 1 Proof. We start by proving the second assertion in the statement of the theorem, for which we express K1 as the intersection of a nested sequence of connected attractors K1 ⊂ · · · ⊂ K1n+1 ⊂ K1n ⊂ · · · ⊂ K1 such that for every n ∈ N there is a continuation between K0 and K1n deﬁned by {Kλ } for λ < 1 but considering K1n instead of K1 at the top level. We shall study the inclusion i : K1n+1 → K1n . Consider Un+1 ⊂ Un such that Un (resp., Un+1 ) is a positively invariant (for ϕ1 ) compact isolating neighborhood of K1n (resp., K1n+1 ) contained in A(K1n ) (resp., A(K1n+1 )), the basin of attraction of K1n (resp., K1n+1 ). We are going to construct a shape deformation retraction from K1n to K1n+1 . (Observe that whenever K1n+1 = K1n then K1n A(K1n+1 ), and hence we cannot construct the shape deformation retraction using ϕ1 .) Since Un and Un+1 are isolating neighborhoods of K1n and K1n+1 there exists ε > 0 such that both are isolating neighborhoods of Kλ (for the ﬂow ϕλ ) lying in the basin of attraction of Kλ for λ ∈ (1 − ε, 1). On the other hand, if we take k0 ∈ N such that ϕ1 (Un × {k0 }) ⊂ intUn and ϕ1 (Un+1 × {k0 }) ⊂ intUn+1 , then if we choose ε suﬃciently small, we can achieve that • ϕλ (Un × [0, k0 ]) ⊂ A(K1n ), • ϕλ (Un+1 × [0, k0 ]) ⊂ A(K1n+1 ), • ϕλ (Un × {k0 }) ⊂ intUn , • ϕλ (Un+1 × {k0 }) ⊂ intUn+1 for every λ ∈ (1 − ε, 1). Consider ˆn = ϕλ (Un × [0, ∞)) = ϕλ (Un × [0, k0 ]), U ˆn+1 = ϕλ (Un+1 × [0, ∞)) = ϕλ (Un+1 × [0, k0 ]) U positively ϕλ -invariant compacta contained in A(Kλ ) (since Un and Un+1 are contained in ˆn ⊂ A(K n ) and U ˆn+1 ⊂ A(K n+1 ), respectively. Consider also A(Kλ )) and such that U 1 1 ˆˆ ˆ U n = ϕ1 (Un × [0, ∞]), ˆ ˆn+1 = ϕ1 (Uˆn+1 × [0, ∞]) U ˆn+1 ⊂ A(K n+1 ), respectively, positively ϕ1 -invariant compacta contained in A(K1n ) and U 1 ˆn+1 ) is a compact neighborhood of K n (resp., K n+1 ) contained in A(K n ) ˆn (resp., U because U 1 1 1 (resp., A(K1n+1 )).

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568

A. GIRALDO AND J. M. R. SANJURJO

We then have the following situation: ˆ ˆn+1 U

ˆˆ - U n 6

I 6@ @

ˆn+1 ˆn -U U r ˆ 6 6 @ n+1 I @ @ R rˆn @ rˆ ˆn+1 Kλ

Un+1 ? rn+1 K1n+1

rˆˆn

@ @ R - Un @ rn I @ @ R ? @ - Kn 1

This is a commutative diagram in the shape category, where all the unlabeled arrows are shape morphisms induced by inclusions, while the labeled ones are shape equivalences induced by inclusions or shape deformation retractions. (We use here the well-known fact that a ﬂow deﬁnes in a natural way a shape deformation retraction from a positively invariant compact neighborhood of an attractor to the attractor itself when the neighborhood lies in its region of attraction.) From this diagram it is possible to deduce that the inclusion i : K1n+1 → K1n is a shape equivalence: a right-inverse is deﬁned by the composition rˆ ˆn+1 rˆn ˆˆ n+1 ˆn −→ ˆn+1 → U Kλ → U , K1n → Un → U n+1 −→ K1

while a left-inverse is deﬁned by the composition rn+1 rˆn ˆn −→ Kλ → Un+1 −→ K1n+1 . K1n → Un → U

As a consequence we obtain a sequence of shape deformation retractions, sn+1

· · · ←− K1n+1 ←− K1n ←− · · · ←− K1 , and, using a classical theorem of Borsuk [5], we have that K1 is a shape deformation retraction of K1 . On the other hand, since the attractor K1 is a continuation of K0 , we have that Sh(K1 ) = Sh(K0 ), and from this follows the second half of the theorem. ˜ 1 as an intersection of a nested sequence of connected To prove the ﬁrst part, express K n+1 n ˜1 ⊂ · · · ⊂ K ⊂ K1 ⊂ · · · ⊂ K11 such that for every n there exists a continuation attractors K 1 {Kλn } with K0n = K0 . We can suppose that Kλn+1 ⊂ Kλn for every λ. We shall study the inclusion K1n+1 → K1n and prove, as before, the existence of a shape deformation retraction from K1n to K1n+1 . This will be a consequence of the two following facts: n +1 (1) If for some λ0 ∈ [0, 1) there exists a shape deformation retraction Kλn0 −→ Kλ0 , then there exists an ε > 0 such that for every λ ∈ (λ0 , λ0 + ε) there exists a shape n +1 deformation retraction Kλn −→ Kλ .

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SINGULAR CONTINUATIONS OF ATTRACTORS

569

(2) If λ0 ∈ (0, 1) satisﬁes that for every λ ∈ [0, λ0 ) there exists a shape deformation retraction Kλn −→ Kλn+1 , then there exists a shape deformation retraction Kλn0 −→ . Kλn+1 0 ⊂ Uλn0 positively To prove (1) we use the same arguments as before to choose Uλn+1 0 ) contained in A(Kλn0 ) invariant (for ϕλ0 ) compact isolating neighborhoods of Kλn0 (resp., Kλn+1 0 )), the region of attraction of Kλn0 (resp., Kλn+1 ), and an ε > 0 such that Uλn+1 (resp., A(Kλn+1 0 0 0 n+1 and Uλn0 are isolating neighborhoods of Kλ and Kλn for every λ ∈ (λ0 , λ0 + ε) and contained in their respective basins of attraction. If k0 is such that ϕλ0 (Uλn0 × {k0 }) ⊂ intUλn0 and × {k0 }) ⊂ intUλn+1 , we can also suppose that ϕλ (Uλn0 × [0, k0 ]) ⊂ A(Kλn0 ) and ϕλ0 (Uλn+1 0 0 × [0, k0 ]) ⊂ A(Kλn+1 ) and ϕλ (Uλn+1 × {k0 }) ⊂ intUλn+1 ϕλ (Uλn0 × {k0 }) ⊂ intUλn0 , and ϕλ (Uλn+1 0 0 0 0 for every λ ∈ [λ0 , λ0 + ε). Now consider, as before, ˆn = ϕλ (U n × [0, ∞)) = ϕλ (U n × [0, k0 ]), U λ0 λ0 ˆn+1 = ϕλ (U n+1 × [0, ∞)) = ϕλ (U n+1 × [0, k0 ]) U λ0 λ0 positively ϕλ -invariant compacta contained in A(Kλn ) and A(Kλn+1 ), respectively (because ˆn ⊂ A(K n ) and U ˆn+1 ⊂ A(K n+1 ), ⊂ A(Kλn+1 )) and such that U Uλn0 ⊂ A(Kλn ) and Uλn+1 λ0 λ0 0 respectively. Consider also ˆˆ ˆ U n = ϕλ0 (Un × [0, ∞)), ˆ ˆn+1 × [0, ∞)) ˆn+1 = ϕλ (U U 0 ), respectively, because positively ϕλ0 -invariant compacta contained in A(Kλn0 ) and A(Kλn+1 0 n+1 n n ˆn+1 ) is a compact neighborhood of K (resp., K ˆn (resp., U U λ0 λ0 ) contained in A(Kλ0 ) (resp., )). A(Kλn+1 0 We then have the following situation: ˆ ˆˆ ˆn+1 - U U n I 6@ @

6

ˆn+1 U

ˆn -U 6 rˆn

6 @ rˆn+1 I @ @ @ R n+1 - Kn rˆ ˆn+1 Kλ rˆˆn λ @ @ R - Un Uλn+1 λ0 0 @ rn I @ ? rn+1 @ R ? @ Kλn0 Kλn+1 0

r The arrows are labeled as in the previous diagram. From this diagram we deduce that the inclusion i : Kλn+1 −→ Kλn is a shape equivalence: a right-inverse is deﬁned by the composition ˆn+1 rn r ˆn+1 r−→ Kλn0 −→ Kλn+1 → Uλn+1 → U Kλn+1 , Kλn → Uλn0 −→ 0 0

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570

A. GIRALDO AND J. M. R. SANJURJO

while a left-inverse is deﬁned by the composition ˆn+1 rˆ ˆn r ˆ ˆn → U ˆn −→ ˆn+1 r−→ Kλn → U Kλn0 −→ Kλn+1 → Uλn+1 → U Kλn+1 . 0 0

As a consequence, for every λ ∈ [λ0 , λ0 + ε) there is a shape deformation retraction −→ Kλn+1 . Suppose now that for every λ ∈ [0, λ0 ) there exists a shape deformation retraction Kλn −→ n+1 Kλ . To prove (2) we make a choice of neighborhoods exactly as in the previous diagram with the only diﬀerence that we take parameter values λ immediately before λ0 (in an interval (λ0 − ε, λ0 )). This produces a commutative diagram Kλn

ˆ ˆn+1 U I 6@ @

ˆˆ - U n 6 ˆn -U 6 rˆn

ˆn+1 U

6 @ rˆn+1 I @ @ @ R n+1 rˆ ˆn+1 Kλ Kλn rˆˆn r @ @ R n+1 - Un Uλ0 λ0 @ rn I @ ? rn+1 @ R ? @ - Kn K n+1 λ0

λ0

→ Kλn0 is a shape equivalence. From this diagram it follows that the inclusion Kλn+1 0 A right-inverse is deﬁned by the composition rˆ ˆn+1 r rˆn ˆˆ n+1 ˆn −→ ˆn+1 → U Kλn −→ Kλn+1 → U Kλn0 → Uλn0 → U n+1 −→ Kλ0 ,

while a left-inverse is deﬁned by the composition r

rˆ

rn+1

n ˆn −→ Kλn −→ Kλn+1 → Uλn+1 −→ Kλn+1 , Kλn0 → Uλn0 → U 0 0

and this proves (2). As a consequence we get a sequence of shape deformation retractions sn+1

· · · ←− K1n+1 ←− K1n ←− · · · ←− K11 , ˜ 1 is a shape deformation retraction and, using again Borsuk’s theorem [5], we have that K 1 1 of K1 . On the other hand, since the attractor K1 is a continuation of K0 , we have that Sh(K11 ) = Sh(K0 ), and from this follows the ﬁrst half of the theorem. 5. Nonextendable continuations of attractors. A Lorenz attractor motivated result. In many interesting situations, there exists a continuation of attractors {Kλ | 0 ≤ λ < 1} that cannot be extended to λ = 1 in any possible way with K1 isolated. On the other hand, it may happen that the continuation {Kλ | 0 ≤ λ < 1} converges upper-semicontinuously to an isolated compact set K1 which is not an attractor.

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SINGULAR CONTINUATIONS OF ATTRACTORS

571

A paradigmatic example of such an evolution can be found in the Lorenz system, for the classical parameters, just after the preturbulence phenomenon [22], between the appearance of the attractor and the loss of stability of the ﬁxed points at the Hopf bifurcation. The Lorenz equations are ⎧ dx ⎪ ⎪ = σ(y − x), ⎪ ⎪ ⎪ ⎨ dt dy = rx − y − xz, ⎪ dt ⎪ ⎪ ⎪ dz ⎪ ⎩ = xy − bz. dt We will ﬁx, as is usually done in the literature, σ = 10 and b = 83 , and study the evolution of the system as r varies. In the following description of the situation we are interested in, we will follow [40] (see [2, 22, 42] for additional information). There exists, for each value of the parameter r, a bounded ellipsoid Er in R3 which all trajectories eventually enter. This implies that there exists a global attractor Kr which can be obtained as the intersection of a nested sequence of topological ellipsoids (obtained as successive images of Er by the ﬂow). We denote by ϕr : Kr × R −→ Kr the ﬂow restricted to this global attractor Kr . For the above-mentioned values σ = 10 and b = 83 , the Lorenz system behaves, as r varies, as follows: • for r = 24.06 . . . (we will write, for brevity, r = 24.06), there is a pair of stable ﬁxed points, C1 and C2 , and if we consider the ﬂow ϕr : Kr × R −→ Kr , there is also a dual repeller of {C1 , C2 } in Kr which contains an unstable strange set that contains the origin; • for 24.06 < r < 24.74 . . . (we will write, for brevity, 24.74), C1 and C2 are still stable ﬁxed points, but the strange set is now an attractor Ar ; • for r = 24.74, C1 and C2 are both unstable. 34

40 35 35

29 30

30

24 25

25 z(t) 19

z(t)

z(t) 20 20

14 15 15 9

15

10 16

12

58

4

10 -20

-20 -16 -12 y(t) -8 4-4 0 0

10

-22

y(t) -10 -12

5 0 14

-5

-10 x(t)

9

-15 20

10

4

5 -2 -1

-6

-11

-16

x(t)

13

8 x(t)

3 18

8

-2 y(t)

-7

-12

-17

Figure 1. Preturbulence (r = 22), coexistence of attractor and stable equilibria (r = 24, 5), Lorenz attractor (r = 26).

For 24.06 < r < 24.74, there is a pair of simple periodic orbits, γ1 and γ2 . Their stable manifolds look, near to the orbits, like cylinders. These cylinders cut oﬀ the origin from the

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stationary points C1 and C2 . Trajectories started inside any of these cylinders converge to the ﬁxed point inside it, while trajectories started outside them converge to Ar . We denote by Rr the set of points in the global attractor Kr which are outside the cylinders or on the surface of the cylinders. Then Rr is a repeller for the restricted ﬂow ϕr : Kr × R −→ Kr . Therefore, if we restrict to 24.06 ≤ r ≤ 24.74, there is a continuation of Lorenz attractors {Ar | 24.06 < r ≤ 24.74} that cannot be extended to r = 24.06. On the other hand, there exists a repeller R24.06 such that (Ar ) converges upper-semicontinuously to it. This is due to the fact that Rr converges upper-semicontinuously to R24.06 and Ar ⊂ Rr if 24.06 < r ≤ 24.74. Consider now the locally compact invariant set Kr = Kr \ {C1 , C2 } obtained by deleting the stable ﬁxed points from the global attractor. Denote by ϕr : Kr × R −→ Kr the ﬂow restricted to this set. Then the sets Rr are maximal compact invariant sets for the ﬂows ϕr in the sense that they contain any other compact invariant set of the ﬂow. On the other hand, the inclusion of an attractor in its region of attraction A(Ar ) is a shape equivalence. For 24.06 < r < 24.74, the region of attraction of the attractor Ar for the ﬂow ϕr : Kr × R −→ Kr restricted to the global attractor Kr is A(Ar ) = Rr \ (γ1 ∪ γ2 ). We see now that, for r close to 24.74, the inclusion of Ar into A(Ar ), the closure of its region of attraction, is a shape equivalence. Observe that A(Ar ) agrees with the repeller Rr deﬁned above. For r ≥ 24.74, if we consider the ellipsoid Er , then all trajectories inside it are attracted by Ar except those which compose the stable manifolds of the ﬁxed points C1 and C2 . These are one-dimensional manifolds whose intersection with Er consists of closed arcs, l1 and l2 , respectively, with their ends in the boundary of Er and such that l1 ∩ l2 = ∅. This fact was used in [39] to prove that the Lorenz attractor has the shape of S 1 ∨ S 1 , the wedge of two circles. Now, if r < 24.74 is close enough to 24.74, and if we denote by Γ1 and Γ2 the stable manifolds of the simple periodic orbits γ1 and γ2 , these are inﬁnite cylinders whose intersection with Er consists of ﬁnite cylinders, L1 and L2 , respectively, with their ends in the boundary of Er and such that L1 ∩ L2 = ∅. If we denote by Er the set of all the points of Er outside these cylinders, then the trajectories of all the points in Er are attracted by Ar . Then we have that the inclusions Ar → Er and Er → Er are shape equivalences. Therefore, the inclusion Ar → Er is a shape equivalence. Now, since the ﬂow induces a shape deformation retraction from Er to Rr , we conclude that the inclusion Ar → Rr = A(Ar ) is a shape equivalence. It is interesting to note, as was shown in [34], that if K is an attractor of a ﬂow on a locally compact space, the inclusion of K into the closure of its region of attraction is not, in general, a shape equivalence. An example is given by the following ﬂow:

p

D

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SINGULAR CONTINUATIONS OF ATTRACTORS

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which has a stable point that attracts all the points of the interior of the shaded set, while the points of its boundary are stationary. Then the stable point is an attractor whose region of attraction has trivial shape, but its closure not, since it divides the plane into an inﬁnite number of components [34]. The following deﬁnition is motivated by the properties of the Lorenz attractor discussed before when the ﬂow is restricted to Kr (the global attractor minus two equilibria). Definition 5.1. Let ϕ : M × R −→ M be a ﬂow on a locally compact metric space, M , and suppose that K ⊂ M is an attractor for ϕ. We say that K is a consistent attractor if the inclusion of K into the closure of its region of attraction is a shape equivalence. We say that K is a quasi global attractor if the closure of its region of attraction is a compact invariant set that contains any other compact invariant set of the ﬂow. We end the paper with a result motivated by and applicable (with slight changes) to the evolution of the Lorenz system between the appearance of the attractor (just after the preturbulence phenomenon) and the Hopf bifurcation. Theorem 5.2. Suppose that ϕλ : M × R −→ M is a parametrized family of ﬂows (parametrized by λ ∈ I, the unit interval) on a locally compact metric space, M , and consider a continuation of connected attractors {Kλ | 0 ≤ λ < 1}. Suppose that {Kλ } converges upper-semicontinuously to K1 such that K1 is a repeller for ϕ1 . Then there exists a subinterval I of I ( 1 ∈ I ) and a family {Kλ | λ ∈ I } such that (i) K1 = K1 , (ii) {Kλ | λ ∈ I } is a continuation of repellers, (iii) Kλ ⊃ Kλ . If, moreover, for some λ0 ∈ I , Kλ0 is a consistent quasi-global attractor, then Sh(Kλ ) = Sh(Kλ ) = Sh(K1 ) for every λ ∈ I \ {1}. Proof. Since repellers continue, the existence of K1 guarantees the existence of a λ0 < 1 and a continuation {Kλ | λ0 ≤ λ ≤ 1} of repellers such that K1 = K1 . We prove now that there exists λ0 > 0 such that Kλ ⊃ Kλ for every λ ≥ λ0 . Consider a compact isolating neighborhood N of K1 such that N ⊂ R(K1 ), the region of repulsion of K1 . Since {Kλ | λ0 ≤ λ ≤ 1} is a continuation, given N there exists λ0 such that, for every λ ≥ λ0 , Kλ ⊂ N ⊂ R(Kλ ), the region of repulsion of Kλ . Moreover, since {Kλ } converges upper-semicontinuously to K1 , we may suppose also that Kλ ⊂ N for every λ ≥ λ0 . We see now that Kλ ⊂ Kλ for every λ ≥ λ0 . Suppose, on the contrary, that for some λ ≥ λ0 there exists x0 ∈ Kλ \ Kλ . Then, since x ∈ Kλ , which is a repeller, the trajectory of x must go out of N , but this is impossible because x ∈ Kλ , which is an invariant subset of N . Therefore, if we take I = [λ0 , 1], the family {Kλ | λ ∈ I } satisﬁes (i), (ii), and (iii). Suppose now that for some λ0 ∈ I , Kλ0 is a consistent quasi-global attractor. Then Kλ0 ⊂ A(Kλ0 ) ⊂ Kλ 0 ⊂ A(Kλ0 ), and this implies that A(Kλ0 ) = Kλ 0 . On the other hand, Sh(Kλ0 ) = Sh(A(Kλ0 )) = Sh(Kλ 0 ) (for Kλ0 consistent). Finally, Sh(Kλ ) = Sh(Kλ 0 ) for every λ ∈ I , and Sh(Kλ ) = Sh(Kλ0 ) for every λ ∈ I \ {1} (by continuation). Therefore, Sh(Kλ ) = Sh(Kλ ) = Sh(K1 ) for every λ ∈ I \ {1}. Corollary 5.3. Consider the Lorenz system for σ = 10, b = 83 , and 24.06 ≤ r ≤ 24.74.

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Then the strange set which exists for r = 24.06 is an unstable set with the same shape as the attractors that exist for 24.06 < r ≤ 24.74. REFERENCES [1] E. Akin, M. Hurley, and J. A. Kennedy, Dynamics of Topologically Generic Homeomorphisms, Mem. Amer. Math. Soc. 164, AMS, Providence, RI, 2003. [2] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos. An Introduction to Dynamical Systems, Springer, New York, 1996. [3] N. P. Bhatia and G. P. Szego, Stability Theory of Dynamical Systems, Grundlehren Math. Wiss. 161, Springer-Verlag, Berlin, 1970. [4] S. A. Bogatyi and V. I. Gutsu, On the structure of attracting compacta, Diﬀerentsial’nye Uravneniya, 25 (1989), pp. 907–909 (in Russian). [5] K. Borsuk, Theory of Shape, Monograﬁe Matematyczne 59, Polish Scientiﬁc Publishers, Warszawa, 1975. [6] T. A. Chapman, On some applications of inﬁnite-dimensional manifolds to the theory of shape, Fund. Math., 76 (1972), pp. 181–193. [7] T. A. Chapman, Shapes of ﬁnite-dimensional compacta, Fund. Math., 76 (1972), pp. 261–276. [8] C. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Reg. Conf. Ser. Math. 38, AMS, Providence, RI, 1976. [9] C. C. Conley, The gradient structure of a ﬂow, Ergodic Theory Dynam. Systems, 8 (1988), pp. 11–26. [10] J. M. Cordier and T. Porter, Shape Theory. Categorical Methods of Approximation, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK, 1989. [11] J. Dydak and J. Segal, Shape Theory: An Introduction, Lecture Notes in Math. 688, Springer-Verlag, Berlin, 1978. [12] R. W. Easton, Geometric Methods for Discrete Dynamical Systems, Oxford Engineering Sci. Ser. 50, Oxford University Press, New York, 1998. [13] S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, Princeton, NJ, 1952. [14] R. Geoghegan and R. Summerhill, Concerning the shapes of ﬁnite-dimensional compacta, Trans. Amer. Math. Soc., 179 (1973), pp. 281–292. [15] A. Giraldo and J. M. R. Sanjurjo, On the global structure of invariant regions of ﬂows with asymptotically stable attractors, Math. Z., 232 (1999), pp. 739–746. ´ n, F. R. Ru´ız del Portal, and J. M. R. Sanjurjo, Some duality properties [16] A. Giraldo, M. A. Moro of non-saddle sets, Topology Appl., 113 (2001), pp. 51–59. [17] M. Gobbino, Topological properties of attractors for dynamical systems, Topology, 40 (2001), pp. 279–298. ¨ nther and J. Segal, Every attractor of a ﬂow on a manifold has the shape of a ﬁnite polyhedron, [18] B. Gu Proc. Amer. Math. Soc., 119 (1993), pp. 321–329. [19] M. Hurley, Attractors: Persistence, and density of their basins, Tran. Amer. Math. Soc., 269 (1982), pp. 247–271. [20] M. Hurley, Properties of attractors of generic homeomorphisms, Ergodic Theory Dynam. Systems, 16 (1996), pp. 1297–1310. ´, R. B. Sher, and G. A. Venema, Complement theorems beyond the trivial range, Illinois J. [21] I. Ivanˇsic Math., 25 (1981), pp. 209–220. [22] J. L. Kaplan and J. A. Yorke, Preturbulence: A regime observed in the ﬂuid model of Lorenz, Comm. Math. Phys., 76 (1979), pp. 93–108. [23] J. A. Kennedy, The topology of attractors, Ergodic Theory Dynam. Systems, 16 (1996), pp. 1311–1322. [24] J. A. Kennedy and J. A. Yorke, Bizarre topology is natural in dynamical systems, Bull. Amer. Math. Soc., 32 (1995), pp. 309–316. ´ and J. Segal, Shape Theory, North–Holland, Amsterdam, 1982. [25] S. Mardeˇsic [26] C. A. Morales, M. J. Pacifico, and E. R. Pujals, Robust transitive singular sets for 3-ﬂows are partially hyperbolic attractors or repellers, Ann. of Math. (2), 160 (2004), pp. 375–432. [27] M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness and domination for surface diﬀeomorphisms, Rev. Mat. Complut., 21 (2008), pp. 293–317.

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[28] T. Porter, Cech homotopy I, J. London Math. Soc., 6 (1973), pp. 429–436. [29] T. Porter, Cech homotopy II, J. London Math. Soc., 6 (1973), pp. 667–675. [30] J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynam. Systems, 8 (1988), pp. 375–393. [31] J. C. Robinson, Global attractors: Topology and ﬁnite-dimensional dynamics, J. Dynam. Diﬀerential Equations, 11 (1999), pp. 557–581. [32] J. C. Robinson, Inﬁnite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Appl. Math., Cambridge University Press, Cambridge, UK, 2001. ´ nchez-Gabites, Dynamical systems and shapes, Rev. R. Acad. Cien. Serie A Mat., 102 (2008), [33] J. J. Sa pp. 127–159. ´ nchez-Gabites and J. M. R. Sanjurjo, On the topology of the boundary of a basin of attraction, [34] J. J. Sa Proc. Amer. Math. Soc., 135 (2007), pp. 4087–4098. [35] J. M. R. Sanjurjo, An Intrinsic Description of Shape, Trans. Amer. Math. Soc., 329 (1992), pp. 625–636. ˇ [36] J. M. R. Sanjurjo, Multihomotopy, Cech spaces of loops and shape groups, Proc. London Math. Soc., 69 (1994), pp. 330–344. [37] J. M. R. Sanjurjo, On the structure of uniform attractors, J. Math. Anal. Appl., 192 (1995), pp. 519–528. [38] J. M. R. Sanjurjo, Morse equations and unstable manifolds of isolated invariant sets, Nonlinearity, 16 (2003), pp. 1435–1448. [39] J. M. R. Sanjurjo, Global topological properties of the Hopf bifurcation, J. Diﬀerential Equations, 242 (2007), pp. 238–255. [40] C. Sparrow, Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Springer, New York, 1982. [41] R. Temam, Inﬁnite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, 2nd ed., Springer-Verlag, New York, 1997. [42] M. Viana, What’s new on Lorenz strange attractors, Math. Intelligencer, 22 (2000), pp. 6–19.

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c 2009 Society for Industrial and Applied Mathematics

SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 8, No. 2, pp. 576–591

Localized Dispersive States in Nonlinear Coupled Mode Equations for Light Propagation in Fiber Bragg Gratings∗ C. Martel†, M. Higuera†, and J. D. Carrasco† Abstract. Dispersion eﬀects induce new instabilities and dynamics in the weakly nonlinear description of light propagation in ﬁber Bragg gratings. A new family of dispersive localized pulses that propagate with the group velocity is numerically found, and its stability is also analyzed. The unavoidable diﬀerent asymptotic order of transport and dispersion eﬀects plays a crucial role in the determination of these localized states. These results are also interesting from the point of view of general pattern formation since this asymptotic imbalance is a generic situation in any transport-dominated (i.e., nonzero group velocity) spatially extended system. Key words. ﬁber Bragg gratings, nonlinear coupled mode equations, envelope equations, multiple scale methods, localized solutions AMS subject classifications. 34E13, 74J30, 74J34, 37K45, 78A60 DOI. 10.1137/070698221

1. Introduction. Fiber Bragg gratings (FBG) are microstructured optical ﬁbers that present a spatially periodic variation of the refractive index. The combination of the guiding properties of the periodic media with the Kerr nonlinearity of the ﬁber results in the very particular light propagation characteristic of these elements, which make them very promising for many technological applications that range from optical communications (wavelength division, dispersion management, optical buﬀers and storing devices, etc.) to ﬁber sensing (structural stress measure in aircraft components and buildings, temperature change detection, etc.); see, e.g., the recent review [12]. The amplitude equations that are commonly used in the literature to model one dimensional light propagation in an FBG are the so-called nonlinear coupled mode equations (NLCME) [24, 7, 9, 8, 1, 10], which, conveniently scaled, can be written as (1.1) (1.2)

+ − + + 2 − 2 A+ t − Ax = iκA + iA (σ|A | + |A | ),

− + − − 2 + 2 A− t + Ax = iκA + iA (σ|A | + |A | ),

where A± are the envelopes of the two counterpropagating wavetrains that resonate with the grating, κ is the strength of the coupling eﬀect produced by the grating, and σ > 0 is ratio of the self to cross nonlinear interaction coeﬃcient (σ = 12 for a cubic Kerr nonlinearity [9]). ∗ Received by the editors July 24, 2007; accepted for publication (in revised form) by B. Sandstede January 29, 2009; published electronically May 7, 2009. This work was supported by Spanish Direcci´ on General de Investigaci´ on under grant MTM2007-64248 and by the Universidad Polit´ecnica de Madrid under grant CCG07-UPM/000-3177. http://www.siam.org/journals/siads/8-2/69822.html † Depto. de Fundamentos Matem´ aticos, E.T.S.I. Aeron´ auticos, Universidad Polit´ecnica de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid, Spain ([email protected], [email protected], [email protected]). The second author’s work was supported by Spanish Direcci´ on General de Investigaci´ on under grant TRA2007-65699.

576

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DISPERSIVE PULSES IN FIBER BRAGG GRATINGS

577

The NLCME can be obtained from the full Maxwell–Lorentz equations using multiple scales techniques in the limit of small grating depth, small light intensity, and slow spatial and temporal dependence of the ﬁeld envelopes (see [10] for a detailed description of this derivation process). It has been recently shown [16, 17] that light propagation in FBG can develop dispersive structures that are not accounted for in the NLCME formulation, and that, to correctly describe the weakly nonlinear dynamics of the system, the NLCME have to be completed with material dispersion terms: (1.3) (1.4)

+ − + + 2 − 2 + A+ t − Ax = iκA + iA (σ|A | + |A | ) + iεAxx ,

− + − − 2 + 2 − A− t + Ax = iκA + iA (σ|A | + |A | ) + iεAxx .

The dispersive nonlinear coupled mode equations (NLCMEd) above are scaled as the NLCME: the characteristic length scale is the slow scale that results from the balance of the advection term with the small eﬀect of the grating, the characteristic time is the corresponding transport time scale (which sets to one the scaled group velocity), and the characteristic size of the wavetrains is the one resulting from the saturation of the small nonlinear terms. The slow envelope assumption forces the dispersive terms to always be small as compared with the advection terms; in other words, second derivatives of the slow amplitudes are much smaller than their ﬁrst derivatives. In the scaled equations this eﬀect is contained in the scaled dispersion coeﬃcient ε (which measures the dispersion to transport ratio), and therefore, in order to be consistent with the slow envelope assumption, the NLCMEd must be considered in the limit ε → 0. The NLCMEd were already analyzed in [6], but they considered only the case ε ∼ 1 that, as explained above, does not correspond to a generic situation from the point of view of large scale pattern formation in extended systems. The NLCMEd can be regarded as asymptotically nonuniform, in the sense that the NLCMEd is an asymptotic model obtained in the ε → 0 (weakly nonlinear, slow envelope) limit that still contains the small parameter ε. This is the unavoidable consequence of simultaneously considering two balances of diﬀerent asymptotic order: one induced by the dominant eﬀect of the transport at the group velocity (balance described by NLCME) and a second one that is associated with the underlying dispersive, nonlinear, Schr¨ odinger-like dynamics of the system. The small dispersive terms in the NLCMEd are essential to describe the dynamics of the system when it develops small dispersive scales δdisp ∼ |ε|. As it was shown in [16], the NLCMEd in the ε → 0 limit constitute a singular perturbation problem (cf. [14]), and the onset of the dispersive scales is not a higher order, longer time eﬀect; it takes place in the same time scale of the NLCME, no matter how small the dispersion coeﬃcient ε is. A solution of the NLCMEd for ε = −10−3 that exhibits small dispersive scales all over the domain is represented in Figure 1: note that, for short time, the dispersive structures just propagate with the group velocity, but, for t ∼ 1, they also interact with each other, giving rise to a very complicated spatio-temporal pattern. The initial condition used in this simulation was a uniform modulus solution with a small and smooth (i.e., associated with an exponentially decaying spectrum) random perturbation that, according to the dispersionless NLCME formulation, was a stable solution (the details of the numerical integration method are given in [16]). In order to ensure that the small scales in Figure 1 are dispersion-induced

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578

C. MARTEL, M. HIGUERA, AND J. D. CARRASCO

2

2

1

1

0

0 0

Ü

1

0

Ü

1

Figure 1. Space-time representation of a solution of the NLCMEd exhibiting small dispersive scales all over the domain (σ = 1/2, κ = 2, ε = −10−3 , and periodicity boundary conditions). See the accompanying ﬁle (69822 01.gif [22.4MB]) for an animation of the onset of the dispersive scales.

scales we have repeated the NLCMEd simulation but with a reduced dispersion ε = −10−3 /4. The result is plotted in Figure 2, whereit can be seen that the small scales again ﬁll the entire domain, but their typical size, δdisp ∼ |ε|, is now approximately one half of that in Figure 1 (see also the corresponding animations 69822 01.gif [22.4MB] and 69822 02.gif [20.1MB]). The results in [10] rigorously establish that the solutions of the NLCME remain asymptotically close to solutions of the original Maxwell equations. This proximity result would appear to be in contradiction with the results just presented in Figures 1 and 2, which show NLCMEd solutions drifting away from the corresponding NLCME solutions because of the destabilization and development of the dispersive scales. There is no contradiction because the result in [10] does not give information about the stability of the solutions involved: when a stable NLCME solution is dispersively unstable, then the result in [10] simply ensures that there is a corresponding close solution of the Maxwell equations, but what happens in the Maxwell equation dynamics is that this solution is also unstable, and the physical system likes to move away from this solution close to the NLCME and develops dispersive scales, as the NLCMEd correctly predict. In other words, this dispersive destabilization is present in the Maxwell equations and is captured by the NLCMEd, but it is simply not seen in the framework of the zero-dispersion NLCME. This point was also numerically veriﬁed in [17], where the solutions of the NLCMEd and the Maxwell equations were computed and compared. In the derivation of the NLCMEd (1.3)–(1.4) from the Maxwell–Lorentz equations, no relative scaling among the diﬀerent small parameters has to be assumed. To make things clear, we brieﬂy sketch below this derivation procedure without scaling. (A detailed derivation

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DISPERSIVE PULSES IN FIBER BRAGG GRATINGS

579

2

2

1

1

0

0 0

Ü

1

Ü

0

1

Figure 2. Space-time representation of a solution of the NLCMEd exhibiting small dispersive scales all over the domain (σ = 1/2, κ = 2, ε = −10−3 /4, and periodicity boundary conditions). See the accompanying ﬁle (69822 02.gif [20.1MB]) for an animation of the onset of the dispersive scales.

can be found in [17], and a similar derivation in the context of the oscillatory instability in extended dissipative systems is given in [19].) The Maxwell–Lorentz equations for the original physical problem of light propagation in an FBG can be written as (using the same nondimensionalization as in [10]) (1.5) (1.6)

∂2E ∂ 2 (E + P ) = , ∂x ˜2 ∂ t˜2 ∂2P = −ωp2 (1 − 2Δn cos(2˜ x))P + ωp2 (n20 − 1)E + ωp2 P 3 , ∂ t˜2

where the only small parameter is the intensity of the grating Δn ∼ ε 1. The basic spatial scale of the periodic grating is λg = π, and the rest of the parameters are of order 1 and independent of ε. We look for weakly nonlinear solutions that are a superposition of the grating resonant wavetrains (1.7)

(E, P ) ∼ (A˜+ ei˜x+iωt + A˜− e−i˜x+iωt ) + c.c. + · · · , ˜

˜

where c.c. stands for complex conjugate. The only assumptions required to obtain the envelope equations are ˜± | |A˜± | 1, · · · |A˜± x ˜x ˜ | |Ax ˜

· · · |A˜± | |A˜± | 1, t˜

and

Δn ∼ ε 1,

that is, small amplitudes that depend slowly on space and time, and small grating strength. The expressions above are inserted into (1.5)–(1.6), and an expansion in powers of the small

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C. MARTEL, M. HIGUERA, AND J. D. CARRASCO

˜± , . . . is obtained. After applying solvability conditions to the resquantities Δn, A˜± , A˜± x ˜ , Ax ˜x ˜ onant problems at each order, we obtain the terms of the resulting envelope equations. We now retain the ﬁrst order nonlinear terms, the ﬁrst order eﬀect of the grating, and the ﬁrst two linear eﬀects (transport and dispersion), and, after rescaling x ˜, t˜, and A˜± as in [10], we end up with the NLCMEd equations (1.3)–(1.4). √ Note that if one sets a priori the scaling x ˜, t˜ ∼ 1/ε and |A˜± | ∼ ε, then the standard NLCME (1.1)–(1.2) are obtained. However, if we try a dispersive nonlinear Schr¨ odinger √ √ (NLS)-like scaling, x ˜ ∼ 1/ ε and |A˜± | ∼ ε, then at ﬁrst order just propagation with the √ group velocity for times t˜ ∼ 1/ ε is obtained. Also, if the length of the domain is of order √ L ∼ 1/ ε and we have periodicity boundary conditions, one ends up, in the longer time scale t˜ ∼ 1/ε, with two averaged NLS equations like the ones analyzed in [15] and [18]. On the other hand, the NLCMEd analyzed in this paper allow us to cover a more general problem: the onset, in the standard NLCME scenario with typical spatial scale x ˜ ∼ 1/ε, of small dispersive √ scales x ˜ ∼ 1/ ε. In this case, all scales in between those two can appear, and in fact they do appear, as is shown in Figures 1 and 2. The main goal of this paper is to show that, in addition to complex spatio-temporal patterns, the dispersion eﬀects can also give rise to new, purely dispersive localized states, which might be of interest from the optical communications point of view. It is interesting to note that the results in this paper apply also to Bose–Einstein condensates in optical lattices (a system that has recently received very much attention [22, 25]) and, in general, to any dissipationless propagative system, extended in one spatial direction, reﬂection- and translation invariant, and with a small superimposed spatial periodic modulation of its background, since the NLCMEd are the appropriate envelope equations for the description of the weakly nonlinear resonant dynamics of this kind of system. In order to show that light propagation in an FBG can happen in the form of dispersive pulses, in section 2 we derive and solve numerically an asymptotic equation for a family of symmetric pulses, and in section 3 we perform some numerical integrations of the complete NLCMEd to show that some of the pulses in this family do propagate as stable localized structures. Finally, some concluding remarks are drawn in section 4. 2. Dispersive pulses. The starting point is the continuous wave (CW) family of constant uniform modulus solutions of the NLCME (1.1)–(1.2), (2.1) (2.2)

iωt+imx , A+ CW = ρ cos θ e

iωt+imx , A− CW = ρ sin θ e

where ρ ≥ 0 is the light intensity ﬂowing through the ﬁber, θ ∈ [− π2 , π2 ] measures the relative amount of both wavetrains, and the frequency and wavenumber of the amplitudes are given by (2.3) (2.4)

σ+1 2 κ + ρ , ω= 2 sin 2θ σ−1 2 κ − ρ cos 2θ. m= sin 2θ 2

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DISPERSIVE PULSES IN FIBER BRAGG GRATINGS

581

A+ A−

|ε| 1

CW x

Figure 3. Sketch of a dispersive pulse on top of a continuous wave.

The CWs with |ω| ∼ 1 and |m| ∼ 1 are approximate solutions of the NLCMEd (up to order ε corrections) and their stability was ﬁrst analyzed in [8] and then completed in [16], where it was found that, for both signs of the dispersion coeﬃcient, there are dispersively unstable CW which are stable in the dispersionless context of the NLCME. The NLCMEd (1.3)–(1.4) have to be considered in the small dispersion limit ε → 0. When looking for small-dispersion induced solutions it seems somehow natural to try to proceed in a similar way as in the semiclassical limit of the NLS, i.e., looking for solutions with a fast dispersive phase extended all over the domain (see, e.g., [4, 11]). But in this case, due to the necessary dominance of the transport terms, one obtains at ﬁrst order just linear propagation with constant group velocity (±1) for the phases, and therefore the phenomenology of the semiclassical limit of the NLS equation (shocks, singularities, etc.) that comes from the resulting nonlinear equation for the phase is not present in the regime explored here. Instead, we look here for a diﬀerent type of solution: small-dispersion induced solutions in which the dispersion eﬀects are not extended but localized. More precisely, we look for dispersive pulses of width ∼ |ε| that propagate on top of a stable uniform CW, as sketched in Figure 3. In order to turn the background CW into a constant, it is convenient to ﬁrst perform in the NLCMEd the change of variables A+ = F + eiωt+imx , A− = F − eiωt+imx , to obtain (2.5) (2.6)

+ , Ft+ − Fx+ + i(ω − m)F + = iκF − + iF + (σ|F + |2 + |F − |2 ) + iεFxx

− . Ft− + Fx− + i(ω + m)F − = iκF + + iF − (σ|F − |2 + |F + |2 ) + iεFxx

A localized dispersive pulse on F + depends on the fast spatial scale X = x/ |ε|, and, according to (2.5)–(2.6), in the short time scale T = t/ |ε| ∼ 1 it just propagates with the group velocity, suggesting that we have to look for solutions of the form F + = F0+ (η, t) + · · · , F − = F0− (η, t) + · · · ,

with η = X + T . Inserting the above ansatz into (2.5)–(2.6) yields − = 0, F0η

which gives

F0− = ρ sin θ,

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582

C. MARTEL, M. HIGUERA, AND J. D. CARRASCO

which means that F − remains in ﬁrst approximation equal to the unperturbed CW. Similarly, the following equation is obtained for F0+ : (2.7)

+ + + i(ω − m)F0+ = iκρ sin θ + iF0+ (σ|F0+ |2 + ρ2 sin2 θ) ± iF0ηη , F0t

where the + (−) sign corresponds to ε positive (negative), together with the boundary conditions (2.8)

F0+ → ρ cos θ

for

η → ±∞,

which ensure that the background CW is recovered away from the pulse. For dispersive pulses propagating over a zero background we have to set to zero ρ, ω, and m in (2.7). A standard NLS equation is then obtained, which is known to exhibit localized pulses (solitons) in the focusing case of positive dispersion (recall that σ > 0). Note that the eﬀect of the grating on the dispersive pulses is felt only through the background CW, and therefore it is completely gone in this case. In the scaling of the original problem of light propagation on an FBG ((1.5)–(1.6)), the only nonzero Fourier spectrum components of these NLS solitons ˜ with correspond, in ﬁrst approximation, to dispersive wavenumbers (k˜ = k˜resonant + Δk, ˜ Δk ∼ |ε|) that are so oﬀ-resonance that they simply do not feel the grating (recall that k˜resonant = ±1; see (1.7)). Equation (2.7) is an NLS equation with a direct forcing term coming from the eﬀect of the grating. The steady solutions of this equation and their stability properties were analyzed in [2, 3], where explicit analytic expressions for the steady pulses were found. In this paper we consider the more general family of traveling localized solutions. More precisely, we look for traveling pulses of the form F0+ = ρ cos θ(1 + a(η + vt)), where v represents, in the original variables, a small correction of the group velocity. The resulting boundary value problem for a, after making use of relations (2.3) and (2.4) and the √ rescaled variable ξ = (η + vt)( σρ cos θ), can be written as (2.9)

v aξ + αa − (|a|2 + a + a ¯)(1 + a), aξξ = −iˆ

a → 0 as ξ → ±∞, √ where vˆ = v/( σρ cos θ) and α = k tan(θ)/(σρ2 cos2 θ). We are restricting our search to the focusing case of positive sign in (2.7) and, in order to have a dispersively stable background CW, we have to consider only the range 0 < θ < π2 (see [16]), which implies that we have to look for pulses in (2.9)–(2.10) only for α > 0. On the other hand, (2.9)–(2.10) remain invariant under the transformation vˆ → −ˆ v and a → a ¯, and therefore we can also set vˆ ≥ 0. The solutions of (2.9)–(2.10) correspond to traveling dispersive pulses and can be regarded as homoclinic orbits (in the variable ξ) connecting the trivial state (i.e., a = 0) back to itself. To analyze the existence of such connections we ﬁrst consider the linearized system around the zero state: (2.10)

(2.11)

du = A∞ u, dξ

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DISPERSIVE PULSES IN FIBER BRAGG GRATINGS

α

583

7

6

5

(b)

4

(a)

X X X X

3

(c)

XX X X

X X X X

2

X

1

0

0

0.5

X X X

1

(d) 1.5

2

2.5

3

3.5

4

vˆ2 Figure 4. Eigenvalues of the trivial state as a function of the parameters α and vˆ2 . Sketches show the distribution of the four eigenvalues in each of the distinct regions.

where uT ≡ (ar , ai , (ar )ξ , (ai )ξ ), ar and ai are the real and imaginary part of a, (ar )ξ = dar /dξ and (ai )ξ = dai /dξ, and the matrix A∞ is given by ⎛ ⎞ 0 0 1 0 ⎜ 0 0 0 1 ⎟ ⎟ (2.12) A∞ = ⎜ ⎝ α − 2 0 0 vˆ ⎠ . 0 α −ˆ v 0 √ √ The eigenvalues of this system are given by λ1± = η± and λ2± = − η± , with v 2 + 2) ± (ˆ v 2 + 2)2 − 4ˆ v 2 α /2. (2.13) η± = 2α − (ˆ Figure 4 shows the behavior of the four eigenvalues λ1± and λ2± in the (α, vˆ2 ) plane. There v 2 : (a) four are four distinct regions, separated by the boundaries α = 2 and α = (ˆ v 2 + 2)2 /4ˆ real eigenvalues, (b) two pairs of complex conjugate eigenvalues, (c) four purely imaginary eigenvalues, and (d) two real eigenvalues along with two purely imaginary eigenvalues. In regions (a) and (b) the unstable and stable manifolds of the origin are two dimensional, while the equilibrium is nonhyperbolic in regions (c) and (d) where the center manifold is, respectively, four- and two dimensional. Homoclinic orbits belong to both the stable and unstable manifold of the origin. We investigate below the presence of homoclinic solutions in cases (a) and (b), where these correspond to the intersections of two dimensional stable and unstable manifolds in a four dimensional space [23]. The problem (2.9)–(2.10) is invariant under the symmetry (2.14)

a→a ¯,

ξ → −ξ,

that comes from the time reversing (Hamiltonian) and spatial reﬂection symmetries of the NLCMEd. We further restrict our search for dispersive pulses to the case of reﬂectionsymmetric pulses, i.e., to pulses that satisfy a(ξ) = a ¯(−ξ).

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584

C. MARTEL, M. HIGUERA, AND J. D. CARRASCO

If we now set the symmetry axis to ξ = 0, we can reduce the problem to a semi-inﬁnite interval ξ ∈ [0, +∞[ with the boundary conditions (2.15)

(ar , ai ) = (a0 , 0)

(2.16)

(ar , ai ) → (0, 0)

and ((ar )ξ , (ai )ξ ) = (0, b0 ) at as

ξ = 0,

ξ → ∞.

Finally, to numerically compute the proﬁles of the symmetric pulses, we replace the inﬁnite interval by a ﬁnite one, [0, L]. Following [13], the resulting boundary conditions at ξ = L are obtained by requiring that the solution be in the subspace spanned by the eigenvectors associated with the decaying eigenvalues of the matrix A∞ (see (2.11)). In summary, the boundary value problem that we integrate numerically is given by (2.9), which we rewrite as a real ﬁrst order system of four equations in ]0, L[ together with the four boundary conditions (2.17)

C0 u = 0 at

(2.18)

C∞ u = 0 at

where uT ≡ (ar , ai , (ar )ξ , (ai )ξ ), (2.19)

C0 =

ξ = 0 and ξ = L,

0 1 0 0 0 0 1 0

,

and C∞ is a matrix whose rows are the left eigenvectors of A∞ associated with the exponentially growing directions √ √ (η+ − α − vˆ2 ) αˆ v / η+ (η+ − α)/ η+ vˆ . (2.20) C∞ = √ √ v / η− (η− − α)/ η− vˆ (η− − α − vˆ2 ) αˆ For each value of α this problem is solved using a shooting method. We start from the known solutions for vˆ = 0 obtained in [3] and apply numerical continuation techniques to locate the propagating pulses with vˆ > 0. This procedure for setting the boundary conditions at ξ = L, rather than simply imposing a(L) = 0, allows the shooting method to converge faster, and the results obtained are found to be essentially independent of L for L 10. The left panel of Figure 5 shows several families of homoclinic orbits represented in the (a0 , vˆ) plane for diﬀerent values of α and corresponding to the case a0 > 0. The dashed line separates the regions where the homoclinic orbit connects to a saddle point and where it connects to a saddle-focus, while the open circles correspond to the solutions shown in the right panels. In Figure 5(II)(a)–(c) the solutions can be seen to develop oscillations as we move towards a0 = 0. This corresponds to moving along a horizontal line in Figure √ 4 (constant α) √ and to the right, approaching region (c) (precisely at a0 = 0, vˆ = α + α − 2) where the eigenvalues become purely imaginary. This oscillatory behavior, however, is not observed for the families of pulse solutions found for a0 < 0, as seen in Figure 6. Instead, these curves display turning points and, to better appreciate these limit points, the results have been plotted in the plane (E, vˆ), where E is the positive quantity

∞ (|a|2 + |aξ |2 ) dξ. E= 0

In this case, as we move along the curves for ﬁxed α and past the turning point, the pulses develop two extra humps that tend to move away from the origin (see Figure 6 (II) (a) and (b)).

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DISPERSIVE PULSES IN FIBER BRAGG GRATINGS

0.6

ar

0.9

α=4

0.8

(II)

(a)

(I)

a0

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ai

0.4 0.2

3.5

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0 −0.1

0

0.7

0.1

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(a)

0

5

10

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−5

0

(b)

5

10

5

10

ξ

ξ 0.2

0.5

0.3

3

0.4

ar

(b)

0.3

ai

0.2 0.1 0

0.1 0 −0.1

−0.1

0.2

−10

(c)

−5

0

0

5

−0.2 −10

10

0.1

0

0.5

1

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2

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3

ai

0.05

ar

0

ξ

0.05

3.5

vˆ

−5

ξ

(c)

0.1

0

0

−0.05

−0.05 −30

−20

−10

0

10

20

30

−30

−20

−10

0

ξ

ξ

10

20

30

Figure 5. (I) The solid lines correspond to homoclinic cycles to the origin with a0 > 0 for the indicated values of α. The dashed line bounds the regions where the origin is a saddle node (on the left) and saddle-focus (on the right). (II) Spatial proﬁles of three pulses for the values marked in (I) with open circles.

ar

(I)

(II)

(a) 0

0.5

ai

−1 −2 −3

5.8

E

−4

(b)

0

−0.5

−5 −10

−5

0

5

10

−10

−5

ξ

5.6

5.4

5

10

(a)

α=4

5.2

3.5

ar

5

3

4.8

0

ξ

(b)

ai

0

1 0.5 0

−2

−0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

vˆ

1.8

2

−4 −10

−5

0

ξ

5

10

−1 −10

−5

0

ξ

5

10

Figure 6. (I) Curves of homoclinic cycles to the origin with a0 < 0 for the indicated values of α. (II) Spatial proﬁles of two pulses corresponding to the values marked in (I) with open circles.

3. NLCMEd simulations. After having found a two-parameter (α, vˆ) family of symmetric dispersive pulses (DP) that can be numerically continued from the solutions for vˆ = 0 obtained in [3], we now proceed to study their stability. The idea is not to perform a complete stability analysis of the family of DP, but to show that stable DP can be found and that the DP can thus be considered as robust realizable localized structures of light propagation in FBG. To do this we select several DP, place them on top of their corresponding background CW, add a small random perturbation, and use them as initial conditions for the full system of NLCMEd (1.3)–(1.4) that we numerically integrate for a certain amount of time with periodic boundary conditions. The numerical method for the integration of the NLCMEd uses a Fourier series in space with NF modes and a fourth order Runge–Kutta scheme for the time integration of the resulting system of ordinary diﬀerential equations for the Fourier coeﬃcients. The stiﬀ linear diagonal terms

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586

C. MARTEL, M. HIGUERA, AND J. D. CARRASCO

7

7

6

6

5

5

4

4

3 0

Ü

3 1

0

Ü

1

Figure 7. Space-time representation of the solution of the NLCMEd (σ = 1/2, κ = 1, ε = 10−5 , and periodicity boundary conditions) for an unstable pulse propagating on top of a CW (ρ = 1 and θ = π/4). The pulse parameters are α = 4 and vˆ = 1, and it corresponds to the point labeled (a) in Figure 6. See the accompanying ﬁle (69822 03.gif [15.8MB]) for an animation of this pulse destabilization.

associated with the small dispersion coeﬃcients are integrated implicitly, and the nonlinear terms are computed in physical space with the usual 2/3 rule to avoid aliasing eﬀects [5]. (The maximum required resolution for the simulations in this paper was NF = 4096 and Δt = .0005.) Unstable DP simply do not persist, and their shape changes, as can be appreciated from the spatio-temporal evolution shown in Figure 7, which corresponds to the pulse labeled (a) in

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DISPERSIVE PULSES IN FIBER BRAGG GRATINGS

587

42

42

41

41

40 0

Ü

40 1

0

Ü

1

Figure 8. Space-time representation of the solution of the NLCMEd (σ = 1/2, κ = 1, ε = 10−5 , and periodicity boundary conditions) for a stable pulse propagating on top of a CW (ρ = 1 and θ = π/4). The pulse parameters are α = 4 and vˆ = 2.6, and it corresponds to the point labeled (a) in Figure 5. See the accompanying ﬁle (69822 04.gif [3.6MB]) for an animation of this propagating pulse.

Figure 6 (see also the corresponding animation 69822 03.gif [15.8MB]). Note that the size of the pulse ﬁrst grows (at t = 5.5 it is larger than at t = 3), and then it decays again at t = 7.0. A few time units later the oscillatory tails spread, and the pulse structure is eventually lost (not shown in the ﬁgure). All DP explored for a0 < 0 propagated over the same simple CW with parameters ρ = 1 and θ = π/4 (cf. (2.1)–(2.2)), which corresponds to α = 4 in (2.14), and all were found to be unstable regardless of the propagation speed vˆ. On the other hand, for a0 > 0 and for the same background CW (i.e., α = 4), the DP are found to be unstable approximately for 0 ≤ vˆ 2.2 and stable for v 2.2. The evolution of two stable pulses is shown in Figures 8 and 9, which correspond to vˆ = 2.6 and vˆ = 3.2, respectively, where the structure of the slightly perturbed pulses is seen to remain now virtually unaltered after more than 40 time units (see also the corresponding animations 69822 04.gif [3.6MB] and 69822 05.gif [3.5MB]). The stability of the pulses was tested by numerical simulation of the complete envelope equations (1.3)–(1.4), using as initial condition the pulse with a small superimposed random perturbation (the details of the spectral numerical integration scheme are given in [16]). For the case of stable pulses they remained practically undistorted for times of the order of t = 40 (Figures 8 and 9). This is a suﬃciently large time since the growth rate of the instabilities is clearly of order 1, as can be seen in Figure 7 where the unstable pulses are already severely distorted at t = 7. Despite the fact that this is not a rigorous stability analysis, we believe that the results of these simulations strongly suggest that some of the pulses found are indeed stable and are therefore robust dynamical states of the system for the time scale t ∼ 1, which

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588

C. MARTEL, M. HIGUERA, AND J. D. CARRASCO

42

42

41

41

40 0

Ü

40 1

0

Ü

1

Figure 9. Space-time representation of the solution of the NLCMEd (σ = 1/2, κ = 1, ε = 10−5 , and periodicity boundary conditions) for a stable pulse propagating on top of a CW (ρ = 1 and θ = π/4). The pulse parameters are α = 4 and vˆ = 3.2, and it corresponds to the point labeled (b) in Figure 5. See the accompanying ﬁle (69822 05.gif [3.5MB]) for an animation of this propagating pulse.

is the relevant one in the NLCME dynamics. Another very interesting feature of the DP that is worth mentioning is the fact that they are somehow transparent to each other: two DP propagating in opposite directions just pass through each other without distortion. This is illustrated in Figure 10 (see also the animation 69822 06.gif [3.9MB]), where two stable DP (corresponding to those in Figures 8 and 9) are sent towards each other and after 40 time units (approximately 80 collisions) they still remain practically undistorted. The reason behind this behavior is the dominant character of the transport eﬀect induced by the group velocity. If we rewrite the NLCMEd for a DP with √ √ short (dispersive) spatial and temporal scales X = x/ ε ∼ 1 and T = t/ ε ∼ 1, they take, at ﬁrst order, the form of two uncoupled wave equations: + A+ T = AX + · · · ,

− A− T = −AX + · · · .

It is clear then that the DP traveling in opposite directions simply propagate through diﬀerent channels, and are (in ﬁrst approximation) completely independent. 4. Conclusions. In this paper we have studied the eﬀect of dispersion in the weakly nonlinear dynamics of light propagation in an FBG. We have shown that the (often neglected) small dispersion terms play a crucial role in the transport-dominated dynamics of light propagation in FBG. The combined eﬀect of propagation at the group velocity and dispersion can give rise to complex spatio-temporal chaotic states, but also to a new family of localized

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DISPERSIVE PULSES IN FIBER BRAGG GRATINGS

589

42

42

41

41

40 0

Ü

40 1

0

Ü

1

Figure 10. Space-time representation of the solution of the NLCMEd (σ = 1/2, κ = 1, ε = 10−5 , and periodicity boundary conditions) showing the simultaneous propagation in opposite directions of the two stable pulses from Figures 8 and 9. See the accompanying ﬁle (69822 06.gif [3.9MB]) for an animation of this pulse interaction.

states (dispersive pulses, DP) that propagate on top of a CW. These DP are approximately advected by the group velocity, but this transport eﬀect does not play any role in the determination of their internal structure, which results basically from a balance between nonlinearity and dispersion. It is also important to emphasize that this type of dynamics is not contained in the standard dispersionless NLCME (1.1)–(1.2) formulation usually applied to model light propagation in FBG. In the context of light propagation on FBG the localized structures that have been most extensively analyzed are the so-called gap solitons (GS). The GS are well-known pulse-like solutions of the NLCME (1.1)–(1.2) that have the striking property that they can propagate at any velocity between zero and the speed of light of the bare ﬁber (see, e.g., [1, 10] and references therein). The new family of localized structures presented in this paper, the DP, appear in the same regime of weakly nonlinear light propagation in FBG where the GS exist (the most frequently analyzed regime). There are robust stable DP, and their dynamics evolve in the same time scale as the GS, but they exhibit several essentially diﬀerent characteristics that are worth mentioning: (i) the GS propagate over the zero intensity state, while the DP propagate on top of a saturated uniform continuous wave; (ii) both GS and DP have amplitudes of the same size, but the DP are much more narrow pulses (in the scalings used in √ this paper, δDP ∼ δdisp ∼ ε 1 and δGS ∼ 1); (iii) the DP propagate with the speed of light of the bare ﬁber (v = ±1) and only on one of the amplitudes, A+ or A− , while the GS have nonzero component on both amplitudes; and, ﬁnally, (iv) the DP have the property that two DP propagating in opposite directions just pass through each other without distortion, which,

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590

C. MARTEL, M. HIGUERA, AND J. D. CARRASCO

a priori, looks like it could be interesting from the point of view of information transmission. The ideas used in the analysis performed in this paper go beyond the context of light propagation on ﬁber gratings and apply to the (systematically not considered in the literature) generic case of pattern formation in propagative spatially extended systems. The main ingredients are (i) the unavoidable diﬀerent asymptotic order of transport (ﬁrst order spatial derivatives) and dispersion/dissipation (second derivatives), and (ii) the need to consider these two eﬀects simultaneously in order to correctly account for the weakly nonlinear dynamics of the system. The envelope equations are necessarily asymptotically nonuniform; that is, the small parameter is not gone from the envelope equation formulation, reﬂecting the impossibility of balancing the two eﬀects mentioned above. The resulting dynamics is much richer, and new patterns (as the pulses obtained in this paper) and instabilities can be found that have shorter spatial scales but that are still slow as compared with the basic linear wavetrains and therefore well accounted for by the envelope equation formalism. All these new dynamical states are not detected if the usual formulation that only takes into account transport eﬀects is applied. Finally, it is also interesting to mention that similar eﬀects have been previously described in the context of the oscillatory instability in dissipative systems [20] and in parametrically forced surface waves [21]. REFERENCES [1] A. Aceves, Optical gap solitons: Past, present and future; theory and experiment, Chaos, 10 (2000), pp. 584–589. [2] I. Barashenkov and Y. S. Smirnov, Existence and stability chart for the AC-driven, damped nonlinear Schroedinger solitons, Phys. Rev. E, 54 (1996), pp. 5707–5725. [3] I. Barashenkov, T. Zhanlav, and M. Bogdan, Instabilities and soliton structures in the driven nonlinear Schroedinger equation, in Nonlinear World Vol. 1, Proceedings of the IV International Workshop on Nonlinear and Turbulent Processes in Physics, V. Bar’yakhtar, V. Chernousenko, N. Erokhin, A. Sitenko, and V. Zakharov, eds., World Scientiﬁc, River Edge, NJ, 1989, pp. 3–9. [4] J. C. Bronsky and D. W. McLaughlin, Semiclassical behavior in the NLS equation: Optical shocksfocusing instabilities, in Singular Limits of Dispersive Waves, NATO Adv. Sci. Inst. Ser. B Phys. 320, N. Ercolani, I. Gabitov, C. Levermore, and D. Serre, eds., Plenum, New York, 1994, pp. 21–38. [5] C. Canuto, H. Hussani, A. Quarteroni, and T. Zang, Spectral Methods in Fluid Mechanics, Springer Ser. Comput. Phys., Springer-Verlag, New York, Berlin, 1988. [6] A. Champneys, B. Malomed, and M. Friedman, Thirring solitons in the presence of dispersion, Phys. Rev. Lett., 80 (1998), pp. 4169–4172. [7] D. Christodoulides and R. Joseph, Slow Bragg solitons in nonlinear periodic structures, Phys. Rev. Lett., 62 (1989), pp. 1746–1749. [8] C. de Sterke, Theory of modulational instability in ﬁber Bragg gratings, J. Opt. Soc. Amer. B Opt. Phys., 15 (1998), pp. 2660–2667. [9] C. de Sterke and J. Sipe, Gap solitons, Progr. Optics, 33 (1994), pp. 203–260. [10] R. Goodman, M. Weinstein, and P. Holmes, Nonlinear propagation of light in one-dimensional periodic structures, J. Nonlinear Sci., 11 (2001), pp. 123–168. [11] S. Jin, C. Levermore, and D. W. McLaughlin, The behavior of solutions of the NLS equation in the semiclassical limit, in Singular Limits of Dispersive Waves, NATO Adv. Sci. Inst. Ser. B Phys. 320, N. Ercolani, I. Gabitov, C. Levermore, and D. Serre, eds., Plenum, New York, 1994, pp. 235–255. [12] R. Kashyap, Fiber Bragg Gratings, Optics and Photonics, Academic Press, New York, 1999. [13] H. Keller, Numerical Solution of Two Point Boundary Value Problems, CBMS-NSF Regional Conf. Ser. in Appl. Math. 24, SIAM, Philadelphia, 1976. [14] J. Kevorkian and J. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Sci. 114, Springer-Verlag, New York, Berlin, 1996.

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[15] E. Knoboch and J. Gibbon, Coupled NLS equations for counterpropagating waves in systems with reﬂection symmetry, Phys. Lett. A, 154 (1991), pp. 353–356. [16] C. Martel, Dispersive destabilization of nonlinear light propagation in ﬁber Bragg gratings, Chaos, 15 (2005), paper 013701. [17] C. Martel and C. Casas, Dispersive destabilization of nonlinear light propagation in ﬁber Bragg gratings: A numerical veriﬁcation, Chaos, 17 (2007), paper 013114. [18] C. Martel, E. Knoboch, and J. Vega, Dynamics of counterpropagating waves in parametrically driven systems, Phys. D, 137 (2000), pp. 94–123. [19] C. Martel and J. Vega, Finite size eﬀects near the onset of the oscillatory instability, Nonlinearity, 9 (1996), pp. 1129–1171. [20] C. Martel and J. Vega, Dynamics of a hyperbolic system that applies at the onset of the oscillatory instability, Nonlinearity, 11 (1998), pp. 105–142. [21] C. Martel, J. Vega, and E. Knoboch, Dynamics of counterpropagating waves in parametrically driven systems: Dispersion vs. advection, Phys. D, 174 (2003), pp. 198–217. [22] H. Sakaguchi and B. Malomed, Dynamics of positive- and negative-mass solitons in optical lattices and inverted traps, J. Phys. B, 37 (2004), pp. 1443–1459. [23] S. Wiggins, Global Bifurcation and Chaos, Appl. Math. Ser. 73, Springer-Verlag, New York, Berlin, 1988. [24] H. Winful and G. Cooperman, Self-pulsing and chaos in distributed feedback bistable optical devices, Appl. Phys. Lett., 40 (1982), pp. 298–300. [25] A. Yulin and D. Skryabin, Out-of-gap Bose-Einstein solitons in optical lattices, Phys. Rev. E, 67 (2003), paper 023611.

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c 2009 Society for Industrial and Applied Mathematics

SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 8, No. 2, pp. 592–623

Experimental and Numerical Investigation of Coexistence, Novel Bifurcations, and Chaos in a Cam-Follower System∗ R. Alzate†, M. di Bernardo†, G. Giordano†, G. Rea†, and S. Santini†‡ Abstract. This paper is concerned with the experimental and numerical analysis of an important class of impacting systems in applications: cam-follower devices. Experiments are reported showing the occurrence of complex dynamical scenarios in these systems including coexistence, discontinuityinduced bifurcations, and chaos. The most notable phenomenon is the sudden transition to chaos detected in the system which is shown to be due to the interruption of a complete chattering sequence. Further insight is gained by performing numerical analysis of an appropriately derived model of the experimental setup. Simulation, continuation, and cell-to-cell mapping algorithms are used to gain a complete picture of the observed dynamics in excellent qualitative agreement with the experimental observations. Key words. experimental nonlinear dynamics, piecewise-smooth dynamical systems, impacting systems, chattering, coexistence, bifurcations, chaos, cam-follower AMS subject classifications. 34C15, 34C28, 34C60, 37G15, 37E05, 70K50 DOI. 10.1137/080723867

1. Introduction and motivation. Impacting systems have been the subject of much research eﬀort in dynamical systems and control for many years. Starting with the pioneering work of Whiston [36], Peterka [25, 26], and Nordmark [18], it has been shown that this class of dynamical systems can exhibit a rich bifurcation scenario involving the occurrence of both classical bifurcations (saddle-node, period-doubling, etc.) and so-called discontinuity-induced bifurcations (DIBs) [9]. DIBs are unique to piecewise-smooth dynamical systems and are associated with the nontrivial interaction between system trajectories and discontinuity boundaries (or manifolds) in phase-space where the states (or vector ﬁeld) become nonsmooth. In impacting systems, the most notable type of DIB is the grazing bifurcation of a limit cycle, observed when, under parameter variations, a limit cycle becomes tangential to the system discontinuity manifold. Grazing bifurcations have been shown to be associated with a wide range of dynamical transitions, including nonsmooth folds and sudden transitions from periodic to chaotic behavior (see [9] and references therein for further details). Another important feature of impacting systems is the possibility for an inﬁnite sequence of impacts to accumulate in ﬁnite time. This phenomenon, also termed chattering or Zeno behavior in the literature [5, 19, 20, 28], has been shown to be the key to uncovering the ∗ Received by the editors May 10, 2008; accepted for publication (in revised form) by T. Kaper February 6, 2009; published electronically May 7, 2009. http://www.siam.org/journals/siads/8-2/72386.html † Dipartimento di Informatica e Sistemistica, Universit` a degli Studi di Napoli Federico II, Via Claudio 21, 80125, Napoli, Italia ([email protected], [email protected], [email protected], [email protected], stefania. [email protected]). ‡ Corresponding author.

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intricate structure of the system dynamics, as, for example, to predict the topology of its basins of attraction or regions where sticking occurs. Sticking in impact oscillators corresponds to the mass remaining in contact with the impacting obstacle over a ﬁnite time interval, and it has been recently related to the occurrence of so-called sliding solutions in piecewise-smooth ﬂows [10]. In [19], it has been proposed that a new type of DIB occurs in impacting systems when, under parameter variations, a complete chattering sequence (leading to sticking) is interrupted. Basically, when one or more parameters are varied, a periodic orbit characterized by an inﬁnite number of impacts accumulating in ﬁnite time suddenly loses its stability as the chattering sequence becomes incomplete with the trajectory escaping the sticking region after a ﬁnite (large) number of impacts. The phenomenon described above has been observed by some authors in the existing literature and given the name of “rising bifurcation” or “chattering interruption.” A reference to this phenomenon can be found in [31], while numerical evidence of its occurrence in a two-degree-of-freedom impacting oscillator is reported in [33, 34, 35]. Therefore a pressing open problem is to fully investigate this novel bifurcation phenomenon which is unique to impacting systems. It is worth mentioning here that, despite its theoretical and numerical observations, this phenomenon has never been experimentally shown to occur. In applications, the occurrence of complex behavior in impacting systems has been recently detected in an important class of devices: cam-follower systems [1, 22, 23]. These are widely used in a large range of mechanical devices, most notably in internal combustion engines. In these systems, an appropriately shaped rotating cam imparts to the follower a desired motion that is used to operate a device of interest (see [21] for further details). In [22, 23] it was observed that cam-follower devices can exhibit complex behavior which was conjectured to be due to chattering and its interruption. The aim of this paper is to provide, for the ﬁrst time in the literature, numerical and experimental evidence of these phenomena by means of a novel cam-follower experimental rig which was developed by the authors and described in [1]. Experimental evidence is contrasted with the numerical predictions of an appropriately derived analytical model of the mechanical rig. It is shown that the interruption of a complete chattering sequence is indeed the mechanism that explains the sudden transition to chaos observed in the device. Also, a coexisting solution branch is detected, showing that a traditional period-doubling cascade coexists with a DIB bifurcation scenario. The simulation and experimental time-series are complemented with continuation results and the computation of the system basins of attraction. The resulting picture is the unfolding of the observed bifurcation behavior and the complete characterization of the dynamics of the system under investigation. Throughout the paper, excellent agreement is shown between the experiments and the numerical predictions. 2. The experimental rig: A brief description. In this section, the experimental rig of interest is brieﬂy described. Further details on the setup can be found in [1]. The rig was designed to investigate the dynamics of cam-follower systems with particular attention to the impacting behavior following the detachment of the follower from the cam. The experimental apparatus is shown in Figure 1, where the cam pushes a rotational follower attached through a spring to a rigid ﬁxed iron frame. Note that the cam-follower system in the rig can be assumed to be stiﬀ and large enough to reduce possible vibrations induced by the inertial force. The cam is also equipped with a ﬂywheel which is designed ad hoc in order to compensate undesired oscillations and torque variations due, for example,

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Figure 1. Schematic of the experimental rig.

to the occurrence of impacts. As explained in [1], the rotational geometry of the rig proposed here makes it easier to measure the follower motion and the cam position directly from the cam-shaft and followershaft rotational angles. Furthermore, the choice of an oscillating arm has the advantage of reducing friction at contact points and, as a consequence, the wearing of pieces and ease of replacement [21]. Also, from a nonlinear dynamics viewpoint, the rotational nature of the follower motion makes it easier to avoid the presence of unwanted stick-slip motion due to friction, which would make the theoretical understanding of the phenomena observed particularly cumbersome. Currently two diﬀerent types of cam can be alternatively mounted on the experiment, which provide, respectively, a simple harmonic motion (eccentric circular cam) and a proﬁle characterized by discontinuities in the acceleration. In this work we will show experiments related to an eccentric circular cam, which is often used to produce motion in pumps or to operate steam engine valves [21]. Other examples of various applications making use of eccentric circular cams can be found in [4, 8, 11, 13, 32]. The use of circular cams in the automotive ﬁeld is instead discussed in [7] and [29]. Further features of the experimental rig described above, together with a table of all physical parameters, can be found in Appendix A. 3. Experimental nonlinear dynamics. Figure 2 shows the complete experimental bifurcation diagram of the system as the angular velocity of the cam, ω, varies between 120 and 200 rpm. Here, the cam-follower exhibits a complex dynamic behavior characterized by different coexisting solutions, several bifurcations, and chaos (see the corresponding multimedia ﬁle showing the experimental bifurcation diagram and the associated cam-follower dynamics, 72386 01.mov [30.1MB]). In order to capture experimentally the two coexisting scenarios, two diﬀerent experimental

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

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runs were performed. First, the cam velocity was swept forward from 120 to 200 rpm. For each value of the cam velocity, 10 seconds of the steady-state behavior of the follower angular position θf were stored, and the stroboscopic points were plotted. In this way, the forward bifurcation diagram was obtained (see Figure 3(a)). A second experiment was then performed with the velocity ω being decreased from 200 to 150 rpm. The result is the backward bifurcation diagram depicted in Figure 3(b). The complete bifurcation diagram, shown in Figure 2, was obtained by overlapping the

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ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI

forward and backward experimental diagrams. It shows two main diﬀerent coexisting routes to chaos, namely: • a sudden transition to chaos seemingly due to the transition from a complete chattering sequence to an incomplete one (labeled as Scenario 1 in the rest of the paper); • a classical period-doubling cascade (labeled as Scenario 2). In what follows, we study each of these scenarios in greater detail, complementing the experimental bifurcation diagram with experimental time-series, phase plane portraits, and stroboscopic maps at the most signiﬁcant values of the cam rotational speed. The stroboscopic map, Π, is obtained experimentally by measuring the follower angular position θf and velocity θ˙f , periodically at each T = 2π/ω. Hence, it is deﬁned as (3.1)

(θfn , θ˙fn ) −→ (θfn+1 , θ˙fn+1 ),

where θfn = θf (nT ) and θ˙fn = θ˙f (nT ). In the rest of the paper, a generic nT -periodic orbit characterized by m impacts per period will be labeled as a P (m, n) orbit. nT -periodic orbits with sticking, characterized by the accumulation of an inﬁnite number of impacts in ﬁnite time, will be labeled as P (∞, n). Note that the remarkable quality of the experimental results presented in this paper is due to the excellent quality of the sensing equipment (a gyroscopic high-quality sensor for the velocity and an optical encoder for the position). Also vibrations due to impacts, usually observed in other experimental setups, are here carefully damped by the design of the ﬂywheel attached to the cam, which absorbs most of the unwanted vibrational modes. (For further details, see [1].) 3.1. Scenario 1: Chattering interruption. For Scenario 1, four regions of diﬀerent qualitative behavior, associated with diﬀerent values of ω, can be detected: 1. Permanent contact (ω < 125), where the cam and the follower stay in contact for all time. 2. Periodic impacting behavior (125 < ω < 155), where the existence of a P (∞, 1) orbit is detected. 3. Sudden transition to chaos due to chattering interruption (155 < ω < 160). 4. Aperiodic motion and chaos (ω > 160). We now give some experimental evidence for each of these scenarios. 3.1.1. Permanent contact—Low velocity regime (ω < 125). The experimental investigation starts by using low values of the cam rotational speed. Experiments conﬁrm the presence of permanent contact between the cam and the follower in this range of ω values. For example, in Figure 4, the dynamics of the follower at a constant cam angular velocity ω = 120 rpm is shown. At this velocity value, the restoring force of the follower is higher than the force exerted by the constraint represented by the cam. In this condition, the cam and the follower remain in contact at all time; hence we have θf = θˆc , where θˆc is the cam angular displacement at the follower joint, a function of the cam angular position θc . It is important to note that permanent contact is experimentally detected up to approximately 125 rpm. We wish to emphasize the extremely small experimental measurement error shown in Figure 4 (ε ≈ 10−3 ).

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS 0.34

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Figure 4. Experimental results. Permanent contact at ω = 120 rpm. Left frame: time history of the cam (dashed line) and follower (solid line) angular position (top panel), with their corresponding diﬀerence Δϑ (bottom panel). Dash-dot line: maximum follower angular displacement. Right frame: phase plane portrait (θf vs. θ˙f ). The red dot represents the stroboscopic sample. 0.7

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Figure 5. Experimental results. P (∞, 1) orbit at ω = 150 rpm. Left frame: time history of the cam (dashed line) and follower (solid line) angular position (top panel), with their corresponding diﬀerence Δϑ (bottom panel). Dash-dot line: maximum follower angular displacement. Right frame: phase plane portrait (θf vs. θ˙f ). The red dots represent the stroboscopic samples.

3.1.2. Periodic impacting behavior (125 < ω < 155). For values of ω ∈ [125, 155] rpm, the follower motion is observed to exhibit 1T -periodic behavior characterized by an inﬁnite number of impacts per period (P (∞, 1) orbit). One example of such a periodic sticking orbit is depicted in Figure 5, where time histories of both the experimental cam and follower angular positions are shown together with their diﬀerence at ω = 150 rpm. Here, it is apparent that during every period an inﬁnite number of impacts accumulate in ﬁnite time (complete chattering) before the sticking phase. 3.1.3. Chattering interruption (155 < ω < 160). For 155 < ω < 160 rpm the system is observed to exhibit a sudden transition to chaos. A closer look at this parameter region

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ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI

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t [s] Figure 6. Experimental results. Chattering interruption. Time history of diﬀerence Δϑ between cam and follower angular position at ω = 155 (a), ω = 158 (b), and ω = 160 rpm (c).

shows that the transition is observed from a complete to an incomplete chattering sequence. Namely, as ω is varied past a critical value, a P (∞, 1) orbit turns into a P (N, 1) solution with N 1. Experimental evidence of such a transition is depicted in Figure 6, where a periodic orbit with sticking (a) is shown to turn into a periodic orbit without sticking (c), resulting in the interruption of the complete chattering sequence. This bifurcation scenario will be studied in further details with additional numerical tools in section 5. Past the critical value of the cam velocity, corresponding to ω = 160 rpm, the follower starts exhibiting aperiodic behavior and sensitive dependence on initial conditions. A representative case of this chaotic behavior is shown in Figure 7, where the time history of the chaotic dynamics together with the corresponding phase plane portrait and stroboscopic points at ω = 165 rpm are depicted. 3.2. Scenario 2: Period-doubling cascade. As mentioned above, in the range 150 < ω < 200, the experimental system also undergoes a coexisting classical period-doubling cascade similar to those reported in the literature on impact oscillators [3, 12]. In particular, for ω ∈ [150, 188] rpm the system exhibits large-amplitude periodic behavior characterized by one impact per period, such as that shown in Figure 8 when ω = 165 rpm. When increasing the cam angular velocity beyond 188 rpm, the follower motion exhibits a period-doubling bifurcation (see Figure 9). The resulting P (2, 2) orbit persists for all the admissible values of the cam velocity. The sharp corner in the P (2, 2) branch observed in the experimental bifurcation diagram at ω ≈ 191 is only due to the transition of one of the impacts characterizing the P (2, 2) solution through the maximum of the cam proﬁle.

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

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Figure 8. Experimental results. P (1, 1) orbits at ω = 165 rpm. Left frame: time history of the cam (dashed line) and follower (solid line) angular position. Dash-dot line: maximum follower angular displacement. Right frame: phase plane portrait (θf vs. θ˙f ). The red dots represent the stroboscopic samples.

The experimental results described so far show several interesting features. In particular, the coexistence of two diﬀerent scenarios is observed, together with the novel sudden transition to chaos caused by the chattering interruption. To better investigate these phenomena, we now turn to the numerical analysis of an appropriate mathematical model of the experimental rig. 4. Mathematical model. To model the experimental rig we use the mathematical derivation presented in [1], where a Lagrangian approach was employed with the angular position of the follower, θf , taken as the generalized coordinate. In particular, we can derive the following model: sin(θf ) d xA 0, ¨ − (yA − d0 − d1 ) +d 2 = (4.1) J θf + K xA tan(θf ) + 2 τ (t), cos(θf ) cos (θf ) cos (θf )

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Figure 9. Experimental results. P (2, 2) orbit at ω = 188 rpm. Left frame: time history of the cam (dashed line) and follower (solid line) angular position. Dash-dot line: maximum follower angular displacement. Right frame: phase plane portrait (θf vs. θ˙f ). The red dots represent the stroboscopic samples.

where J and K represent, respectively, the inertia of the follower body and the stiﬀness of the spring; τ (t) is the periodic forcing exerted by the cam; (xA , yA ) are the coordinates of the spring hooking point; d is the half height of the follower from the rotation plane; d0 is the relaxed spring length; and d1 is the distance between the ending points of a mechanical device used to avoid the deformation of the spring by rotation. Further details on the notation and the parameter values are reported in Appendix B, together with a schematic of the system. When the right-hand side of (4.1) is equal to zero, the system is in the unconstrained mode of operation. Conversely, if the right-hand side is equal to τ (t) = 0, the system evolution becomes forced or constrained. The transition between such modes at the discontinuity boundary is given by the following restitution law: d ˆ ˙ − θ˙f (t+ k ) = −r θf (tk ) + (1 + r) ω dθ θc , c

(4.2)

where r ∈ [0, 1] is the restitution coeﬃcient (inelastic impact), ω is the rotational velocity of the cam (in rpm) and θˆc is the angular projection of the contact point along the cam. Notice that, as shown in [1], θˆc can be computed as a function of the rotation-angle, θc , for a circular cam proﬁle as (4.3) θˆc (θc ) = arcsin

d+R (e cos(θc ) + x0 )2 + (e sin(θc ) + y0 )2

e sin(θc ) + y0 − arctan − e cos(θc ) + x0

,

where e and R are the eccentricity and radius of the cam and (x0 , y0 ) the coordinates of the center of rotation in the reference plane. All model parameters including the coeﬃcient of restitution r were identiﬁed as reported in [1] and are included in Appendix B.

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Remarks. • Note that the model presented above, whose derivation is carefully described in [1], includes damping only through the coeﬃcient of restitution law at the impacts. The lack of an explicit damping term was motivated by the modelling choice of keeping the model as simple as possible and minimizing friction at the follower joint in the experimental setup. As shown in this paper, the model provides good qualitative and quantitative predictions of the experimental dynamics. • As explained in [1], all model parameters were identiﬁed either using the physical parameters of the experimental setup or by means of appropriate identiﬁcation procedures. Most notably, the parameter r in the model was identiﬁed as a function of the rotational cam velocity ω and estimated via a least-squares optimization method. (See [1] for further details.) 5. Numerical analysis. Using the model described above, we now explore the system dynamics in greater detail starting from the derivation of the bifurcation diagram. 5.1. Bifurcation diagram. The bifurcation diagram predicted by the numerical simulations of the analytical model is shown in Figure 10 (see also the time-series in Figure 11). The qualitative agreement between the numerical and the experimental diagrams is remarkable. The simulated diagram contains all the features of the cam-follower system dynamics observed experimentally. In particular, the model captures the coexistence of diﬀerent solutions, the

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Figure 11. Time series of the most meaningful follower dynamics observed in the bifurcation diagram in Figure 10. Table 1 List of the diﬀerent values of ω at which the main dynamical events occur respectively in experiments (ωexp ) and simulations (ωsim ). Main dynamical events Permanent contact Detachment P (∞, 1) orbits Chattering route to chaos Chaos P (1, 1) orbits P (2, 2) orbits

ωexp (rpm) < 125 125 ]125, 155] ]155, 160] [160, 170] ]150, 188] ]188, 200]

ωsim (rpm) < 125 125 ]125, 152] [152.6, 154] [154, 154.5] [135, 151] [152, 170]

nonsmooth chattering route to chaos, and the smooth period-doubling cascade.1 The only detectable diﬀerence between the experimental and numerical bifurcation diagrams is related to the ranges of ω in which some of the scenarios described above are observed to take place. In Table 1, the quantitative comparison of the experimental and simulated behavior of the cam-follower system is reported. Namely, the most signiﬁcant dynamical sce1 Actually, the existence of the period-doubling cascade was not detected experimentally at ﬁrst. It was only because of the model prediction that ad hoc experiments were carried out to conﬁrm its existence in the real physical system!

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

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Figure 12. Numerical results. P (1, 1) orbit at ω = 145 rpm. Left frame: cam (dashed) and follower (solid) trajectories. Dash-dot line: maximum follower angular displacement. Right frame: phase portrait.

narios are summarized together with the values of ω at which they occur in the experiments, ωexp, and in the simulations, ωsim . Despite the good qualitative agreement, results highlight that the most signiﬁcant mismatch occurs at high velocity regimes when all the nonlinear eﬀects, neglected in the model, such as friction, the presence of bearings, or ﬂuctuations in the torque acting on the cam, become more relevant. Other sources of uncertainty are the coeﬃcient of restitution and the unavoidable presence of unmodelled dynamics. For example, the coeﬃcient of restitution, which is an index of how elastic a collision is, is theoretically an unknown nonlinear function of the approach speed and the materials’ physical characteristics. Since the impact velocity is not easy to measure directly with a suﬃcient degree of accuracy, r can be estimated in the whole range of the operating conditions of the system with only a limited degree of accuracy. Furthermore, all the geometrical parameters necessary to describe the follower motion are obtained in the permanent contact region in the absence of impacts. In this operating condition, the spring works in its linear region, and thus, by supposing a constant elastic coeﬃcient, a good agreement between experiment and model can be achieved for the unconstrained follower motion [1]. Obviously, the hypothesis of a linear and constant spring introduces some approximation into the model, which explains the mismatch between the predicted values of ω for higher velocity regimes or when the follower works around its maximum displacement. This is particularly relevant for the period-doubling bifurcation that occurs only for large amplitude impacting solutions close to the saturation of the follower position (maximum displacement). This means that all sensing equipment and actuators on board the experimental setup are pushed to work at the boundary of the operating region they were designed for. This explains the apparent mismatch between the numerical predictions for this bifurcation and the experimental observations. 5.2. Period-doubling cascade. The coexisting (large amplitude) smooth period-doubling cascade can be observed numerically for ω ∈ [135, 160] rpm, in accordance with the experimental observations reported in section 3. The fundamental branch experiences a ﬁrst perioddoubling close to ω = 152 rpm. Another period-doubling is then detected at ω ≈ 157.5 rpm. Figures 12, 13, and 14 illustrate this situation by showing the time-series for the P (1, 1), P (2, 2), and P (4, 4) orbits also detected experimentally. Notice the excellent qualitative agreement between the experiments shown in Figures 8 and 9 and the numerical predictions in

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ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI 0.7

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0

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θ˙f [rad/s]

Figure 13. Numerical results. P (2, 2) orbit at ω = 152 rpm. Left frame: cam (dashed) and follower (solid) trajectories. Dash-dot line: maximum follower angular displacement. Right frame: phase portrait. 0.7

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0.6 0.55

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θf , θˆc [rad]

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97

t [s]

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98

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−1

0

1

2

3

θ˙f [rad/s]

Figure 14. Numerical results. P (4, 4) orbit at ω = 158 rpm. Left frame: cam (dashed) and follower (solid) trajectories. Dash-dot line: maximum follower angular displacement. Right frame: phase portrait.

Figures 12 and 13. Note that the orbit in Figure 14 cannot be observed experimentally, as its amplitude is greater than the maximum follower displacement allowed in the experiment. To isolate with greater accuracy the period-doubling bifurcation points, numerical continuation routines were adapted in order to cope with the discontinuous impacting nature of the cam-follower system. The results of such a continuation are reported in Figure 15 and were obtained using TCHAT, a novel toolbox for AUTO developed by Thota and Dankowicz (see [30] for further details). As shown in Table 2, the computed multipliers of the periodic orbit along the P (1, 1) branch show the occurrence of two smooth bifurcations. In particular at ω ≈ 135.946 rpm, one real multiplier is observed to cross the point +1, indicating the occurrence of a fold bifurcation through which the whole P (1, 1) branch originates. Also at ω = 151.65 rpm another real multiplier crosses the unit circle at −1. This conﬁrms that a ﬂip bifurcation is causing the period-doubling observed in both the numerical and experimental bifurcation diagrams. This scenario can therefore be fully explained in terms of classical smooth bifurcations similar to those already studied in impact oscillators [12]. An interesting feature shown in Figure 15 is the seemingly global bifurcation involving the unstable solutions branching from the fold and the chaotic evolution born as a result of the chattering interruption.

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

605

0.5

0.48

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θf [rad]

0.44

0.42

0.4

0.38

0.36

0.34

0.32 130

135

140

145

ω [rpm]

150

155

160

Figure 15. Continuation results. Period-doubling cascade: continuation prediction (solid and dashed lines) vs. stroboscopic points (dotted line). Table 2 Eigenvalue evolution across the P (1, 1) branch computed by using a continuation algorithm. ω value (rpm) 135.946698873817 135.946850039257 135.947353924057 135.949033540057 135.951832900057 147.941373499943 148.962048199932 150.186857839919 151.656629407902 153.420355289483

Eigenvalue 1 0.212856564951 0.214199564174 0.216351137284 0.220281698370 0.224549930832 −0.467436119698 − 0.300839993821i −0.521622451717 − 0.214489798539i −0.467765740979 −0.353243069838 −0.297944998464

Eigenvalue 2 0.996008664393 0.989769009976 0.979943000383 0.962513414400 0.944309362903 −0.467436119698 + 0.300839993821i −0.521622451717 + 0.214489798539i −0.703735852712 −0.970265166132 −1.206154145399

This explains the abrupt disappearance of that chaotic attractor when ω = 154.5 rpm. We move now to the analysis of Scenario 1 which is instead organized by DIBs unique to impacting dynamical systems [9]. 5.3. Chattering route to chaos. The other scenario detected experimentally and captured by the numerical bifurcation diagram emerges from the permanent contact solution shown in Figure 16. It consists of a branch of P (∞, 1) solutions undergoing a sudden tran-

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ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI

θf , θˆc [rad]

0.7 0.34

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θf [rad]

Δθ = θf − θˆc [rad]

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x 10

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t [s]

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0

0.2

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θ˙f [rad/s]

Figure 16. Numerical results. Permanent contact at ω = 120 rpm. Left frame: cam and follower trajectories (top) and their corresponding diﬀerence Δθ (bottom). Dash-dot line: maximum follower angular displacement. Right frame: phase portrait. 0.7

θf , θˆc [rad]

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Δθ = θf − θˆc [rad]

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t [s]

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26

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1

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2

θ˙f [rad/s]

Figure 17. Numerical results (equivalent to Figure 5 above). P (∞, 1) orbit at ω = 148 rpm. Left upper frame: cam (dashed) and follower (solid) trajectories; lower frame: their corresponding diﬀerence Δθ. Dash-dot line: maximum follower angular displacement. Right frame: phase portrait.

sition to chaos when ω ≈ 152.67 rpm. The experimental investigation reported in section 3 strongly suggested that the mechanism causing such a transition was the interruption of a complete chattering sequence. The numerically predicted P (∞, 1) orbit is shown in Figure 17 (again in perfect qualitative agreement with the experimental time-series shown in Figure 5). Careful numerical simulations reported in Figure 18 conﬁrm that the interruption of a chattering sequence is indeed the key phenomenon to explain the observed jump to chaos. Further conﬁrmation is provided in Figure 19, where the system attractors are plotted before and after the transition to chaos (see the corresponding multimedia ﬁle showing the evolution of the system attractors on the impact stroboscopic map as ω is varied over the range of interest 72386 02.mov [901KB]). As ω is increased through the critical value, we observe the emergence of a ﬁngered attractor (Figure 19(c)) typical of discontinuous dynamical systems often associated with the occurrence of grazing bifurcations. The chaotic time-series exhibited by the system when ω = 154.1 rpm is shown in Figure 20.

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

0.04

607

(a)

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Δθ = θf − θˆc [rad]

0 85.9

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110.7

89.1

89.2

89.3

(c)

0.04 0.02 0 110.1

t [s]

110.8

110.9

111

111.1

Figure 18. Numerical results (equivalent to Figure 6 above). Time evolution for the diﬀerence Δθ between cam and follower angles, showing (a) complete chatter at ω = 151 rpm, (b) slightly interrupted chatter at ω = 153.5 rpm, and (c) highly interrupted chatter at ω = 153.9 rpm.

The derivation of the numerical bifurcation diagram close to the transition to chaos shown in Figure 21 reveals that a cascade of grazing bifurcations is taking place in a small neighborhood of the transition point. Such an intricate cascade cannot be observed experimentally, given that it accumulates over an extremely thin range of ω (about 0.003 rpm) further below the degree of resolution available experimentally. A more in-depth explanation of this cascade will be presented elsewhere [2]. Here we choose to analyze only those phenomena that are relevant over a realistic range of ω. Therefore, we look now at the extremely important issue of understanding how large the regions of asymptotic stability (basins of attraction) are for diﬀerent coexisting solutions and how these regions evolve under variation of the cam rotational speed ω. 6. Coexistence. A fundamental characteristic of the cam-follower system is the coexistence of diﬀerent attractors, as is evident from both the experimental and the numerical bifurcation diagrams. In particular, by looking at the numerical bifurcation diagram in Figure 10, it can be highlighted that • for ω ∈ [135, 151] rpm, the multi-impacting orbits P (∞, 1) coexist with the periodic motion P (1, 1) characterized by one impact per period; • for ω ∈ [152, 152.6] rpm, the P (∞, 1) coexists with the P (2, 2) solutions; • for ω ∈ [152.6, 154] rpm, the chaotic motion originating from the interruption of the chattering sequence coexists with the branch of P (2, 2) orbits. A more careful Monte Carlo–based bifurcation diagram in the region ω ∈ [152, 160] is reported in Figure 22. Here, we see that there are actually three coexisting cascades occurring within this range: the interruption of the chattering sequence and the period-doubling cascades mentioned above together with the period-doubling of a three-periodic solution which has not been detected experimentally.

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ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI

−0.2

−0.2

−0.4

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Δθ˙f [rad/s]

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φ [rad] (a) 0

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2

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φ [rad] (c) Figure 19. Numerical results: chattering route to chaos. Impact maps showing relative velocity at impacts (vertical axis) versus impact phase (horizontal axis) along P (∞, 1) chattering orbits: (a) complete chattering sequence at ω = 152.66 rpm, (b) incomplete chattering sequence at ω = 152.882 rpm, (c) chaotic impacting orbit at ω = 153.45 rpm.

0.7

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θf [rad]

θf , θˆc [rad]

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0

0.5

1

1.5

2

2.5

θ˙f [rad/s]

Figure 20. Numerical results. Chaotic motion at ω = 154.1 rpm. Left frame: cam (dashed) and follower (solid) trajectories. Dash-dot line: maximum follower angular displacement. Right frame: phase portrait.

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

609

0.3372

θf [rad]

0.3372

0.3372

0.3372

0.3372

152.6765

152.677

152.6775

ω [rpm]

152.678

152.6785

Figure 21. Numerical results: chattering route to chaos. Detailed stroboscopic map close to aperiodicity onset for the multi-impacting orbit. Chain of chaotic windows with periodic frames in between.

(a)

(b)

Figure 22. Monte Carlo bifurcation diagram showing coexistence of solutions. The bifurcation diagram has been obtained by plotting stroboscopic samples (of θf in (a) and θ˙f in (b)) of the last 50 forcing cycles of 100, for 20 initial conditions distributed uniformly at several parameter values within the range ω ∈ [152, 160] rpm.

6.1. Basins of attraction. The simultaneous existence of such diﬀerent attractors makes the system behavior dependent on the choice of initial conditions. Basins of attractions (BA) are an invaluable tool for characterizing the coexistence and stability of diﬀerent solutions [24]. By adapting the cell-to-cell mapping method presented in [17] to the cam-follower system of interest (see also Appendix C), several BAs were computed for diﬀerent values of the bifurcation parameter ω. Figure 23 shows the BA of both the period-one coexisting solutions P (∞, 1) and P (1, 1) reported in Figure 24 when ω = 145 rpm. We notice that both solutions are associated with well-deﬁned BAs with the P (∞, 1) orbit associated with a consistent region

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ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI 8

6

θ˙f (0) [rad/s]

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A

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B

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θf (0) [rad] Figure 23. BA of the two period-one coexisting solutions at ω = 145 rpm. The blue areas correspond to the P (∞, 1) basin, and light blue ones to the P (1, 1) basin. The point A labels the initial condition [0.42092, 1.84]T , while the point B corresponds to [0.33704, 0.13333]T . 0.65

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θf , θˆc [rad]

θf , θˆc [rad]

0.3 0.5

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39

39.5

40

40.5

t [s]

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0.2

39

39.5

40

40.5

41

t [s]

Figure 24. Time-series of the follower angular position θf corresponding to initial conditions A and B in Figure 23. Left frame: P (1, 1). Right frame: P (∞, 1).

of asymptotic stability for low values of initial positions and velocity. As ω is increased, such coexistence persists with the BA of the P (∞, 1) orbit becoming more and more intricate and assuming a clearer fractal structure, as can be noticed from Figures 25 and 27. In Figure 25, we notice the presence of the P (2, 2) solution originating form the ﬂip of the P (1, 1) orbit, as shown in the related time-series depicted in Figure 26. In Figures 27 and 28, the fractalization of the basins becomes increasingly clear, with the dark blue region corresponding now to the basin of the chaotic solution originating from the

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

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8

6

θ˙f (0) [rad/s]

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D2 D1

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C

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θf (0) [rad] Figure 25. BA of the two period-one coexisting solutions at ω = 152.3 rpm. The blue areas correspond to the P (∞, 1) basin, and light blue ones to the P (2, 2) basin. The point C labels the initial condition [0.33704, 0.24]T , while D1 corresponds to [0.37253, 1.7867]T and D2 corresponds to [0.430593, 1.9466]T . 0.34

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θf , θˆc [rad]

θf , θˆc [rad]

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0.2

17.5

18

18.5

t [s]

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0.2

17.5

18

18.5

19

19.5

t [s]

Figure 26. Time-series of the follower angular position θf corresponding to initial conditions C and D in Figure 25. Left frame: P (∞, 1). Right frame: P (2, 2).

interruption of the complete chattering sequence. At higher values of ω, the BAs now associated with the coexistence of periodic and chaotic attractors show a high degree of mixing and fractalization; see, for example, Figure 29, obtained by computing the BA at ω = 156 rpm. Here, we see the coexistence of the P (2, 2) solution (blue region) with the P (6, 6) solution uncovered in the bifurcation diagram shown in Figure 22. Note that the latter solution is associated with a thin BA (yellow region in Figure 29) explaining the fact that it was not detected experimentally. For the sake of com-

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ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI 8

6

θ˙f (0) [rad/s]

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F1 F2 E

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θf (0) [rad] Figure 27. BA of the two period-one coexisting solutions at ω = 154.2 rpm. The blue areas correspond to the basin of the chaotic regime originated by chattering interruption, and light blue ones to the P (2, 2) basin. The time-series related to the initial conditions E ( [0.36365, 0.008]T ) and F 1 ( [0.34623, 1.704]T ) are in Figure 28. 0.36

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θf , θˆc [rad]

θf , θˆc [rad]

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t [s]

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37

37.2

37.4

37.6

37.8

38

38.2

38.4

38.6

38.8

t [s]

Figure 28. Time-series of the follower angular position θf corresponding to initial conditions E and F in Figure 27. Left frame: chaos. Right frame: P (2, 2).

pleteness we show the BA when ω = 160.5 rpm in Figure 31. We notice that the regions of initial conditions associated with periodic solutions have become extremely thin or even disappeared. The zoom of the basin at ω = 160.5, shown in Figure 32, reveals a ring-like structure of the BA which is highly reminiscent of the BA structure for impacting dynamical systems with chattering predicted analytically in the classical paper by Budd and Dux [5]. This oﬀers further evidence that the interruption of a chattering sequence is undoubtedly at

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

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θ˙f (0) [rad/s]

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H6 H5 G2 H4 G1 H3

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θf (0) [rad] Figure 29. BA at ω = 156 rpm. The blue areas correspond to the P (2, 2) basin, while the orange ones correspond to higher periodic regimes. As an example, the time-series related to the initial conditions G1 ( [0.33268, 1.656]T ) and H 1 ( [0.4226893, −2.552]T ) are in Figure 30. 0.65

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θf , θˆc [rad]

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36

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38

t [s]

Figure 30. Time-series of the follower angular position θf corresponding to initial conditions G and H in Figure 29. Left: P (2, 2). Right: P (6, 6).

play in causing the sudden transition to chaos which was observed both experimentally and numerically. 7. Conclusions. We have discussed the complex dynamics of impacting dynamical systems, focusing in particular on cam-follower devices, a particularly relevant class of mechanical devices in applications. Using a recently developed experimental testbed rig, we uncovered the coexistence of intricate bifurcation scenarios involving both classical bifurcations (also observed in smooth dynamical systems) and discontinuity-induced bifurcations. Namely, the

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ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI 8

6

θ˙f (0) [rad/s]

4

2

0

−2

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−8

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1

1.1

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1.3

θf (0) [rad] Figure 31. BA at ω = 160.5 rpm. The dark area corresponds to the chaotic regimes. 3

2

θ˙f (0) [rad/s]

1

0

−1

−2

−3

0.35

0.4

0.45

0.5

0.55

0.6

θf (0) [rad] Figure 32. Zoom of the BA at ω = 160.5 rpm. The θf range is restricted to [0.3, 0.8]; θ˙f to [−4, 3.2].

experimental analysis of the system under investigation showed the coexistence of a perioddoubling cascade and a scenario involving the sudden transition from a periodically sticking solution to chaos. Experimental evidence was presented for the ﬁrst time supporting the conjecture that such a transition is due to the interruption of a complete chattering sequence.

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

615

The experimental results were conﬁrmed by numerical simulations of an appropriately derived model of the cam-follower rig. In particular, excellent qualitative agreement was obtained between the numerical predictions and the experiments. The numerical analysis conﬁrmed that indeed chattering (and its interruption) is the key to unfolding the system behavior and explaining the sudden transition to chaos detected experimentally. Continuation of the P (1, 1) solution coexisting with the chattering orbit conﬁrmed the presence of a classical period-doubling cascade in the model, matching with great accuracy the experimentally observed dynamics. To gain a greater insight into the system behavior and better characterize the coexistence of several solutions, Monte Carlo–based simulations of the bifurcation diagrams were complemented with the computation of basins of attraction at diﬀerent values of the cam rotational speed. To this end, a cell-to-cell algorithm was adapted to the cam-follower system of interest. It was shown that, as ω increases, the BAs of coexisting solutions gain an increasingly fractal structure showing a topology highly reminiscent of that predicted in the classical paper by Budd and Dux on chattering in impact oscillators [6]. Further work is needed to characterize analytically the observed phenomena and, in particular, to understand the detailed structure of the transition from periodic solutions with chattering to chaos. We conjecture that, as shown in Figures 6, 18, and 21, such a transition is due to a cascade of grazing bifurcations accumulating on an extremely small parameter range. Work in progress is aimed at characterizing this transition as well as at generalizing the observed scenarios to a wider class of impacting dynamical systems. Appendix A. Experimental rig implementation. Additional details on the material and the dimensions of each individual components of the rig are given in Table 3. Instruments are listed in Table 4. Table 3 Details on materials and dimensions. Part Cams, ﬂywheels, and follower contact surface Follower body Cam internal radius Cam external radius Cam eccentricity Flywheel radius Follower length Follower width Follower height

Description Made of low-alloy hardened stainless steel UNI 38NiCrMo4 Made of aluminium Al 30 mm 60 mm 15 mm 80 mm 600 mm 16 mm 60 mm

A.1. Implementation. The physical implementation of the experimental rig described above is depicted in subﬁgures (a) and (b) of Figure 33, where the mechanical device is shown to be appropriately coupled to electronic systems for the acquisition, storage, and processing of experimental data. The main features of the experimental setup can be summarized as follows: - The cam motion is controlled by a brushless motor driven through an embedded controller. Notice that the angular position and velocity of the cam and the driving

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616

ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI Table 4 Additional details on the instrumentation. Device Servo (motor-driver) system Data acquisition system Follower position encoder Cam position resolution Follower position resolution

Description Sanyo-Denki Q-series [27] dSPACE ACE-kit ACE1104CLP [15] HENGSTLER RI-58 D [14] 5000 pulses per revolution 10000 pulses per revolution

(a)

(b) Figure 33. Pictures of the experimental rig: (a) view from above, (b) front view.

motor are assumed to be identical because of a rigid coupling connecting the cam to the motor shaft. - The measures of the cam and the follower angular position are obtained through high resolution optical encoders mounted respectively on the cam and follower shaft. - The follower angular velocity is measured by a gyroscope sensor. - A photodiode object reﬂector sensor implements the triggering of signals during the

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

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Figure 34. Picture of the trigger mounted on cam-follower system.

acquisition of data. It is composed of a “phototransistor” coupled with a “photodiode” with the aim of detecting the passage of reﬂecting objects. In our case, it marks the preﬁxed angular position of the cam identiﬁed by a lucid bolt. The optical devices are equipped with appropriate electronic circuits in order to supply a two-level digital output (“0” on the passage for the ﬁxed point, “1” elsewhere). See Figure 34. - Reliable AD/DA conversion and signal processing are implemented through DSPACE [15], a widely used commercial data acquisition integrated hardware-software system (16 bit, 250 MHz, PCI interface). The hardware is based on a DSpace CDS1104 R&D Controller Board (Motorola PowerPC processor technology). It is equipped with a Real-Time Interface (RTI) library that allows us easily to develop experiments from a Simulink model. A Simulink scheme is cross-compiled, thus allowing us (through ControlDesk) to run the experiment and acquire the signals. - The signals are processed and analyzed using MATLAB [16]. - The maximum cam velocity during the experimental analysis is limited to ω = 200 rpm. For higher velocity regimes the follower displacement is very close to the maximum admissible value, and the energy dissipated in the impacting behavior can even destroy the experimental rig. A more extensive description of the experimental rig can be found in [1]. A.2. Automation of the experimental activity. The aim of experiment automation is to perform all the experimental analysis in a simple way even in the absence of an expert experimentalist. An automatic procedure allows the device to capture all the data necessary for the analysis and put it in a format appropriate to process and analyze automatically with MATLAB. The whole process is performed through three consecutive fundamental steps: initialization, acquisition, and data analysis.

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ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI

Figure 35. Examples of plots available from the graphical user interface.

In the initialization phase the cam is automatically rotated in order to place the follower in a preﬁxed angular position at the start of each experiment; then, a reset of all variables of the experiment is carried out. A graphical user interface was developed in order to allow the choice of the various experimental parameters, e.g., the range of cam rotational speed in rpm (initial and ﬁnal value); the step (unit) for the progressive increment of the cam velocity; the transient duration in seconds (transient time); the recording duration in seconds (capture time). After the acquisition phase all the captured data are automatically processed and analyzed with MATLAB during the data analysis phase. Examples of the kind of plots produced at the end of the automatic experimental run are shown in Figure 35. Notice that the three steps are reported with three diﬀerent graphical interfaces (see Figure 36). All the operations described in the three steps, included the management of the interfaces, happens in an automatic way through the execution of a script in Python language. The direct access to variables of the experiment deﬁned in the Simulink model is performed through the ControlDesk Test Automation libraries package [15], which provides the features for remote control of the complete hardware-in-the-loop environment. The key feature is the ability to build up and generate ﬂexible sequences of signals that can be executed in real time. These signals are used as test stimuli, references, or to deﬁne the expected behavior. Measured data can be also recorded by external data loggers. Appendix B. Notation and further system details. All the notation used in section 4 to describe the model is summarized in Table 5. Symbols are related to the schematics of the system in Figure 37, while the values of the model parameters are reported in Table 6. Values of the coeﬃcient of restitution r as a linear function of the cam rotational speed are in Figure 38. Details on the identiﬁcation procedure and validation results can be found in [1].

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

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Figure 36. Graphical interface of the experiments.

Table 5 Summary of the model notation. Symbol (tˆ, n ˆ) (x0 , y0 ) (˜ x, y˜) (˜ xc , y˜c ) θf θc θˆc d d0 d1 d2 e K J pA = (xA , yA ) pB = (xB , yB ), pP = (xP , yP ) pC = (xc , yc ) pE = (xE , xE ) pF = (xf , yf ) pG = (xg , yg ) x α , yα ρ Σ h d

Description Reference system attached to the follower Coordinates of the rotational center in the (x, y) system Reference system obtained by translating the axes (x, y) to (x0 , y0 ) Reference system pivoted at the cam with origin (x0 , y0 ) Angular position of follower with counterclockwise sense of rotation Angular position of cam with counterclockwise sense of rotation Angular displacement of the follower joint when in contact with the cam Half height of follower Relaxed spring length Distance between pB and pP Distance between pP and pE Cam eccentricity Spring stiﬀness Follower moment of inertia Hooking point for spring Ending points of the mechanical element that prevents rotation of spring Point on the cam surface which is nearest to Σ Intersection between vertical line passing through pA and g(x) = tan(θf )x Point of follower surface that will impact on the cam Geometric center of the rotating cam x and y coordinates of point pα Distance between the origin of the axis (x, y) and pE Boundary of follower surface that becomes in contact with cam Distance between pC and Σ Intersection between Σ and the axis y (it is equal to −d/ cos(θf ))

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ALZATE, DI BERNARDO, GIORDANO, REA, AND SANTINI

(a)

(b) Figure 37. Schematic diagram of the cam-follower system: (a) unconstrained mode, (b) constrained mode. All the labels are deﬁned in Table 5. Table 6 Model parameters. Parameter R cam radius e cam eccentricity (x0 , y0 ) center of rotation of the cam d half of the follower height J moment of inertia of the follower K spring coeﬃcient xa x-coordinate of the spring hanging point ya − d0 − d1 spring elongation distance

Value 0.045 m 0.015 m (0.249, 0) m 0.021 m 0.043 kg · m2 105 N/m −0.031 m 0.173 m

Appendix C. Cell-to-cell mapping. In order to implement the cell-to-cell algorithm in [17], the dynamics of the system are described by using a Poincar´e map. The region of feasible initial conditions is subdivided into a large number N of small cells. All unfeasible initial

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BIFURCATIONS AND CHAOS IN CAM-FOLLOWER SYSTEMS

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0.6

0.55

r

0.5

0.45

0.4

0.35 130

135

140

ω [rpm]

145

150

Figure 38. Coeﬃcient of restitution r as a function of the cam velocity ω: experimental points (asterisks), linear approximation (solid line).

conditions are regarded as a small number m of very large cells, so-called sink cells. The mapping is applied to each center point (initial condition), and the cell containing the image is then located. All of the N + m cells point to initial conditions inside one of the other cells, except the sink cells, which point to themselves by deﬁnition. Starting with cell 1, a sequence of cells is mapped by following the pointers deﬁned by the system ﬂow. This sequence ends either in a sink cell or in a repetitive cycle. This cycle can consist of one self-repeating cell (a ﬁxed point, which could be a sink cell) or a number of cells. The repetitive cycle is identiﬁed, and all cells in the sequence are labeled as belonging to the BA of that cycle. Then the procedure is repeated with all N cells. For further details, see [17]. Acknowledgments. The authors would like to thank several colleagues for the interesting discussions and all the comments they made on preliminary versions of this manuscript. In particular, we thank Dr. Ivan Merillas (Barcelona, Spain), Dr. Gustavo Osorio (Manizales, Colombia), Dr. Petri Piiroinen (NUI, Galway), and Dr. Joanna Mason (Bristol). A special thanks goes to Dr. Phani Thota (Bristol, UK), who used the continuation software package TCHAT that he developed to obtain the continuation results depicted in Figure 15, and to Mr. Silvio Massimino, who produced the multimedia ﬁle showing the experimental bifurcation diagram. REFERENCES [1] R. Alzate, M. di Bernardo, U. Montanaro, and S. Santini, Experimental and numerical veriﬁcation of bifurcations and chaos in cam-follower impacting systems, Nonlinear Dynamics, 50 (2007), pp. 409– 429. [2] R. Alzate, M. di Bernardo, and P. Piiroinen, Transition from complete to incomplete chattering in impacting systems: The case of a representative cam-follower device, in preparation. [3] S. R. Bishop, Impact oscillators, Phil. Trans. R. Soc. London, 347 (1994), pp. 347–351. [4] G. Brooker, Development of a w-band scanning conscan antenna based on the twist-reﬂector concept, in Proceedings of the 2nd International Conference on Microwave and Millimeter Wave Technology, Beijing, China, 2000, pp. 436–439.

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[5] C. Budd and F. Dux, Chattering and related behaviour in impact oscillators, Phil. Trans. R. Soc. London A, 347 (1994), pp. 365–389. [6] C. Budd and F. Dux, Intermittent behaviour in an impact oscillator close to resonance, Nonlinearity, 7 (1994), pp. 1191–1224. [7] J. A. Caton, Operating Charactheristing of a Spark-Ignition Engine Using the Second Law of Thermodynamics: Eﬀect of Speed and Load, SAE paper 2000-01-0952, 2000. [8] A. Chae and J. Hollerbach, Dynamic stability issues in force control of manipulators, in Proceedings of the IEEE International Conference on Robotics and Automation, Raleigh, NC, 1987, IEEE Press, Piscataway, NJ, Vol. 4, pp. 890–896. [9] M. di Bernardo, C. J. Budd, A. R. Champneys, and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci. 163, Springer, New York, 2008. [10] M. di Bernardo, P. Kowalczyk, and A. Nordmark, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings, Phys. D, 170 (2002), pp. 175–205. [11] D. J. Dickrell, D. B. Dooner, and W. G. Sawyer, The evolution of geometry for a wearing circular cam: Analytical and computer simulation with comparison to experiment, J. Tribology, 125 (2003), pp. 187–192. [12] S. Foale, Analytical determination of bifurcations in an impact oscillator, Proc. R. Soc. London, 347 (1994), pp. 354–364. [13] H. R. Hamidzadeh and M. Dehghani, Dynamic stability of ﬂexible cam follower systems, Proc. Institution of Mechanical Engineers, Part K: J. Multi-Body Dynamics, 213 (1999), pp. 45–52. [14] Hengstler, Incremental Shaft Encoders: Type RI 58, technical report, HENGSTLER, 2001, online at http://www.hengstler.de/. [15] Hengstler, DS 1104 R&D Controller Board: Features, technical report, dSPACE, 2005, online at http:// www.dspaceinc.com/ww/en/inc/home.cfm. [16] D. J. Higham and N. J. Higham, Matlab Guide, SIAM, Philadelphia, 2000. [17] C. S. Hsu, Global analysis by cell mapping, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), pp. 727–771. [18] A. Nordmark, Grazing Conditions and Chaos in Impacting Systems, Ph.D. thesis, Department of Mechanics, Royal Institute of Technology, Stockholm, Sweden, 1992. [19] A. Nordmark and R. Kisitu, On Chattering Bifurcations in 1 dof Impact Oscillator Models, preprint, Royal Institute of Technology, Stockholm, Sweden, 2003. [20] A. B. Nordmark and P. T. Piiroinen, Simulation and stability analysis of impacting systems with complete chattering, Nonlinear Dynamics, (2009), to appear. [21] R. Norton, Cam Design and Manufacturing Handbook, Industrial-Press, New York, 2002. [22] G. Osorio, M. Di Bernardo, and S. Santini, Chattering and complex behavior of a cam-follower system, in Proceedings of the European Nonlinear Dynamics Conference (ENOC), Eindhoven, The Netherlands, 2005. [23] G. Osorio, M. di Bernardo, and S. Santini, Corner-impact bifurcations: A novel class of discontinuity-induced bifurcations in cam-follower systems, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 18–38. [24] E. Ott, Chaos in Dynamical Systems, 2nd ed., Cambridge University Press, Cambridge, UK, 2002. [25] F. Peterka, Part 1: Theoretical analysis of n-multiple (1/n)-impact solutions, Acta Tech. CSAV, 26 (1974), pp. 462–473. [26] F. Peterka, Results of analogue computer modelling of the motion. Part 2, Acta Tech. CSAV, 19 (1974), pp. 569–580. [27] Sanyodenki, SANMOTION Q: AC Servo Systems, technical report, Sanyo-Denki, 2002, online at http:// www.sanyo-denki.com/. [28] S. W. Shaw and P. J. Holmes, Periodically forced linear oscillator with impacts: Chaos and long-period motions, Phys. Rev. Lett., 51 (1983), pp. 623–626. [29] R. H. Sherman and P. N. Blumberg, The Inﬂuence of Induction and Exhaust Processes on Emissions and Fuel Consumption in the Spark Ignited Engine, SAE paper 770990, 1997. [30] P. Thota, Analytical and Computational Tools for the Study of Grazing Bifurcations of Periodic Orbits and Invariant Tori, Ph.D. dissertation, Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA, 2007; available online at http://scholar.lib.vt.edu/theses/available/ etd-02142007-140350/.

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[31] C. Toulemonde and C. Gontier, Sticking motions of impact oscillators, Eur. J. Mech. A Solids, 17 (1998), pp. 339–366. [32] A. M. Urin, Experience with the use of pk-proﬁle joints in agricultural machines, Research, Design, Calculations, and Experience of Operation of Chemical Equipment Technology, Chemical and Petroleum Engineering, 34 (1998), pp. 534–536. [33] D. J. Wagg, Rising phenomena and the multi-sliding bifurcation in a two-degree-of-freedom impact oscillator, Chaos Solitons Fractals, 22 (2004), pp. 541–548. [34] D. J. Wagg, Periodic sticking motion in a two-degree-of-freedom impact oscillator, Internat. J. NonLinear Mech., 40 (2005), pp. 1076–1087. [35] D. J. Wagg, Multiple non-smooth events in multi-degree-of-freedom vibro-impact systems, Nonlinear Dynam., 43 (2006), pp. 137–148. [36] G. S. Whiston, The vibro-impact response of a harmonically excited and preloaded one-dimensional linear oscillator, J. Sound Vib., 115 (1987), pp. 303–324.

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c 2009 Society for Industrial and Applied Mathematics

SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 8, No. 2, pp. 624–640

The Two-Fold Singularity of Discontinuous Vector Fields∗ M. R. Jeﬀrey† and A. Colombo‡ Abstract. When a vector ﬁeld in R3 is discontinuous on a smooth codimension one surface, it may be simultaneously tangent to both sides of the surface at generic isolated points (singularities). For a piecewise-smooth dynamical system governed by the vector ﬁeld, we show that the local dynamics depends on a single quantity—the jump in direction of the vector ﬁeld through the singularity. This quantity controls a bifurcation, in which the initially repelling singularity becomes the apex of a pair of parabolic invariant surfaces. The surfaces are smooth except where they intersect the discontinuity surface, and they divide local space into regions of attraction to, and repulsion from, the singularity. Key words. Filippov, sliding, singularity, nonsmooth, discontinuous AMS subject classifications. 34C23, 37G10, 37G35 DOI. 10.1137/08073113X

1. Introduction. A piecewise-smooth dynamical system contains discontinuities that approximate sudden changes in the governing vector ﬁeld. These systems have enjoyed widespread application in recent years, from control theory and nonlinear oscillators to economics and biology. Nevertheless, research into the theory of piecewise-smooth dynamics is at a relatively early stage. Such systems consist of a ﬁnite set of ordinary diﬀerential equations, (1.1)

˙ = Gi (X) , X

X ∈ R i ⊂ Rn ,

whose right-hand sides are vector ﬁelds Gi deﬁned on disjoint regions Ri and smoothly extendable to the closure of Ri . The Ri are separated by an n − 1 dimensional set Σ which we call the switching surface. The union of Σ and all Ri covers Rn . The literature in real-world piecewise-smooth problems is now extensive, and we refer the reader to [2, 3, 6, 7, 10] for an overview. Our concern will be vector ﬁelds with no constraint on the degree of discontinuity across the switching surface, so-called Filippov systems [5], where the continuous ﬂow deﬁned by (1.1) may be nondiﬀerentiable and irreversible, and may contain sliding orbits which are conﬁned to the switching surface. The theory of singularities in piecewise-smooth systems has proven to be a rich source of novel dynamics, particularly near points where the vector ﬁeld is tangent to the switching ∗

Received by the editors July 24, 2008; accepted for publication (in revised form) by J. Meiss February 20, 2009; published electronically May 14, 2009. http://www.siam.org/journals/siads/8-2/73113.html † Applied Nonlinear Mathematics Group, Department of Engineering Mathematics, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK (mike.jeﬀ[email protected]). The work of this author was supported by the EPSRC grant Making It Real. ‡ DEI, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy ([email protected]). With thanks to the hospitality of the University of Bristol during this work. 624

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THE TWO-FOLD SINGULARITY

625

surface [3, 4, 11], commonly referred to as “fold” points where the surface is smooth. Here we discuss a particular problem at the very heart of nonsmooth dynamics—points where a vector ﬁeld is tangent to both sides of a switching surface in systems of three dimensions. This “twofold” problem has been most notably brought to the fore by Teixeira [14, 15, 16], including a case we call the “Teixeira singularity” which epitomizes the current state of nonsmooth singularity theory. In [15], the Teixeira singularity is shown to violate conditions set out for a particular deﬁnition of structural stability in a nonsmooth system, and asymptotic stability is determined only under limited conditions of hyperbolicity. This singularity is our main subject of interest because, despite these results, confusion still surrounds this pivotal point of nonsmooth dynamics. The reason is that much of our intuition fails in the face of discontinuities. Indeed, there is not yet even a consensus on the deﬁnition of topological equivalence in nonsmooth systems, or how deﬁnitions of structual stability (e.g., [1, 7, 10, 12, 15]) reﬂect the robustness of dynamics in a nonsmooth model. Here we study the dynamics directly, without reliance on these deﬁnitions, revealing explicit behavior that should be reﬂected in general theories on structural stability. Adopting a transparent geometric approach, we study the dynamics around the Teixeira singularity and reveal the simplicity characterizing its local behavior. The interesting case is when the ﬂows of two ﬁelds Gi and Gj on either side of Σ consist locally of orbits which always return to Σ, spiraling around the singularity between impacts and giving rise to intricate dynamics (see Figure 1). Then the dynamics depends on the relative directions of the vector ﬁelds Gi and Gj at the singularity, that is, the quantity tan θ i / tan θ j , where θ i,j are the angles subtended by Gi,j at the singularity, to an arbitrary reference direction in Σ. When Gi and Gj are antiparallel at the singularity, a bifurcation takes place: on one side of the bifurcation all local trajectories reach the sliding region of Σ in ﬁnite time, and on the other side two invariant manifolds separate the local state space into regions of attraction to, and repulsion from, the singularity. In section 2 we state the problem in terms of standard concepts and state the central result, Theorem 1. In section 3 we provide a local coordinate expression. In section 4 we derive a map that essentially treats the switching surface as a Poincar´e section of the ﬂow, revealing a bifurcation of invariant manifolds and proving the theorem. Dynamics on the invariant manifolds is studied in section 5. The preservation of straight lines in the system is key to dealing with the discontinuity, exposing a strict relation between dynamics crossing the switching surface and sliding dynamics on the switching surface which is found in section 6. Near the bifurcation small nonlinear eﬀects must be included, as discussed in section 7. 2. The two-fold problem. Consider a region in which the vector ﬁeld (1.1) is discontinuous along a smooth codimension one switching surface Σ. Let (2.1) Σ = X ∈ R3 : h (X) = 0 in terms of a scalar valued function h (X). Deﬁnition 1. In a dynamical system ˙ − (X) for h (X) < 0 , ˙ = X ˙ + (X) for h (X) > 0, X X

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626

M. R. JEFFREY AND A. COLOMBO

Σcr

S− esc

Σ

X

x

z x

s−

φ+ S+

s+

Σsl

y

Σcr

φ−

Figure 1. Coordinates and tangency sets.

˙ ± ∈ R3 are smooth vector fields, a point Xp ∈ Σ is a two-fold singularity if where X (2.2)

˙ ± (Xp ) · ∇h (Xp ) = 0 and X

˙ ± (Xp ) · ∇)2 h (Xp ) = 0. (X

We begin with a local coordinate description of a three dimensional piecewise-smooth dynamical system in which the two-fold singularity is generic. In the neighborhood of any point on Σ, we can choose a coordinate x perpendicular to Σ such that (2.3) Σ = X ∈ R3 : x = 0 . We can distinguish coordinates (y, z) in Σ by writing a general vector in R3 as (2.4)

X = [x, x] = [x, (y, z)] ,

x ∈ R2 .

The corresponding dynamical system is + + ˙ = [x˙ (X) , x˙ (X)] , x > 0 . (2.5) X [x˙ − (X) , x˙ − (X)] , x < 0 Generically, there exist tranverse n − 2 dimensional tangency sets S ± given by (2.6) S ± = X ∈ Σ : x˙ ± = 0 . We can choose the x = (y, z) coordinates such that (2.7)

S + = {X ∈ Σ : y = 0} ,

S − = {X ∈ Σ : z = 0} ;

see Figure 1. The tangency sets S ± are perpendicular in these coordinates, intersecting at the singularity p ∈ Σ where the x component of both vector ﬁelds vanishes, x˙ ± p = 0, at ± ± x = y = z = 0. They possess unique (up to sign) normal unit vectors S = [0, s ] satisfying S+ × S− = 0. The tangency sets partition Σ into four regions. Where x˙ + < 0 < x˙ − we have the sliding region Σsl , and where x˙ − < 0 < x˙ + we have the escaping (or “unstable sliding”) region Σesc . If we choose the sign of s± such that x˙ ± · s± > 0 (as in Figure 1), these can be written as Σesc = x ∈ Σ : s± · x < 0 . (2.8) Σsl = x ∈ Σ : s± · x ≥ 0 ,

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THE TWO-FOLD SINGULARITY

627

Orbits pass from one ﬂow to the other by traversing Σ in the two crossing (or “sewing”) regions (2.9)

Σcr = x ∈ Σ : s+ · x < 0 ≤ s− · x or s− · x < 0 ≤ s+ · x .

Throughout this paper we distinguish between “orbits” that may meet the switching surface only in Σcr and “sliding orbits” that are contained in Σsl,esc . More precisely, we give the following deﬁnition based on [1]. Deﬁnition 2. • An orbit is a piecewise-smooth curve γ ⊂ R3 whose segments in x > 0 are trajectories ˙ =X ˙ − , whose intersec˙ =X ˙ + and whose segments in x < 0 are trajectories of X of X tions with x = 0 consist of crossing points or tangency points, such that γ is maximal with respect to these two conditions. • A sliding orbit is a smooth curve γ ⊂ Σ such that γ is a trajectory of the Filippov sliding vector field expressed in (6.1)–(6.2). Two-folds may contain two diﬀerent forms of tangency point. Deﬁnition 3. Tangency of the vector field to Σ is x± ) = ±1, and • visible on S ± if sign (¨ ± x± ) = ∓1. • invisible on S if sign (¨ That is, “visible” implies that the orbit tangent to Σ at S + (or S − ) extends locally into the region x > 0 (or x < 0). We will refer to a two-fold singularity consisting of two coincident ˙± invisible tangencies as the Teixeira singularity. In this case both the smooth ﬂows of X consist locally of orbits which always return to Σ, spiraling around the singularity between impacts, giving rise to intricate dynamics. Then we may deﬁne a second-return map φ that maps a point from Σ, through one smooth vector ﬁeld until it hits Σ, and then through the other vector ﬁeld until it impacts Σ again. The following have been proven in [15]. T1: The two-fold singularity is structurally stable if and only if at least one of the tangency sets is visible. Thus the Teixeira singularity is structurally unstable. T2: The Teixeira singularity is asymptotically stable provided that (i) the second-return map φ is hyperbolic, and (ii) the Filippov sliding vector ﬁeld is hyperbolic with the phase portrait of an attracting node and the eigendirection associated with the eigenvalue of smaller absolute value in the sliding region Σsl . Structural stability of a piecewise-smooth system, deﬁned in [1, 3, 9], in short requires that orbits, sliding orbits, and switching surfaces of a system be mapped through a homeomorphism onto those of all neighboring systems in the parameter space. T2 is a paraphrasing of the statement “the U[Teixeira]-singularity is asymptotically stable provided that it is an S-singularity,” and for a precise deﬁnition we refer the reader to [15]. We will deﬁne the Filippov sliding vector ﬁeld in (6.1). In the cases where at least one of the tangency sets is visible, the dynamics is rather straightforward, and asymptotic stability then relies only on the form of the Filippov sliding vector ﬁeld, which has been further considered in [16]. Henceforth we will be interested only in the case of coincident invisible tangencies, the Teixeira singularity. According to T1 and T2, the Teixeira singularity is structurally unstable, with asymptotic stability determinable only when the return map and Filippov ﬁeld are hyperbolic.

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M. R. JEFFREY AND A. COLOMBO

Our aim is to shed light on this problem without recourse to T1 and T2 by an explicit study of the local dynamics. We will show that the vector ﬁeld is indeed structurally unstable at a certain parameter value and unfold the resulting bifurcation. This allows us to determine regions over which the singularity is attracting or repelling. This last issue must be considered with care, since the singularity is not a stationary point of the vector ﬁeld. The convergence of sliding orbits upon the singularity causes confusion over the local dynamics, which we will study in detail, revealing regions of attraction to and repulsion from the singularity instead of asymptotic stability. The central result, which will be proven in section 4, is the following theorem. Theorem 1. A two-fold singularity can be expressed in a local approximation as ˙ − for x < 0 ˙ = X ˙ + for x > 0, X X in coordinates X = [x, y, z], where Σ = X ∈ R3 : x = 0 , and (2.10)

˙ + = −ya, 1, V + , X

˙ − = zb, V − , 1 X

for a, b, V ± ∈ R. For the Teixeira singularity a, b > 0, this satisfies the following: (i) If V + V − > 1 and V ± < 0, every orbit of (2.10) crosses Σ an infinite number of times. There exist a pair of invariant surfaces that meet at the singularity. (ii) If V + V − < 1 or V + > 0 or V − > 0, every orbit of (2.10) crosses Σ a finite number of times. A bifurcation takes place at V + V − = 1 for V ± < 0. Furthermore, the following will be shown. (i) If V + V − > 1 and V ± < 0, one of the invariant surfaces is asymptotically attractive, and encloses the escaping region Σesc within the domain of repulsion of the singularity; the other invariant surface is asymptotically repulsive and encloses the sliding region Σsl within the domain of attraction of the singularity. (ii) If V + V − < 1 or V + > 0 or V − > 0, sliding orbits are repelled from the singularity, and (ii.i) if V + > 0, every orbit crosses Σ at most once from x < 0 to x > 0, (ii.ii) if V − > 0, every orbit crosses Σ at most once from x > 0 to x < 0, and (ii.iii) if 0 < V + V − < 1 and V ± < 0, every orbit crosses Σ at least once before impacting the sliding region. The topology and dynamics of the locally invariant surfaces for (i) will be determined in sections 4–5. 3. Local approximation. To determine the fate of orbits in the neighborhood of the singularity, we derive a local approximation for the vector ﬁeld. First, local cubic tangencies to Σ are prohibited by conditions (3.1)

x ¨+ < 0,

x ¨− > 0,

˙ ± = 0. Under these assumptions, and there are assumed to be no local equilibria, i.e., X in a suﬃciently small neighborhood of the singularity, the vector ﬁelds’ projection onto the

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THE TWO-FOLD SINGULARITY

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switching surface Σ is approximately constant. That is, by a Taylor expansion up to linear order in the x-direction and zeroth order in the y, z-directions, we can express the vector ﬁeld as + + − − (3.2) x˙ , x˙ ≈ −ya, v+ , x˙ , x˙ ≈ zb, v− , where v+ = (1, V + ) and v− = (V − , 1) are nonzero vector constants, and a, b > 0. Henceforth we can set a = b = 1 without loss of generality. More details are given in the appendix. The ﬂows of each vector ﬁeld in this parabolic approximation map points on Σ, through the smooth regions x > 0 and x < 0, to return points on Σ according to (3.3)

φ+ : {Σ : y < 0} → {Σ : y > 0} ,

φ− : {Σ : z < 0} → {Σ : z > 0}

and given explicitly by (3.4)

φ+ : x → x − 2yv+ ,

φ− : x → x − 2zv− .

As observed by Teixeira [15], this deceptively simple map is the key to understanding coincident invisible tangencies. In what follows we study its geometry. The overlap of the domains of φ± is the escaping region Σesc , and the overlap of their ranges is the sliding region Σsl . The local dynamics is given by an alternating series of iterations of the maps φ+ and φ− . Any point on the escaping region can only be a start point of the series, and any point in the sliding region can only be an end point of the series. The term “escaping” refers to the ˙ ± move away from Σesc . In this fact that, arbitrarily close to Σesc with x = 0, orbits of X section we will regard points arbitrarily close to Σesc as in fact being on Σesc . In section 6, we will deal with the sliding orbits that apply to points exactly in Σesc with x = 0. Our goal here is to understand under what conditions all points on the escaping and crossing regions eventually reach the sliding region, and what happens when these conditions are not met. To label points on Σ let m ∈ Z, then let every x2m be mapped by φ+ , and let every x2m−1 be mapped by φ− , so that (3.5)

s+ · x2m < 0,

s− · x2m−1 < 0.

We can rewrite the maps (3.4) locally as oblique reﬂections in the line S ± : ⎧ ⎧ + − ⎨ x2m+1 − x2m = v , ⎨ x2m − x2m−1 = v , + − |x2m+1 − x2m | |v+ | |x2m − x2m−1 | |v− | φ : (3.6) φ : ⎩ + ⎩ − s · (x2m+1 + x2m ) = 0, s · (x2m + x2m−1 ) = 0. The upper condition speciﬁes that the direction of reﬂection is v± , while the lower condition speciﬁes that the start and end points have the same perpendicular distance from S ± . This is illustrated in Figure 2. Projecting the upper equation along s+ (for φ+ ) or s− (for φ− ) and then eliminating the quantities |xi+1 − xi |, we obtain (3.7)

x2m+1 − x2m = −2

s+ · x2m + v , s+ · v+

x2m − x2m−1 = −2

s− · x2m−1 − v . s− · v−

Notice that both denominators are equal to 1 as given by (3.2).

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630

M. R. JEFFREY AND A. COLOMBO

φ+

x2m

v+

v−

v+

φ−

S−

v− x2m+1

arctanV +

x2m−1 S+

arctanV −

Figure 2. Local mapping: In the parabolic approximation, points are reﬂected in S + and S − . In this example the point x2m−1 is reﬂected obliquely in S − to x2m , and then in S + to x2m+1 , remaining in the crossing regions. The sliding region is shaded.

T’4 T4

T2 T’2

T0

φ+ T’0

φ−

φ+

T’1 T1

T’3

T3

...

φ−

Figure 3. The ﬁrst four iterates of the escaping region on Σ. In this example they map repeatedly into the crossing regions.

It is tempting to consider these two dimensional maps in their obvious cartesian form, but it is not clear how to apply principles of asymptotic stability from smooth dynamical systems. For example, Teixeira [15] remarks that the second-return map xm → xm+2 (the map φ+ ◦ φ− or φ− ◦ φ+ ) is nonhyperbolic if, in the notation of Theorem 1, 0 < V + V − < 1. It is unclear whether this condition says anything about the stability of the system because orbits only cross Σ over a ﬁnite time period before entering the sliding region. Instead we can exploit the fact that the maps (3.7) preserve straight lines through the origin. That is, any point on the line x2m = R (cos θ2m , sin θ2m ) for R ∈ R variable maps to a point on another line, x2m+1 = R (cos [θ2m + f (θ2m )] , sin [θ2m + f (θ2m )]). So to study the images of Σsl and Σesc under successive iterations of φ± , we need only consider the rotation of their boundaries, which are straight lines through the origin, as illustrated in Figure 3. This rotation constitutes the angular behavior of (3.7). In the next section we shall see that, under certain conditions, this angular map has two ﬁxed points, corresponding to invariant manifolds of the second return map derived from (3.7). In section 5 we will study the radial behavior of (3.7) restricted to the two invariant manifolds. 4. The tangent map. Let us introduce the quantities (4.1)

T2m =

s− · x2m , s+ · x2m

T2m−1 =

s+ · x2m−1 , s− · x2m−1

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THE TWO-FOLD SINGULARITY

631

which are, respectively, the tangent T2m of the angle made with s+ by a vector x2m in the domain of φ+ and the tangent T2m−1 of the angle made with s− by a vector x2m−1 in the domain of φ− . These domains are deﬁned in (3.3). We deﬁne corresponding quantities for the vectors v± as V+ =

(4.2)

s− · v+ , s+ · v+

V− =

s+ · v− . s− · v−

Tm is positive for points xm in the escaping and sliding regions, negative in the crossing regions, and zero on the boundary of Σesc . Moreover, it is well deﬁned, except on the boundary of Σsl (where sliding dynamics apply; see section 6), whereas V ± are always well deﬁned due to conditions (3.1). From (3.7) and (4.1)–(4.2) we obtain maps for T2m and T2m−1 : T2m+1 =

(4.3)

2V

+

1 , − T2m

T2m =

2V

−

1 . − T2m−1

Clearly, a positive V + or V − implies that points in the crossing regions (Tm < 0) are mapped into the sliding region (Tm > 0) after at most two iterations, or one iteration if both V ± are positive. More precisely, note that an iterate of (4.3) lies in the crossing regions only if it satisﬁes 2V + < T2m < 0,

(4.4)

2V − < T2m−1 < 0.

We therefore have the following lemma. Lemma 4.1. The following statements hold for the Teixeira singularity, as expressed in (2.10) with a, b > 0: (i) if V + > 0, every orbit crosses Σ at most once from x < 0 to x > 0, (ii) if V − > 0, every orbit crosses Σ at most once from x > 0 to x < 0, and (iii) if 0 < V + V − < 1 and V ± < 0, every orbit crosses Σ at least once before impacting the sliding region. Proof. (i) If V + > 0, (4.4) implies that any T2m is mapped by (4.3) to T2m+1 > 0, a termination point in the sliding region. Therefore, there is at most one crossing point T2m in the region y < 0 < z, where orbits cross from x < 0 to x > 0. (ii) If V − > 0, (4.4) implies that any T2m−1 is mapped by (4.3) to T2m > 0, a termination point in the sliding region. Therefore, there is at most one crossing point T2m−1 in the region z < 0 < y, where orbits cross from x > 0 to x < 0. (iii) If 0 < V + V − < 1 and V ± < 0, then for any Tm > 0 we have from (4.3) that (4.5)

Tm+1 =

1 1 <0 = 2V ± − Tm −2 |V ± | − |Tm |

for V + if m is even or V − if m is odd. Therefore, an iterate Tm exists for all orbits; thus there always exists at least one crossing point.

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632

M. R. JEFFREY AND A. COLOMBO

V

+ (a)

τ m+2

τm

(b) (c) 1

τS τU

V +V −=1

V

−

Figure 4. Tangent mapping and bifurcation diagram: (a) No invariant manifolds for V + V − < 1, (b)

+ − + − + − τS = τ and τU = 1/ τ V V in V V > 1. τm = 0 is bifurcation along V V = 1, (c) two ﬁxed points

the boundary of Σesc . Lines with τm = 2 − 1/ 2V + V − map to the τm+2 graph asymptotes (dashed) which are the boundaries of Σsl . The bound τm < 2 ensures the existence of the intermediate step τm+1 .

More insight is given by the second-return maps (T2m → T2m+1 followed by T2m+1 → T2m+2 and vice versa). These are a pair of M¨obius transformations expressible concisely as (4.6)

τm+2 =

2V

τm − 2 + V − (τ − m

2) + 1

for T2m = V + τ2m and T2m−1 = V − τ2m−1 , shown in Figure 4. For this to make sense, the intermediate (m + 1)th iterate must lie in the crossing region, limiting the angle subtended by Tm to the boundaries of the escaping region, with the conditions (4.4). From this second-return map (4.6) we have the result as stated in Theorem 1: (i) If V + V − > 1 and V ± < 0, every orbit of (2.10) crosses Σ an infinite number of times. There exist a pair of invariant surfaces that meet at the singularity. (ii) If V + V − < 1 or V + > 0 or V − > 0, every orbit of (2.10) crosses Σ a finite number of times. Proof of Theorem 1. The local approximation (2.10) is obtained by a constant scaling of the Teixeira singularity vector ﬁeld given in [15]. Then consider the second-return map (4.6). If V + V − > 1, the map has two ﬁxed points—one at τ with eigenvalue (1 − 2/τ )−2 < 1 which is therefore stable (asymptotically attracting), and one at 1/ (τ V + V − ) with eigenvalue (1 − 2V + V − τ )−2 > 1 which is therefore unstable (asymptotically repelling), where 1 (4.7) τ = 1 − 1 − + − . V V Note that τ > 0 and 1/ (τ V + V − ) > 0. Then the following hold:

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THE TWO-FOLD SINGULARITY

633

(i) If V + V − > 1 and V ± < 0, the equilibria V ± τ < 0 and V ± / (τ V + V − ) < 0, of the Tm → Tm+2 maps from (4.6), lie in the crossing regions. From (4.6), the Tm → Tm+2 maps are monotonic; therefore, all trajectories tend asymptotically toward the equilibria either in forward or in reverse time and thus cross Σ an inﬁnite number of times. The smooth segments of orbits starting and ending at crossing points along the {Tm , Tm+1 } directions thus form invariant surfaces; the surfaces intersect Σ along lines through the singularity given by x = l (1, V + τ ) and x = l (1, V + / (τ V + V − )) for l ∈ R. (ii) If V + V − > 1 and V ± > 0, then the equilibria V + τ > 0 and V ± / (τ V + V − ) > 0, of the Tm → Tm+2 maps, lie in the sliding or escaping regions, so the equilibria are outside of the range of (4.6). If V + V − < 1, there are no real-valued equilibria. In either case there are then no admissible limit points (i.e., in the crossing regions), so all trajectories intersect the sliding and escaping regions after ﬁnitely many iterations. More explicitly, from (4.7), the maps T2m → T2m+2 and T2m−1 → T2m+1 , respectively, have stable equilibria TS+ and TS− given by TS− TS+ 1 = − =1− 1− + − (4.8) + V V V V and unstable equilibria TU+ and TU− given by (4.9)

TU− TU+ = =1+ V+ V−

1−

1 V

+V −

.

± < 0; otherwise, we have one of the cases where These exist only in the crossing regions, TU,S V + or V − are positive. They are invariant manifolds of the second-return maps derived from (3.7). From (4.4) it is clear that they divide Σ such that the stable manifolds TS± enclose ± the escaping region, while the unstable TU enclose the sliding region. It is easy to manifolds + − show from (4.8)–(4.9) that each pair TS , TU and TU+ , TS− forms a straight line through the origin since TS± TU∓ = 1. At V + V − = 1, the invariant manifolds of each map coalesce and annihilate in what we refer to as the “nonsmooth diabolo” bifurcation for reasons that will become apparent.

5. Dynamics on the invariant manifolds. Now consider the dynamics of points on the invariant manifolds (4.8)–(4.9). A point xm has radial coordinate (5.1) Rm = (s+ · xm )2 + (s− · xm )2 . By combining this with (3.7) and iterating twice (and using (4.6) to simplify), we ﬁnd the radial maps + 2 1 + T2m+2 2V − T2m 2 2 , R2m R2m+2 = 2 T2m+2 1 + T2m − 2 1 + T2m+1 2V − T2m−1 2 2 (5.2) , R2m−1 R2m+1 = 2 T2m+1 1 + T2m−1

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634

M. R. JEFFREY AND A. COLOMBO

+ TS

+ TU

(a) V +V −>1

− TS

− TU (b) V +V −=1

(c) V +V −<1

Figure 5. Invariant manifold bifurcation: (a)–(c) correspond to the parameter values in Figure 4. (a) TS± and TU± form two stable and two unstable manifolds in the crossing regions, which are respectively repelling and attracting with respect to the singularity at the center. (b) The manifolds coalesce in a line of ﬁxed points. (c) All points map from the escaping region to the sliding region in ﬁnite time.

which on the invariant manifolds simplify to (5.3)

1 ± 1 − 1/V + V − Rm+2 = , Rm 1 ∓ 1 − 1/V + V −

taking the upper signs for TS± and the lower signs for TU± . Since V + V − > 1, Rm+2 > Rm (5.4)

Rm+2 < Rm

on the stable manifolds TS± , and on the unstable manifolds TU± .

Therefore, points in TU move toward the singularity, while points in TS move away from it. This is illustrated in Figure 5. We can easily extend this picture out of the switching surface Σ, since each {xm , xm+1 } pair contains the start and end points of a smooth orbit segment in the ﬂow of (3.2). Thus the invariant manifolds form two continuous, parabolic (i.e., quadratic, of the form x ∝ V + y 2 + V − z 2 − 2V + V − yz), invariant surfaces U and S, which are smooth except at their intersections TU± and TS± with Σ. They form a nonsmooth diabolo (Figure 6), an attractive cone S which encloses Σesc , and a repelling cone U which encloses Σsl , both with apex at the ± . origin and nondiﬀerentiable where they intersect Σ at edges along TU,S 6. Dynamics in the sliding region. We have determined the qualitative dynamics of orbits in the system (3.2), exclusive of any dynamics on the switching surface that occurs before ejection from the escaping region Σesc , and after impact with the sliding region Σsl . The system (3.2) does not specify the vector ﬁeld in these regions. To address this we adopt the Filippov convention, (6.1)

˙ = [0, f ] X

for all X ∈ Σsl ∪ Σesc ,

where, recalling that we have set a = b = 1,

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THE TWO-FOLD SINGULARITY

635

+ TU

U

b a

+ TS

− TU

c

− T S S Figure 6. The nonsmooth diabolo: Invariant manifolds near a two-fold singularity. The three qualitatively diﬀerent types of orbit are shown. a: An orbit starting near the inside of U spirals in toward the singularity and hits the sliding region (shaded). b: An orbit starting near the outside of U initially spirals inward toward the singularity, then spirals out away from the singularity, and tends asymptotically toward S. c: An orbit spirals outward from the escaping region and away from the singularity, approaching S asymptotically.

˙ − x˙ + − ∇h · X ˙ + x˙ − ∇h · X f= ˙−−X ˙+ ∇h · X (6.2)

=

zv+ + yv− . z+y

In the sliding region Σsl the denominator of (6.2) is strictly positive, so a coordinate transformation that preserves sliding orbits scales out the denominator and we consider locally (6.3)

˜f = zv+ + yv− .

This has Jacobian determinant − 1 ˜ V (6.4) D f = 1 V+

= V − V + − 1;

recall that v+ = (1, V + ) and v− = (V − , 1). Then, (6.5)

sign D˜f = sign V + V − − 1 =

± +1 ⇔ TU,S < 0,

± ∈ / R. −1 ⇔ TU,S

The equivalence on the right holds for V ± < 0, the parameter regime where the invariant manifold bifurcation occurs.

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636

M. R. JEFFREY AND A. COLOMBO

(a)

(b)

(c)

Figure 7. Sliding dynamics and invariant manifolds. (a)–(c) correspond to Figure 5. The normalized sliding vector ﬁeld (6.3) has (a) an attracting node at the singularity for V + V − > 1, (b) a line of equilibria extending from the singularity for V + V − = 1, and (c) a saddle at the singularity with unstable manifold in Σsl (shaded) for V + V − < 1. These lead to the true sliding vector ﬁeld (6.2) as depicted. ± Thus the existence of the invariant manifolds TU,S for V + V − > 1 coincides with the existence of a node in ˜f at the singularity. To see that the node is attractive, observe that ˜f ± ± ± < 0, so the sum of the points into the sliding region and V ± = ss∓ ·v ·v± < 0, and therefore V eigenvalues of D˜f is simply (6.6) Tr D˜f = V + + V − < 0.

Also from (6.5), the absence of invariant manifolds for V + V − < 1 coincides with the existence of a saddle in ˜f at the singularity. The eigenvectors of D˜f are

(6.7) w± = 1, ω± − V − , and the corresponding eigenvalues are 2 + − 1 (6.8) ω± = 2 V + V ± (V + − V − ) + 4 . The ﬁrst component of w± is positive, while the second component is positive for ω+ and negative for ω− . That is, (6.9)

w+ ∈ Σsl ,

w− ∈ Σcr .

Thus the saddle’s unstable separatrix direction w− lies in the sliding region, and the stable separatrix direction w+ lies in the crossing region. In the exact sliding vector ﬁeld f the sliding orbits are the same as the node (V + V − > 1) and saddle (V + V − < 1) orbits of ˜f , but they reach/depart the singularity in ﬁnite time. That is, the singularity is not an equilibrium point of the sliding vector ﬁeld f . Furthermore, to obtain the sliding vector ﬁeld in the escaping region Σesc , the time direction must be reversed from that of the normalized ﬁeld. This is illustrated in Figure 7, including the presence of a line of equilibria when V + V − = 1. 7. Small perturbations near the bifurcation. Here we investigate the eﬀect of nonlinear terms on the local dynamics when V + V − ≈ 1. To understand what happens to the line of equilibria in the sliding region we may appeal to center manifold theory: choose a coordinate

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THE TWO-FOLD SINGULARITY

637

λ>0 (a)

(b)

(c)

(b)

(c)

λ<0 (a)

Figure 8. Perturbed sliding dynamics, showing the eﬀect of positive perturbation λ > 0 (top) and negative perturbation λ < 0 (bottom). Triangular arrowheads indicate nonlinear (in)stability. (a)–(c) correspond to parameter values in Figure 7.

u along the center manifold—the line of equilibria in Figure 7(b). The one-dimensional sliding ﬁeld on the center manifold, f˜ = ω+ u, is structurally unstable at the bifurcation point ω+ = 0. A nonlinear perturbation gives the transcritical bifurcation normal form (7.1)

f˜ = ω+ u + λu2 .

This introduces an equilibrium in the sliding region: a saddlepoint that exists when V + V − < 1 for a positive perturbation λ > 0, and an attracting node that exists when V + V − > 1 for a negative perturbation λ < 0, illustrated in Figure 8. In the escaping region Σesc the same analysis follows; note that the transformation to ˜f reverses the time direction there. As we pass through the bifurcation at V + V − = 1, the ﬁeld ˜f undergoes a transcritical bifurcation, as the second equilibrium moves between the sliding and escaping regions. Note, however, that in the true sliding vector ﬁeld f , the singularity is no longer an equilibrium, and sliding orbits reach it in ﬁnite time. Regarding crossing orbits, the line of ﬁxed points in Figure 5(b) illustrates the presence of a structural instability in the second-return maps, coexisting with the center manifold in the sliding region. This occurs because the condition V + V − = 1 means that the reﬂection vectors (1, V + ) and (V − , 1) are colinear, with the same direction but opposite orientation. A full investigation of higher order behavior is beyond the scope of this paper. One way forward is to consider the eﬀect of higher order terms on the second-return maps. At V + V − = 1 the eigenvalues of the Jacobians of these maps satisfy the Takens–Bogdanov condition [8, 13] that they are both unity. The eigenvalues for V + V − < 1 lie on the unit circle, so the origin is nonhyperbolic, but this does not imply structural instability because orbits evolve under the map only for a ﬁnite time, after which they reach the sliding region. 8. Concluding remarks. A vector formulation has been employed here wherever the analysis applies in general coordinate systems, for example, when S ± are nonorthogonal. The

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M. R. JEFFREY AND A. COLOMBO

vector expressions also generalize naturally to higher dimensions. Speciﬁcally, the vector directions ∇h and S± are well deﬁned in n dimensions, respectively, by the codimension one switching surface Σ and the codimension two tangency sets S ± . Only the components ∇h · X and s± · X play an essential role in a neighborhood where the maps (3.7) are valid. The unfolding parameter V + V − and the codimension two manifolds S and U will form a foundation for study in n > 3, where dynamics is possible within the tangency sets S ± , and parallel to the n − 3 dimensional singular set S + ∩ S − . Theorem 1(i) characterizes the attractivity of the two-fold singularity when the two vector ˙ ± form an obtuse angle at the singularity, in the plane spanned by orthogonal s+ ﬁelds X and s− , measured on the side of the sliding region (the condition V + V − > 1). The state space contains regions of attraction and repulsion whose boundaries are a pair of invariant parabolic surfaces. A stable surface S encloses the escaping region, and orbits are repelled from the singularity as they approach it. An unstable surface U encloses the sliding region and orbits are attracted to the singularity as they depart it, until they impact the sliding region or approach S. Both surfaces are smooth except at their intersections with the crossing regions. Orbits of the sliding vector ﬁeld take the form of a stable node at the singularity, though the node is reached in ﬁnite time, and the vector ﬁeld is undeﬁned at the singularity itself. As the obtuse angle increases and the vector ﬁelds pass the point V + V − = 1 where they are colinear at the singularity, a bifurcation occurs which destroys the two invariant surfaces, and all orbits and sliding orbits ﬂow away from the singularity. In this case (see Theorem 1(ii)), the two vector ﬁelds form an obtuse angle measured on the side of the escaping region (the condition V + V − < 1). All orbits originating close to the escaping region will eventually impact the sliding region, where the sliding vector ﬁeld takes the form of a saddlepoint with its unstable manifold in the sliding region. If one or both of the vector ﬁelds points into the sliding region at the singularity, V ± ≥ 0, then all orbits reach the sliding region by crossing Σ at most once, after which they are repelled ˙ + is perpendicular to its from the singularity. The case V + = 0 means that the vector ﬁeld X tangency set S + (similarly for the “−” case). At the bifurcation point V + V − = 1 the vector ﬁelds are anticolinear at the singularity. The Teixeira singularity system then becomes an unfolding of the “fused-focus” in planar nonsmooth systems [9]. ˙ + is parallel to its own tangency set S + and forms a line When V + → ∞, the vector ﬁeld X of cusps; that case is too degenerate to be of interest here (similarly for V − ). If the tangency sets S ± become tangent to each other at a codimension two point, we can appeal to our analysis for some basic intuition. The codimension two point splits under perturbation into a pair of two-fold singularities identiﬁed by Teixeira [16], one of the saddle type V + V − > 1 and one of the focal type V + V − < 1, and the dynamics around this singularity certainly merits further investigation. The bifurcation in the sliding vector ﬁeld observed in the bottom row of Figure 8 is related to Teixeira’s “Q5-singularity case 2” [16]. We have shown how it occurs necessarily in the unfolding of the bifurcation. Is the Teixeira singularity stable? On the question of asymptotic stability, we reiterate that the singularity is not a stationary point of the vector ﬁeld. Asymptotic stability can only refer to the normalized sliding vector ﬁeld (6.3), whose dynamics are diﬀerent from the true

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THE TWO-FOLD SINGULARITY

639

system. Instead we ﬁnd that state space is separated into regions of attraction and repulsion, as given by the two regimes of Theorem 1 and the sliding dynamics of section 6. The invariant surfaces are locally asymptotically stable (S) and unstable (U), except at the singularity. The dynamics at the singularity is not uniquely deﬁned in the Filippov convention, which we have followed here. It is our position that structural stability in nonsmooth dynamical systems is not yet on as sound a footing as in smooth systems, and the Teixeira singularity is vital to this continuing investigation. To this end we have described the local dynamics and shown that it varies smoothly with the parameter V + V − , except at the bifurcation. This implies that the unfolding of the singularity in the parameter V + V − is structurally stable in the usual sense— intuitively that nearby orbits have the same topology in terms of the number of crossings, the tendency toward S and away from U, the impact in Σ, and so on. We have shown that the characteristic dynamics of the Teixeira singularity involves bifurcations simultaneously in the crossing regions (equivalently, out of the switching surface) and in the sliding/escaping regions. A single parameter V + V − quantiﬁes the relative direction of the vector ﬁelds at the singularity (or the jump in direction of the overall nonsmooth vector ﬁeld through the singularity), controlling the bifurcation and determining domains of attraction. We have shown how the system behaves under perturbation at the bifurcation, and the eﬀect of higher order terms here is currently in progress. Also of interest for applications is a closer look at the dynamics of the Tm map, including the number of iterations in each orbit and the dynamics around near misses of the sliding boundary, and comparison of this with physical models. Appendix A. An explicit expression for the vector fields. At the singularity the x component of both vector ﬁelds vanishes, and locally we can expand to ﬁrst order in the coordinates, giving ± ± ± ± ± (A.1) x˙ , x˙ ≈ x · a± + xb± x , xA + xb + c ± ± ± in terms of constant scalars b± x , vectors a , b , and 2 × 2 matrices A . We are interested only + − in quadratic tangencies, so we must impose the condition cy , cz = 0, and in general we will assume that c± are nonparallel. Transversality of the tangent sets S ± requires that a± are also nonparallel. The unit vectors s± satisfy

(A.2)

s± · a± = 0,

and the choice of coordinates giving (2.7) is (A.3)

y = −x · a+ − xb+ x,

z = x · a− + xb− x.

This is a diﬀerentiable coordinate transformation given the condition + + + − s ·a s · a (A.4) s− · a+ s− · a− = 0. The analysis thereafter applies on a neighborhood of the singularity satisfying

and s− · xA± + xb± c± (A.5) s+ · xA± + xb± c± y z.

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M. R. JEFFREY AND A. COLOMBO

Writing

(A.6)

x˙ + , x˙ + ≈ −y, c+ , − − x˙ , x˙ ≈ z, c− ,

we can rescale time independently in the x > 0 and x < 0 systems without altering the − piecewise-smooth system topologically, letting t → t/c+ y for x > 0 and t → t/cz for x < 0, + − resulting in (2.10) by setting a = 1/cy and b = 1/cz . REFERENCES [1] M. E. Broucke, C. Pugh, and S. Simic, Structural stability of piecewise smooth systems, Comput. Appl. Math., 20 (2001), pp. 51–90. [2] R. Casey, H. de Jong, and J. L. Gouze, Piecewise-linear models of genetic regulatory networks: Equilibria and their stability, J. Math. Biol., 52 (2006), pp. 27–56. [3] M. di Bernardo, C. J. Budd, A. R. Champneys, and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Springer-Verlag, New York, 2008. [4] M. di Bernardo, F. Garofalo, L. Glielmo, and F. Vasca, Switchings, bifurcations, and chaos in dc/dc converters, IEEE Trans. Circuits Systems I Fund. Theory Appl., 45 (1998), pp. 133–141. [5] A. F. Filippov, Diﬀerential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. [6] J. Y. Hung, W. B. Gao, and J. C. Hung, Variable structure control—a survey, IEEE Trans. Industrial Electronics, 40 (1993), pp. 2–22. [7] M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. [8] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer-Verlag, New York, 1998. [9] Yu. A. Kuznetsov, S. Rinaldi, and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), pp. 2157–2188. [10] R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, SpringerVerlag, New York, 2004. [11] H. E. Nusse, E. Ott, and J. A. Yorke, Border-collision bifurcations: An explanation for observed bifurcation phenomena, Phys. Rev. E (3), 49 (1994), pp. 1073–1076. [12] N. S. Simic, K. H. Johansson, J. Lygeros, and S. Sastry, Structural stability of hybrid systems, in Proceedings of the European Control Conference, Porto, Portugal, 2001, pp. 3858–3863. [13] F. Takens, Forced oscillations and bifurcations, in Applications of Global Analysis I, Comm. Math. Inst. Rijksuniv. Utrecht 3, Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, pp. 1–59. [14] M. A. Teixeira, On topological stability of divergent diagrams of folds, Math. Z., 180 (1982), pp. 361–371. [15] M. A. Teixeira, Stability conditions for discontinuous vector ﬁelds, J. Diﬀerential Equations, 88 (1990), pp. 15–29. [16] M. A. Teixeira, Generic bifurcation of sliding vector ﬁelds, J. Math. Anal. Appl., 176 (1993), pp. 436– 457.

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c 2009 Society for Industrial and Applied Mathematics

SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 8, No. 2, pp. 641–675

Spatially Periodic Patterns of Synchrony in Lattice Networks∗ Ana Paula S. Dias† and Eliana Manuel Pinho† Abstract. We consider n-dimensional Euclidean lattice networks with nearest neighbor coupling architecture. The associated lattice dynamical systems are inﬁnite systems of ordinary diﬀerential equations, the cells, indexed by the points in the lattice. A pattern of synchrony is a ﬁnite-dimensional ﬂowinvariant subspace for all lattice dynamical systems with the given network architecture. These subspaces correspond to a classiﬁcation of the cells into k classes, or colors, and are described by a local coloring rule, named balanced coloring. Previous results with planar lattices show that patterns of synchrony can exhibit several behaviors such as periodicity. Considering suﬃciently extensive couplings, spatial periodicity appears for all the balanced colorings with k colors. However, there is not a direct way of relating the local coloring rule and the coloring of the whole lattice network. Given an n-dimensional lattice network with nearest neighbor coupling architecture, and a local coloring rule with k colors, we state a necessary and suﬃcient condition for the existence of a spatially periodic pattern of synchrony. This condition involves ﬁnite coupled cell networks, whose couplings are bidirectional and whose cells are colored according to the given rule. As an intermediate step, we obtain the proportion of the cells for each color, for the lattice network and any ﬁnite bidirectional network with the same balanced coloring. A crucial tool in obtaining our results is a classical theorem of graph theory concerning the factorization of even degree regular graphs, a class of graphs where lattice networks are included. Key words. lattice dynamical systems, coupled cell networks, balanced colorings, patterns of synchrony, spatially periodic patterns AMS subject classifications. 82B20, 34C15, 37L60, 05C90 DOI. 10.1137/080725969

1. Introduction. 1.1. Context. The theory of coupled cell systems, developed by Stewart, Golubitsky, and Pivato [11] and Golubitsky, Stewart, and T¨ or¨ ok [7], applies to the case of a lattice network, GL , where an inﬁnite number of identical cells are disposed on a lattice L. Each cell is a system of ordinary diﬀerential equations with phase space Rd , with d ∈ N, and is coupled to the same number of cells in its neighborhood such that the whole system is invariant by the translations of L. In this work we consider n-dimensional Euclidean lattices with nearest neighbor coupling architecture. If J is the set of the nearest neighbors of the origin, then a lattice dynamical ∗

Received by the editors May 30, 2008; accepted for publication (in revised form) by M. Silber March 11, 2009; published electronically May 14, 2009. This research was supported in part by Centro de Matem´ atica da Universidade do Porto (CMUP), financed by FCT through the programmes POCTI and POSI, with Portuguese and European Community structural funds. http://www.siam.org/journals/siads/8-2/72596.html † CMUP and Departamento de Matem´ atica Pura, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal ([email protected], [email protected]). The second author was supported by FCT Grant SFRH/BPD/ 29975/2006. 641

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ANA PAULA S. DIAS AND ELIANA MANUEL PINHO

system is deﬁned by the inﬁnite number of ordinary diﬀerential equations: (1.1)

x˙ l = f (xl , xl+g1 , . . . , xl+gv ) with l ∈ L,

where v = #J is the valence or degree, g1 , . . . , gv ∈ J , and f is invariant under all permutations of the variables under the bar. Lattice networks are constant degree graphs, called regular graphs. This follows the general deﬁnition of lattice networks and lattice dynamical systems considered by Antoneli et al. [2], where it is assumed that the coupling structure is determined by the distances between lattice points. Patterns of synchrony are ﬁnite-dimensional ﬂow-invariant subspaces for all lattice dynamical systems with a given network architecture and are formed by setting equal the coordinates in diﬀerent cells. The general theory of coupled cell systems [7, 11] shows that ﬁnding patterns of synchrony is equivalent to ﬁnding balanced colorings of cells, where the color of the cell l ∈ L deﬁnes the number of its neighbors with each color. In a nearest neighbor coupling architecture, each balanced coloring corresponds to a coloring matrix A = (aij ). The entry aij is the number of cells with color j that are coupled to a cell of color i, with i, j ∈ U = {1, . . . , k}. Here U is the set that indexes the k diﬀerent colors of the balanced coloring. Our study concerns balanced colorings for a general number of colors k. Equation (1.1) can be restricted to the ﬂow-invariant subspace deﬁned by identifying the cell coordinates of cells with the same color, denoted by xi with i ∈ U. Thus, the restriction is deﬁned by the ﬁnite-dimensional system with k equations: ⎧ ⎪ ⎨ x˙ 1 = f (x1 , y11 , . . . , y1v ) , .. (1.2) . ⎪ ⎩ x˙ k = f (xk , yk1 , . . . , ykv ) , where the arrays (yi1 , . . . , yiv ) have aij entries equal to xj . This system can be described by a quotient network, a coupled cell network with k cells, one of each color, having adjacency matrix A. See section 1.4 for an illustration of these concepts. There are several results concerning patterns of synchrony in lattice networks. In what follows, we consider that a family of patterns is a set of patterns in a lattice network having the same balanced coloring rule, for a given architecture, and that a spatially periodic pattern of synchrony corresponds to a balanced coloring of an n-dimensional lattice network that is periodic along n linearly independent directions. Golubitsky, Nicol, and Stewart [6] describe an inﬁnite family of two-color patterns of synchrony on planar square lattice systems with nearest neighbor coupling, and Wang and Golubitsky [12] classify all possible two-color patterns of synchrony of planar square and hexagonal lattice networks with two diﬀerent architectures—nearest or both nearest and next nearest neighbor couplings. For nearest neighbor coupling architecture they show that there are two (resp., three) inﬁnite continuum families in the square (resp., hexagonal) lattice and eight (resp., ten) isolated patterns. Moreover, these include spatially periodic and nonperiodic colorings. The technique followed in this enumeration was strongly based on the following: (i) the enumeration of all possible two-cell networks with one type of edge and where each cell

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receives four (resp., six) inputs; (ii) checking which of these two-cell networks was corresponding to a balanced coloring of the ﬁxed planar lattice with nearest neighbor coupling. From the tractable point of view this method, followed for k = 2 and n = 2, does not generalize for all n or k. Antoneli et al. [2] study k-color patterns of synchrony in Euclidean lattices, proving that, for planar lattices, they are periodic if the couplings are suﬃciently extensive. 1.2. Our results. Our aim is to relate the local structure, given by the rule of the balanced coloring, with the long-range behavior of the patterns of synchrony, for a general n-dimensional lattice and a general balanced k-coloring. Given an n-dimensional Euclidean lattice L, with nearest neighbor coupling architecture, and a k × k matrix A, our main result, Theorem 5.4, states a necessary and suﬃcient condition for the existence of a spatially periodic pattern of synchrony in the lattice network, corresponding to the balanced coloring deﬁned by A. The condition involves the bidirectional networks having the coloring matrix A, networks that have all the couplings bidirectional and of even valence. These networks admit a factorization into 2-factors—a decomposition of the couplings which produces networks with the same cells and valence two. This deﬁnes a set Σ of permutations of their cells, one permutation for each 2-factor. If there is a homomorphism between the group generated by these permutations and the lattice L, then we say that Σ and J are identiﬁable. Let the set DA be formed by the sets of permutations Σ associated with all the factorizations of all the possible ﬁnite bidirectional networks with coloring matrix A. We state our main result, which will be proved later, as follows. Theorem 5.4. Let A be a coloring matrix with even valence v, and let L be a Euclidean lattice with #J = v. The lattice network with nearest neighbor coupling architecture GL has a periodic balanced coloring with the coloring matrix A if and only if DA has an element identiﬁable with J . We highlight some features of our results and the underlying methods. One important point is that, given a k-coloring rule and an n-dimensional Euclidean lattice network with nearest neighbor coupling architecture, the existence of a periodic pattern of synchrony is translated into a local problem, involving ﬁnite networks with special properties. These ﬁnite networks are easily handled, and standard methods in matrix theory can be used to treat them. In particular, if we approach the problem by imposing restrictions upon the fundamental domain of possible patterns of synchrony, the set of ﬁnite networks to be considered is ﬁxed. These restrictions may concern dimension and geometry and arise naturally, for example, in problems with periodic boundary conditions. See the second example in this section and the proof of Theorem 5.4 (from lattice to ﬁnite bidirectional networks). Another aspect is the constructive nature of the method. The determination of an element in DA identiﬁable with J leads immediately to the construction of a periodic pattern of synchrony, and the same is true in the converse direction. We present several examples of these constructions in this section and in section 6. Moreover, the proof of Theorem 5.4 is itself constructive. A crucial tool in obtaining our results is a classical theorem of graph theory concerning the factorization of even degree regular graphs, a class of graphs which includes lattice networks, suggesting the possibility of applying other graph theory techniques in related problems.

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ANA PAULA S. DIAS AND ELIANA MANUEL PINHO

Finally, the method for establishing the existence of periodic patterns of synchrony stands for lattices of any dimension and k-colorings for all k. Moreover, as described in section 6, knowing patterns of synchrony for a Euclidean lattice network gives information about patterns in lattices with diﬀerent dimensions or diﬀerent geometry. 1.3. Article structure. We continue the introduction with two examples illustrating our results and the methods we have used to prove them. The rest of the paper is organized in the following way. In section 2 we introduce the notation and background on coupled cell networks and lattice networks used in this work. Section 3 speciﬁes the deﬁnition of balanced coloring of the cells in a network to the class of identical-edge homogeneous networks. Moreover, we introduce a matrix formulation of balanced coloring and formalize the notions of symmetry and spatial periodicity of balanced colorings. Sections 4 and 5 are the core of our work. Section 4 is dedicated to the construction and the decomposition of ﬁnite bidirectional networks. We state results concerning the ratio of the cells for each color on a balanced coloring of both ﬁnite bidirectional and lattice networks, and ensure the factorization into 2-factors of the considered ﬁnite bidirectional networks. In section 5 we prove our main result, Theorem 5.4. Under the conditions stated in this theorem, we show how to relate a periodic balanced coloring in a ﬁnite bidirectional network to the same balanced coloring in a lattice network, using a projection of the lattice into the ﬁnite network. Finally, in section 6, we remark on several consequences of Theorem 5.4, where some “inclusions” involving balanced colorings with diﬀerent dimensions are established. Several examples of periodic patterns of synchrony for the standard cubic lattice are presented. 1.4. First example. We consider an inﬁnite number of identical cells disposed on a lattice and ask whether it is possible to color these cells with two diﬀerent colors, say black and grey, such that the following rule is followed: each black cell has four grey neighbor cells, and each grey cell has two black neighbors, where the neighbors of each cell are its closest cells of the lattice. If each cell in the lattice is a system of ordinary diﬀerential equations, coupled to the nearest neighbor cells that are identical systems, this coloring problem corresponds to the definition of a ﬂow-invariant subspace of the total phase space; see [2] for a complete description of these systems. Applying the main result of this paper (Theorem 5.4), we ensure that the answer to that question is aﬃrmative for the planar square, the planar hexagonal, and the standard cubic lattices. Moreover, we have a method to construct a periodic two-coloring of each of these lattice networks respecting the given rule. 1.4.1. The planar square lattice network. Formally, we begin with a planar square lattice L = {(1, 0), (0, 1)}Z and consider the nearest neighbor coupling architecture—each element l ∈ L interacts with the elements in l + J = {l ± l1 , l ± l2 }, where l1 = (1, 0) and l2 = (0, 1); see Figure 1. Given the lattice and the architecture above, the two-dimensional lattice network that we consider is the one represented in Figure 2, where the couplings are bidirectional: each cell receives four inputs from its neighbor cells—we say the valence of the graph is 4.

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Figure 1. The nearest neighbors of l ∈ L are the elements in the black dots.

Figure 2. A portion of the two-dimensional lattice network for a square lattice architecture with nearest neighbor coupling.

Choosing a phase space Rd for each point in the lattice, a lattice dynamical system consistent with the given lattice architecture is described by (1.3)

x˙ l = f (xl , xl+l1 , xl−l1 , xl+l2 , xl−l2 ) .

Here l ∈ L, xl ∈ Rd , and f is invariant under all permutations of the variables under the bar. 1.4.2. The balanced coloring. Consider a set of two diﬀerent colors, black and grey, indexed by the elements in U = {1, 2}, and the rule each black cell receives four inputs from grey cells and each grey cell receives two inputs from black cells. This is case 42 considered by Wang and Golubitsky [12, p. 633], and we will use this terminology throughout. This rule can be described by the adjacency matrix 0 4 A= , 2 2 where the (i, j) entry is the number of inputs that each cell of color i receives from cells of color j or, equivalently, by a coupled cell network with two cells, one for each color, as in Figure 3. We now construct a periodic two-coloring of the lattice network respecting the above rule. We show in Theorem 5.4 that this coloring is related to the decomposition of a ﬁnite network whose couplings are bidirectional and whose cells are colored under the same rule. In sections

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ANA PAULA S. DIAS AND ELIANA MANUEL PINHO

Figure 3. Each black cell receives four inputs from grey cells, and each grey cell receives two inputs from grey cells and two inputs from black cells.

1.4.3 to 1.4.6 we choose a ﬁnite network with bidirectional couplings, decompose it, and build a periodic coloring of the planar square lattice. Observe that equations (1.3), when restricted to the ﬂow-invariant subspace deﬁned by identifying the cell coordinates of cells with the same color, form the ﬁnite-dimensional system (1.4)

x˙ b = g (xb , xg , xg , xg , xg ) , x˙ g = g (xg , xg , xb , xg , xb ) ,

where xb and xg denote the coordinates, respectively, of black and grey cells and belong to Rd , for some d ∈ N. 1.4.3. Proportion of cells for each color. By Lemmas 4.3 and 4.8, we obtain the proportion pi of cells of color i ∈ U in this balanced two-coloring of the square lattice network with nearest neighbor coupling architecture, as well as in the balanced coloring of a ﬁnite network with bidirectional couplings and valence 4. Speciﬁcally, it is the ith coordinate of the left eigenvector of the matrix A, associated with the eigenvalue 4, whose coordinates sum to 1: (p1 , p2 )A = 4(p1 , p2 ). The eigenvector such that p1 + p2 = 1 is T

p =

1 2 , 3 3

,

meaning that the ratio of black and grey cells in a balanced coloring corresponding to case 42 is p1 /p2 = 1/2. 1.4.4. The finite networks with bidirectional couplings. We construct coupled cell networks, with bidirectional couplings, corresponding to case 42. By Lemma 4.3, the number of cells for each color in these networks follows the proportion given by p. We choose the networks having the smallest number of cells. Since 3pT = (1, 2), three is that smallest number. Moreover, there are two possible networks with three cells; see Figure 4. Let {c1 , c2 , c3 } be the set of cells, where c1 is the black cell and c2 and c3 are the grey cells. The adjacency matrices B1 and B2 of these bidirectional coupled cell networks have in the (i, j)-position the number of inputs that cell ci receives from cell cj : ⎞ 0 2 2 B1 = ⎝ 2 0 2 ⎠ 2 2 0 ⎛

⎞ 0 2 2 B2 = ⎝ 2 2 0 ⎠ . 2 0 2 ⎛

and

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Figure 4. Two coupled cell networks where all the couplings are bidirectional. Each black cell receives four inputs from grey cells, and each grey cell receives two inputs from grey cells and two inputs from black cells.

Figure 5. The decomposition of the couplings for the two bidirectional networks of Figure 4. Both G1 and G2 are decomposed into bidirectional networks with valence 2.

1.4.5. Decomposition in cycles. By Lemma 4.4, each of these matrices can be decomposed into the sum of two (half the valence) symmetric matrices whose lines sum to 2. For this example we have ⎛ ⎞ ⎛ ⎞ 0 1 1 0 1 1 B1 = ⎝ 1 0 1 ⎠ + ⎝ 1 0 1 ⎠ 1 1 0 1 1 0 and

⎛

⎞ ⎛ ⎞ 0 2 0 0 0 2 B2 = ⎝ 2 0 0 ⎠ + ⎝ 0 2 0 ⎠ , 0 0 2 2 0 0

corresponding to the decomposition of the couplings shown in Figure 5. Each one of the matrices obtained can be written as the sum Mσ + MσT for some permutation matrix Mσ associated with a permutation σ ∈ S3 . Here 3 is the number of cells in the bidirectional coupled cell network that we are considering. For example, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 1 1 0 0 1 0 1 0 ⎝ 1 0 1 ⎠ = ⎝ 1 0 0 ⎠ + ⎝ 0 0 1 ⎠. 1 1 0 0 1 0 1 0 0

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648

ANA PAULA S. DIAS AND ELIANA MANUEL PINHO

If Mσ1 is the matrix associated with the permutation σ1 = (c1 c2 c3 ), ⎛

Mσ1

⎞ 0 0 1 = ⎝ 1 0 0 ⎠, 0 1 0

and taking σ2 = σ1 , we have that B1 is the sum

B1 = Mσ1 + MσT1 + Mσ2 + MσT2 = 2 Mσ1 + MσT1 . 1.4.6. The periodic pattern. Since σ1 and σ2 commute, then, by Theorem 5.4, we can construct a periodic balanced two-coloring of the lattice network, as shown in Figure 6. Beginning with any cell ci in any position l ∈ L, the cell in position l + lj will be σj (ci ), for j = 1, 2.

Figure 6. The permutations σ1 and σ2 are associated, respectively, with the generators l1 and l2 of the lattice.

The resulting colored pattern is represented in Figure 7, together with its noncolinear periods 3l1 and 2l1 + l2 . By Theorem 5.4, these periods correspond to the compositions of the permutations σ13 = and σ12 ◦ σ2 = , where denotes the identity element of S3 . 1.4.7. The pattern is always found. A similar decomposition for the matrix B2 results in

B2 = Mσ3 + MσT3 + Mσ4 + MσT4 ,

where σ3 = (c1 c2 )(c3 ) and σ4 = (c1 c3 )(c2 ). Since these permutations do not commute, the above construction is not possible. However, Theorem 5.4 ensures that, if there is a periodic pattern satisfying the required balanced coloring condition on the lattice, then there is always a ﬁnite bidirectional network whose decomposition allows its construction; see the next example in section 1.5.

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Figure 7. The periodic pattern obtained for case 42 with the two noncolinear periods 3l1 and 2l1 + l2 .

l +l 2

l +l 3 l -l 1 l

l +l 1

l -l 3

l -l 2

(0,1,0) l 2 (0,0,1) l3 (0,0,0)

l1

(1,0,0)

Figure 8. The nearest neighbors of l ∈ L are the elements in the black dots.

1.4.8. The planar hexagonal and the standard cubic lattice networks. Now we repeat the above construction for two lattice networks with valence 6. Let L be the standard cubic lattice L = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}Z with nearest neighbor coupling architecture J = {±l1 , ±l2 , ±l3 }, where l1 = (1, 0, 0), l2 = (0, 1, 0), and l3 = (0, 0, 1). See Figure 8.

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Figure 9. The nearest neighbors of l ∈ L are the elements in the black dots.

C1

C2

C3

Figure 10. A bidirectional coupled cell network. Each black cell receives four inputs from grey cells, and each grey cell receives two inputs from grey cells and two inputs from black cells.

Consider also the planar hexagonal lattice L=

(1, 0),

√ 1 3 , 2 2

Z

with √ nearest neighbor coupling architecture J = {±l1 , ±l2 , ±l3 }, where l1 = (1, 0), l2 = (1/2, 3/2), and l3 = l2 − l1 . See Figure 9. The coloring rule corresponding to case 42 is now described by the adjacency matrix 2 4 A= , 2 4 and the proportion of cells with each color is T

p =

1 2 , 3 3

as in the previous example with valence 4. One ﬁnite bidirectional network with three cells Figure 10, has adjacency matrix B: ⎞ ⎛ ⎞ ⎛ ⎛ 2 2 2 0 1 1 B =⎝ 2 2 2 ⎠=⎝ 1 0 1 ⎠+⎝ 1 1 0 2 2 2

,

corresponding to case 42, represented in ⎞ ⎛ ⎞ 0 1 1 2 0 0 1 0 1 ⎠+ ⎝ 0 2 0 ⎠. 1 1 0 0 0 2

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As in the previous case, the ﬁrst two matrices in the decomposition of B are associated with the permutations σ1 and σ2 : σ1 = σ2 = (c1 c2 c3 ). The third matrix, 2Id3 , is associated with σ3 = . These permutations commute, and, by Theorem 5.4, this condition ensures that we can build a periodic balanced coloring of the given cubic lattice if we associate each permutation with a generator. See the resulting pattern in Figure 11.

(0,1,0)

(0,0,1) (1,0,0) (0,0,0)

Figure 11. A periodic pattern obtained for case 42.

For the hexagonal lattice, the elements l1 , l2 , and l3 are not linearly independent, l3 = l2 − l1 . By Theorem 5.4, we can associate the permutation σi with the direction li if σ3 = σ2 ◦ σ1−1 . This condition is veriﬁed, and thus we built the periodic pattern in Figure 12, one of the two balanced colorings corresponding to case 42 in the planar hexagonal lattice with nearest neighbor architecture, described by Wang and Golubitsky [12, p. 636]. In section 6 we present other examples of periodic balanced colorings of the standard cubic lattice with nearest neighbor coupling architecture. 1.5. Second example. In this section we follow the converse direction of the previous example. We begin with a spatially periodic pattern with a balanced coloring in a planar hexagonal lattice and then construct a ﬁnite network with bidirectional couplings and with the same coloring rule as the lattice network. 1.5.1. The periodic coloring in the lattice. Let L be the planar hexagonal lattice described above, √ 3 1 , , L = (1, 0), 2 2 Z

with √ nearest neighbor coupling architecture J = {±l1 , ±l2 , ±l3 }, where l1 = (1, 0), l2 = (1/2, 3/2), and l3 = l2 − l1 ; see Figure 9.

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Figure 12. A periodic pattern obtained for case 42.

Figure 13. The periodic pattern for case 52 with the two noncolinear periods l1 + 2l2 and 3l2 − 2l1 . The fundamental domain, whose periodic repetition forms the pattern, is in light grey.

Consider a coloring of the cells on this lattice with two colors, black and grey, such that each black cell receives ﬁve inputs from grey cells and one input from a black cell, and each grey cell receives two inputs from black cells and four inputs from grey cells. This is case 52 considered by Wang and Golubitsky [12, p. 636], and we present in Figure 13 the only possible pattern respecting this coloring rule, as proved in [12, Theorem 1.7]. We can identify all the cells of the hexagonal lattice that diﬀer by a period of the colored pattern, obtaining seven diﬀerent types of cells that compose the fundamental domain—a portion of the pattern that is periodically repeated. Let {c1 , . . . , c7 } be the set of diﬀerent cells up to periods and see how they form the pattern in Figure 15 below. 1.5.2. The finite network. Now we associate a permutation of the cells c1 , . . . , c7 with each direction of the couplings in the lattice network. See Figure 14.

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Figure 14. The directions of the nearest neighbor couplings and the associated permutations. Here l3 = l2 − l1 , where l1 and l2 generate the planar hexagonal lattice.

3

2 1

1

2

3

Figure 15. The sequence of the cells c1 , . . . , c7 associated with the permutations σ1 = (c1 c2 c6 c3 c7 c4 c5 ), σ2 = (c1 c3 c5 c6 c4 c2 c7 ), and σ3 = (c1 c6 c7 c5 c2 c3 c4 ) along each direction of the lattice and for a ﬁnite coupled cell network.

In Figure 15 we see that the permutations σ1 = (c1 c2 c6 c3 c7 c4 c5 ), σ2 = (c1 c3 c5 c6 c4 c2 c7 ), and σ3 = (c1 c6 c7 c5 c2 c3 c4 ) describe the sequence of cells along the given directions. These sequences deﬁne the couplings of a ﬁnite coupled cell network with the set of cells {c1 , . . . , c7 }. The inverse permutations, σ1−1 = (c1 c5 c4 c7 c3 c6 c2 ), σ2−1 = (c1 c7 c2 c4 c6 c5 c3 ), and σ3−1 = (c1 c4 c3 c2 c5 c7 c6 ), correspond to the sequences of cells in the pattern along the opposite directions. If we consider both σi and σi−1 , then the couplings of the ﬁnite network are all

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ANA PAULA S. DIAS AND ELIANA MANUEL PINHO

bidirectional. We obtain the coupled cell ⎛ 0 ⎜ 1 ⎜ ⎜ 1 ⎜ B=⎜ ⎜ 1 ⎜ 1 ⎜ ⎝ 1 1 given by

network in Figure 16, having adjacency matrix ⎞ 1 1 1 1 1 1 0 1 1 1 1 1 ⎟ ⎟ 1 0 1 1 1 1 ⎟ ⎟ 1 1 0 1 1 1 ⎟ ⎟ 1 1 1 0 1 1 ⎟ ⎟ 1 1 1 1 0 1 ⎠ 1 1 1 1 1 0

B = Mσ1 + MσT1 + Mσ2 + MσT2 + Mσ3 + MσT3 .

The coloring of this ﬁnite coupled cell network follows the rule of the original coloring of the hexagonal lattice. Therefore, if we were looking for a coloring of the hexagonal lattice for case 52, our results imply that the existence of this ﬁnite network would give us a decomposition ensuring a periodic pattern on the lattice network with the given balanced coloring rule.

Figure 16. The coupled cell network with bidirectional couplings obtained from case 52 of a hexagonal lattice.

2. Definitions—Coupled cell networks. In this section we describe the notation and concepts concerning the coupled cell networks structure. 2.1. Coupled cell network. We use the following deﬁnition of coupled cell network with a countable number of cells in [2]. Definition 2.1 (see [2, Definition 2.1]). A coupled cell network G consists of 1. a countable set C of cells, 2. an equivalence relation ∼C on cells in C, 3. a countable set E of edges or arrows, 4. an equivalence relation ∼E on edges in E (the edge type of edge e is the ∼E -equivalence class of e), 5. (local ﬁniteness) a head map H : E → C and a tail map T : E → C such that for every c ∈ C the sets H−1 (c) and T −1 (c) are ﬁnite, 6. (consistency condition) equivalent arrows having equivalent tails and equivalent heads; that is, if e1 ∼E e2 in E, then every H(e1 ) ∼C H(e2 ) and T (e1 ) ∼C T (e2 ). Here we use the notation G = (C, E, ∼C , ∼E ).

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2.2. Input set of a cell and homogeneous coupled cell network. We consider networks that may have multiple edges and self-coupling. Thus, we follow the deﬁnition of an input set of a cell and of isomorphic input sets in [7, Deﬁnitions 2.2 and 2.3], as follow. Definition 2.2. Let c ∈ C. The input set of c is I(c) = {e ∈ E : H(e) = c} with a ﬁxed ordering. An element of I(c) is called an input edge or input arrow of c. We say that c receives m inputs from d if m = #{e ∈ I(c) : T (e) = d}. Definition 2.3. Two input sets I(c) and I(d) are isomorphic if there is a bijection β : I(c) → I(d) that preserves the input edge type: i ∼E β(i)

for all i ∈ I(c).

Observe that if two cells have isomorphic input sets, then, in particular, they are in the same ∼C -equivalence class. Definition 2.4. A coupled cell network is homogeneous if all the cells have isomorphic input sets. From now on we consider identical-edge homogeneous networks, that is, homogeneous coupled cell networks with only one edge type. We denote these by G = (C, E). For this class of networks, the input sets I(c) have only one edge type and a ﬁxed cardinality called valence, v = #I(c), for all c ∈ C. 2.3. Lattice and nearest neighbors. An n-dimensional lattice L is a set L = {l1 , . . . , ln }Z ⊂ Rn , where the n elements l1 , . . . , ln ∈ Rn , the generators of the lattice, are linearly independent. Let |l| denote the Euclidean norm of l ∈ Rn . The set {|l| : l ∈ L} is a countable set, and the possible Euclidean distances of l ∈ L to the origin are r0 , r1 , r2 , . . . such that r0 = 0 and ri < ri+1 for all i ∈ N. Considering the set of cells J deﬁned by J = {l ∈ L : |l| = r1 }, the nearest neighbors of l ∈ L are the elements on l + J = {l + g : g ∈ J }. Definition 2.5. An n-dimensional Euclidean lattice is a lattice L = {l1 , . . . , ln }Z such that |li | = r1 for all i ∈ {1, . . . , n}. Observe that if L = {l1 , . . . , ln }Z is a Euclidean lattice, then we have {l1 , . . . , ln } ⊂ J . 2.4. n-dimensional lattice network. We deﬁne an n-dimensional lattice network with nearest neighbor coupling architecture, a special case of the deﬁnition given by Antoneli et al. [2, Deﬁnition 2.5] for more general architectures. Definition 2.6. An n-dimensional lattice network with nearest neighbor coupling architecture consists of

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1. an n-dimensional lattice L, 2. a homogeneous coupled cell network GL whose cells are indexed by L, 3. I(c) = {e : T (e) ∈ c + J } for all c ∈ L, 4. all the arrows in I(0) having the same edge type. The valence of a lattice network GL is a geometric property of L and equals #J for the nearest neighbor architecture. Since −L = L and |−l| = |l|, the set J has even cardinality. If L = {l1 , . . . , ln }Z is a Euclidean lattice, then we can choose ln+1 , . . . , lv/2 ∈ L such that J = {±l1 , . . . , ±ln , ±ln+1 , . . . , ±lv/2 }. All the couplings in the lattice networks are bidirectional. 3. Definitions—Balanced colorings. In this section we specify the deﬁnitions of balanced equivalence relation on C, with k equivalence classes, and of the quotient network given in [7, sections 4 and 5] to the class of identical-edge homogeneous networks. Moreover, we introduce a matrix formulation of balanced coloring that will be used in the remaining sections. 3.1. Balanced k-colorings. Definition 3.1. Let G = (C, E) be a coupled cell network, and suppose that we color the cells in C with k diﬀerent colors. Let the colors be indexed by the set U = {1, . . . , k}. The k-coloring ξ of the network G is the function ξ : C −→ U, c −→ ξ(c), where ξ(c) is the color of cell c. In order to use matrices we deﬁne an analogous vector notation, ξ : C −→ {ei , i ∈ U }, = eξ(c) , c −→ ξ(c) where {ei , i ∈ U } is the canonical basis of Rk . Unless otherwise stated, all the vectors are column vectors. We will refer to the coloring as without distinction, since they are equivalent deﬁnitions by the isomorphism i ←→ ei ξ or ξ, between U and {ei , i ∈ U }. Definition 3.2. Let G = (C, E) be an identical-edge homogeneous coupled cell network with a k-coloring ξ. For any ﬁnite subset F ⊂ E we deﬁne the vector q(F, ξ) = (q1 , . . . , qk )T whose ith coordinate is the number of edges in F with tail cell of color i: qi = #{e ∈ F : ξ(T (e)) = i}. In vector notation, this is q(F, ξ) =

(e)). ξ(T

e∈F

Now we express the concepts of balanced coloring and quotient network in terms of the vectors q(I(c), ξ), c ∈ C, for a coloring ξ of an identical cell homogeneous network G = (C, E). Definition 3.3. Let G = (C, E) be an identical-edge homogeneous coupled cell network. A k-coloring ξ of G is balanced if, for all cells c, d ∈ C, ξ(c) = ξ(d)

implies

q(I(c), ξ) = q(I(d), ξ).

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Thus, in a balanced k-coloring we identify each color i ∈ U with a particular set of colors for the tails T (I(c)), where c is any cell with color i. This induces the next deﬁnition. 3.2. Quotient networks. Definition 3.4. Let G = (C, E) be an identical-edge homogeneous coupled cell network with input sets I(c), for c ∈ C, and ξ a balanced k-coloring of G with colors in U = {1, . . . , k}. Let G0 = (C0 , E0 ) be the coupled cell network with k cells, C0 = {c1 , . . . , ck }, with input sets I0 (c), for c ∈ C0 , and a k-coloring ξ0 such that ξ0 (ci ) = i for i ∈ U. If, for all i ∈ U, q(I0 (ci ), ξ0 ) = q(I(c), ξ) for any c ∈ C such that ξ(c) = i, then G0 is the quotient network of G by ξ. The above deﬁnition implies, in particular, that the valences of G and G0 are the same. 3.3. Adjacency matrix. Definition 3.5. Let G = (C, E) be an identical-edge homogeneous network with C = {c1 , . . . , cm }. The adjacency matrix of G is the m × m matrix B = (bij ), where bij is the number of edges from cell cj to cell ci . The adjacency matrix characterizes G, and thus we write GB . The sum of all the entries in a line of the adjacency matrix B of GB is the valence v. We refer to the adjacency matrix B of a network GB = (C, E), assuming we have ﬁxed an ordering of the set C. Unless otherwise stated, given a k-coloring ξ of GB , with the colors indexed by U = {1, . . . , k}, we enumerate the cells of C by C = {c1 , . . . , cm } such that ξ(ci ) ≤ ξ(cj ) for i < j. Consider the k-coloring ξ0 of a network G0 with k cells c1 , . . . , ck and with adjacency matrix A = (aij ). The entry aij is the number of inputs from the j-color cell that the i-color cell receives. Therefore, the ith line of the adjacency matrix of G0 is the transpose of the vector q(I0 (ci ), ξ0 ). Let G = (C, E) be a coupled cell network with a balanced k-coloring ξ. If G0 is the quotient of G, then, for any cell c ∈ C with color i, the vector q(I(c), ξ) is the transpose of the ith line of A. Thus, any cell of color i receives aij inputs from cells with color j. Definition 3.6. A k-coloring matrix is the adjacency matrix A of a network G0 with k cells, one of each color. If G is a network having a balanced k-coloring with quotient network G0 , then we say that G has the k-coloring matrix A. 3.4. Matrix formulation of a balanced coloring. Lemma 3.7. Let ξ be a balanced k-coloring of an identical-edge homogeneous coupled cell network G = (C, E) having the k-coloring matrix A. Therefore, = (d)) for all c ∈ C. ξ(T (3.1) AT ξ(c) d∈I(c)

For the particular case of a lattice network GL , with nearest neighbors indexed by J and k-coloring matrix A, the above equation is = + g) for all l ∈ L. ξ(l (3.2) AT ξ(l) g∈J

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Proof. If ξ(c) = i, then the transpose of AT ξ(c) is the ith line of the matrix A, i.e., q(I(c), ξ). By deﬁnition this vector equals d∈I(c) ξ(T (d)). For (3.2), notice that (3.3)

{l + g : g ∈ J } = {T (d) : d ∈ I(l)}

for all l ∈ L. 3.5. Permuting the colors. Suppose we have a network G = (C, E) with a k-coloring matrix A and corresponding to a balanced k-coloring ξ, with colors indexed by U = {1, . . . , k}. Let σ be a permutation of the elements of U, and let σ · ξ be the coloring of G such that (σ · ξ)(c) = σ(ξ(c)) for c ∈ C. Thus, σ · ξ is also a balanced k-coloring and has the k-coloring matrix Mσ AMσT , where Mσ is the k×k permutation matrix corresponding to σ; i.e., Mσ ei = ej if j = σ(i), for i, j ∈ U. To prove the statement above, let A = (aij ), and let Aσ be the adjacency matrix of the coloring after permuting the colors by σ. The entry aij is the number of inputs from cells with color j that each cell of color i receives, for the initial balanced k-coloring. In the new coloring, σ · ξ, the cells with color σ(i) receive aij inputs from the cells with color σ(j); i.e., the matrix Aσ has the value aij on the position (σ(i), σ(j)). If A commutes with the permutation matrix Mσ , then Mσ AMσT = A, since Mσ−1 = MσT for permutation matrices. This permutation σ ∈ Sk does not change the k-coloring matrix A. Therefore, if XG,k is the space of balanced k-colorings of the network G, there is a natural action of the permutation group Sk : Sk × XG,k −→ XG,k , (σ, ξ) −→ σ · ξ, where we denote the identity permutation by . for c ∈ C. Equation (3.1) becomes, for Mσ ξ(c), Using vector notation, (σ · ξ)(c) = Mσ ξ(c) = ATσ Mσ ξ(c)

(d)) Mσ ξ(T

for all c ∈ C

d∈I(c)

and is another way of showing that Aσ = Mσ AMσT . Remark 3.8. We describe balanced k-colorings of a network up to these permutations, i.e., up to the Sk -orbits on XG,k . For colorings of a lattice network GL we can deﬁne other symmetries, inherited from the symmetries of the lattice L. This is done in the next subsections. 3.6. Spatial transformation of a coloring for lattice networks. Let GL be a lattice network. Since the cells are indexed by L, we deﬁne a k-coloring ξ of this lattice network as ξ : L −→ U, l −→ ξ(l), where ξ(l) is the color of the cell in the position l ∈ L, which we call cell l.

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The pattern associated with ξ is the set Ψξ = {(l, ξ(l)), l ∈ L}. ˙ O(n) be the symmetry group of the lattice L, where H is the ˙ H ⊂ Rn + Let Γ = L + holohedry of L. We denote the elements in Γ by γ = (t, δ), where t is a translation belonging to L and δ is an orthogonal matrix, and they transform x ∈ Rn into γx = t + δx. See [3] for some remarks relating the symmetries of the lattice and the symmetries of the equations governing the corresponding lattice dynamical systems. Notice that the set J ⊆ L is H-invariant. This follows from the fact that J is deﬁned by a distance and the holohedry is the group of orthogonal transformations such that H(L) = L. The usual action of Γ at the spaces of functions XG,k is the scalar action: (γ · ξ)(l) = ξ(γ −1 l) = (ξγ −1 )(l)

for all γ ∈ Γ and l ∈ L.

Suppose that we have a pattern Ψξ1 , Ψξ1 = {(l, ξ1 (l)), l ∈ L}, and we transform the coloring by a symmetry of the lattice γ ∈ Γ, obtaining ξ2 = γ · ξ1 . The new pattern is Ψξ2 = {(l, ξ2 (l)), l ∈ L} = {(l, ξ1 (γ −1 l)), l ∈ L}. This set equals {(γl, ξ1 (l)), γl ∈ γL} and, since γL = L, Ψξ2 = {(γl, ξ1 (l)), γl ∈ L}. Thus, Ψξ2 is the initial pattern after a Euclidean transformation. Remark 3.9. We describe patterns up to these transformations. 3.7. Periodicity. For the nearest neighbor coupling architecture, the structure of the lattice network GL is deﬁned by L, and so we use the notation XL,k for the space of the balanced k-colorings of a lattice network GL . The group acting on XL,k is, then, Θ = Γ × Sk , where, ˙ H with the following action: as before, Γ = L + Θ × XL,k −→ XL,k , ((γ, σ), ξ) −→ (γ, σ) · ξ, where ((γ, σ) · ξ)(l) = σ(ξ(γ −1 l)) for all l ∈ L. We say that a pattern Ψξ has a symmetry (γ, σ) ∈ Θ if the associated coloring is (γ, σ)invariant: (γ, σ) · ξ(l) = ξ(l) for all l ∈ L. In particular, we say that Ψξ has a period t ∈ L, t = 0, if ((t, Idn ), ) · ξ(l) = ξ(l − t) = ξ(l) for all l ∈ L.

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660

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Definition 3.10. Let ξ be a balanced k-coloring of an n-dimensional lattice network GL . The pattern Ψξ is periodic if it has periods along n noncolinear directions. If we denote the ˜ set of its periods by L, L˜ = {˜l1 , . . . , ˜ln }Z ⊆ L, ˜ for n noncolinear elements ˜l1 , . . . , ˜ln ∈ Rn , then we also say that Ψξ is L-periodic. ˜ ˜ ˜ ˜ Let Ψξ be an L-periodic pattern for L = {l1 , . . . , ln }Z , and consider the n-dimensional parallelepiped {˜l1 , . . . , ˜ln }[0,1) . The intersection of this parallelepiped with L is a set C, iso˜ The fundamental domain of the pattern Ψξ is its restriction morphic to the quotient L/L. to {(l, ξ(l)) : l ∈ C}. In fact the pattern is a regular repetition of the fundamental domain, along the n noncolinear directions ˜ l1 , . . . , ˜ln . Given a lattice L and a k-coloring matrix A, we prove in Theorem 5.4 a necessary and suﬃcient condition for the existence of a periodic pattern Ψξ associated with a balanced k-coloring ξ with matrix A. 4. Decomposition of finite bidirectional netwoks. Our main result (Theorem 5.4) relates the existence of spatially periodic balanced colorings of a lattice network, for a given coloring matrix A, to the decomposition of ﬁnite coupled cell networks with the same coloring matrix A and having all the couplings bidirectional. In this section we study networks with bidirectional couplings in two major steps—stating the proportion of cells for each color in a balanced coloring (for both ﬁnite bidirectional and lattice networks) and ensuring the factorization of the considered ﬁnite bidirectional networks. Definition 4.1. A bidirectional network is a coupled cell network where all the couplings are bidirectional. In a bidirectional network, given any two distinct cells, there is a direct path between them. Therefore, the adjacency matrix of a bidirectional network is an irreducible matrix (see section 4.4). Moreover, if ξ is a balanced k-coloring of a bidirectional network with the k-coloring matrix A, then A is also an irreducible matrix. Bidirectional networks have symmetric adjacency matrices, described in the next lemma. 4.1. Number of cells for bidirectional networks. Lemma 4.2. Let ξ be a k-coloring of a ﬁnite bidirectional identical-edge homogeneous network GB = (C, E) with colors indexed by the set U = {1, . . . , k}. Let A = (aij ), with i, j ∈ U, be the corresponding k-coloring matrix. Let pi be the proportion of cells with color i, for i ∈ U, and let #C = m. Then the adjacency matrix B of the network has the following block structure: ⎞ ⎛ B11 B12 · · · B1k ⎜ B21 B22 · · · B2k ⎟ ⎟ ⎜ (4.1) B=⎜ . .. .. ⎟ , .. ⎝ .. . . . ⎠ Bk1 Bk2 · · · Bkk where, for all i, j ∈ U,

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1. Bij is an mpi × mpj matrix, 2. the sum of any line of Bij is aij , T, 3. Bji = Bij 4. the elements in the diagonals of Bii are even numbers. Proof. To prove 1 and 2, note that for all i, j ∈ U the entry aij is the number of inputs from cells with color j that a cell of color i receives. Now we have mpi cells of color i and mpj cells of color j. The inputs that the ﬁrst ones receive from the second ones are the entries of an mpi × mpj matrix Bij , whose lines sum to aij because the coloring is balanced; see [1, Theorem 3.5]. In a bidirectional network, each edge e ∈ E with T (e) = c and H(e) = d, for c, d ∈ C, implies the existence of another edge f ∈ E in the opposite direction, T (f ) = d and H(f ) = c. T for all i, j ∈ U. In particular, self-couplings arise in pairs, Thus, B is symmetric and Bji = Bij and the k square matrices Bii have even numbers in their diagonals. Let A be a k × k matrix with ﬁxed valence v (that is, the sum of the entries of any line is v), and consider the possible identical-edge homogeneous bidirectional networks GB = (C, E), with ﬁnite C, having A as a k-coloring matrix. The proportion of cells for each one of the k colors that GB must have is stated in Lemma 4.3 below. Lemma 4.3. Let G = (C, E) be a ﬁnite bidirectional identical-edge homogeneous network with a k-coloring ξ, with colors in U = {1, . . . , k}, and corresponding to a k-coloring matrix A = (aij ), i, j ∈ U, with valence v. Suppose that #C = m and that pi is the proportion of cells with color i, for i ∈ U = {1, . . . , k}. Then the vector pT = (p1 , . . . , pk ) ∈ [0, 1]k is a left eigenvector of A generating the one-dimensional eigenspace that corresponds to the eigenvalue v. Moreover, the entries of A satisfy the condition (4.2)

pi aij = pj aji

for all i, j ∈ U.

Proof. By Lemma 4.2, the adjacency matrix of G has the block structure (4.1). Thus, the summation of all the entries of Bij is mpi aij for all i, j ∈ U. Since B is symmetric and T for all i, j ∈ U, the sum of all the entries of these two matrices must coincide, Bji = Bij implying condition (4.2). Summing the two sides, we obtain k i=1

mpi aij =

k

mpj aji

i=1

= mpj

k

aji

i=1

= mpj v, where the last equality follows from the deﬁnition of valence. Now the left-hand side summation is the jth component of the product mpT A. Therefore pT A = vpT . Since A is an irreducible matrix (see section 4.4), the valence v has multiplicity one; see, for example, [5, Lemma 5.1.1]. Therefore pT generates the eigenspace of A corresponding to the eigenvalue v, and it is the unique vector in this space whose coordinates sum to 1.

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Observe that Lemma 4.3 deﬁnes the proportion of cells for each color, in a bidirectional network G = (C, E) that admits a k-coloring corresponding to a ﬁxed k-coloring matrix. It follows, in particular, that there is a minimum number of cells #C. Moreover, condition (4.2) restricts the structure of the matrices that can be k-coloring matrices for identical-edge homogeneous bidirectional networks. The results stated in the previous lemma are also valid for lattice networks; see section 4.4, below. 4.2. Decomposition of B. A bidirectional identical-edge homogeneous network G = (C, E) is a graph whose vertex set is C and whose undirected edges are the bidirectional couplings. For a ﬁxed valence v, it is a v-regular graph, i.e., a graph such that all the vertices are incident with exactly v edges. A 2-factor of G is a vertex-disjoint union of 2-regular subgraphs of G that covers C. Since v is even, these graphs have a factorization into v/2 graphs that are 2-regular. This is a major tool in our work and is stated in Lemma 4.4, below. Lemma 4.4 (see [10]). Every v-regular graph, with v even, has a factorization into 2factors. Outline of the proof (see [4, pp. 72, 75, 229]). Let G be a 2k-regular graph with the set of vertices C = {v1 , . . . , vm }; without loss of generality, assume that G is connected. Let G1 be an Euler tour in G. Form a bipartite graph G with bipartition (X, Y ), where X = {x1 , . . . , xm } and Y = {y1 , . . . , ym }, by joining xi to yj whenever vi immediately precedes vj on G1 . Show that G is 1-factorable using Hall’s theorem, and hence that G is 2-factorable. Lemma 4.4 can be reformulated in the following way. A bidirectional identical-edge homogeneous network GB = (C, E), with even valence v, can be decomposed into v/2 bidirectional networks GB1 , . . . , GBv/2 , where, for V = {1, . . . , v/2}, 1. GBi = (C, Ei ), for i ∈ V, v/2 2. the disjoint union ∪i∈V Ei is E or, equivalently, i=1 Bi = B, 3. GBi is a bidirectional network with valence 2, for i ∈ V. Lemma 4.5. Let #C = m ∈ N, and let GB = (C, E) be a bidirectional identical-edge homogeneous network with adjacency matrix B and even valence v. Then, for V = {1, . . . , v/2}, there is a set of permutations Σ = {σi , σi−1 : i ∈ V} ⊂ Sm such that, for all c ∈ C, (4.3)

{T (e) : e ∈ I(c)} = {σ(c) : σ ∈ Σ}.

Equivalently, B can be written as a sum (4.4)

B=

v/2

i=1

Mσi + MσTi ,

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where Mσ is the m × m permutation matrix associated with σ ∈ Σ. Proof. Consider a factorization of G into 2-factors, GB1 , . . . , GBv/2 . For all i ∈ V, the network GBi = (C, Ei ) is a vertex-disjoint union of cycles that covers C. If we choose one direction for each cycle, then we deﬁne a digraph associated with a permutation σi of the cells in C. The inverse permutation, σi−1 , corresponds to the converse digraph, obtained by reversing the direction of the cycles. The union of these digraphs is GBi , and therefore the set of cells coupled to any cell c ∈ C can be described by Σ, as in (4.3). If e ∈ Ei , then either H(e) = σi (T (e)) or H(e) = σi−1 (T (e)) and, equivalently, Bi = Mσi + MσTi , since Mσ−1 = MσTi . Thus (4.4) follows from ∪i∈V Ei = E, which is equivalent to i v/2 B = B. i i=1 4.3. The set DA . Lemma 4.5 states that every ﬁnite bidirectional identical-edge homogeneous network with even valence can be associated with sets of permutations Σ. In fact, each Σ deﬁnes only one coupled cell network; see the proof of Lemma 4.5. However, if a bidirectional network has various decompositions into 2-factors, then it will have diﬀerent associated sets Σ. Thus, we can identify each of these permutations sets by C and by the partition of the couplings: Σ = Σ(C, E1 , . . . , Ev/2 ). Definition 4.6. Let A be a k × k matrix with ﬁxed valence v, with v even, and let V = {1, . . . , v/2}. The set DA is Σ = Σ(C, E1 , . . . , Ev/2 ) : G = (C, ∪i∈V Ei ) is a bidirectional . DA = network with coloring matrix A Remark 4.7. The elements in the set DA represent all the decompositions of all the ﬁnite bidirectional networks having k-coloring matrix A. If A is not irreducible, then there are no connected bidirectional networks with k-coloring matrix A, and thus the set DA is empty. Now consider condition (4.2), i.e., pi aij = pj aji for all i, j ∈ {1, . . . , k}, where (p1 , . . . , pk )T is any left eigenvector of A = (aij ) associated with v. If the entries of A do not satisfy this condition, then there are no bidirectional networks having k-coloring matrix A, and DA is also an empty set. 4.4. Proportion of colors for lattice networks. Now we present a result concerning the ratio of the cells for each color on a balanced coloring of a lattice network. Lemma 4.8 below extends Lemma 4.3 for lattice networks, a class of bidirectional networks with an inﬁnite number of cells. Consider a random walk through the colored cells of a lattice network GL where the possible steps are the bidirectional nearest neighbor couplings between cells. Suppose that GL has a balanced k-coloring with k-coloring matrix A and that, at each step, we register the color i of the cell, with i ∈ U = {1, . . . , k}. Thus, indexing the steps by time t = 0, 1, 2, . . . , we obtain a sequence of random variables {Xt }, where Xt ∈ U is the state-space for time t. Being in a cell of color i, the probability of going to a cell of color j in the next step is pij = aij /v, the number of inputs from cells of color j that a cell of color i receives, over the valence. This probability does not change with t and does not depend on the previous steps. Therefore this random walk is a Markov chain with transition matrix 1 A = (pij ) , v

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where the entry pij is the conditional probability (4.5)

pij =

P [a cell has color i and receives an input from a cell of color j] . P [a cell has color i]

A cell of any color intercommunicates with cells of all the colors in U, via the bidirectional couplings. Therefore, this is an irreducible Markov chain (see [8, Theorem II.1.1]). Let pT = (p1 , . . . , pk ) ∈ [0, 1]k be the left eigenvector of A, 1 pT A = pT ⇔ pT A = vpT , v

k normalized so that i=1 pi = 1, called the invariant probability vector. By Isaacson and Madsen [8, Theorems III.2.2. and III.2.4.] this vector exists and is the limit distribution of the colors, independent of the starting point of the random walk. In another interpretation, pi is the fraction of time that the process is expected to spend in cells of color i, for a large number of steps; see [9, section 4.2]. Thus, pi represents the proportion of the cells with color i ∈ U in the whole coloring: P [a cell has color i]. The probability P [a cell has color i and receives an input from a cell of color j] equals, by the above considerations, pi pij = pi aij /v for all i, j ∈ {1, . . . , k}. This probability is the proportion of couplings between the cells of color j and the cells of color i in the whole set of couplings of the lattice network. Since the couplings are bidirectional, it equals P [a cell has color j and receives an input from a cell of color i] = pj pji = pj aji /v. It follows that pi aij = pj aji for all i, j ∈ {1, . . . , k}. This proves the next lemma. Lemma 4.8. Let U = {1, . . . , k}, and let GL be a lattice network with nearest neighbor coupling architecture having a balanced k-coloring with k-coloring matrix A = (aij ). Let pi be the proportion of cells having color i, for i ∈ U, and v the valence. Then pT = (p1 , . . . , pk ) ∈ [0, 1]k is the unique vector whose components sum to 1 and that is a left eigenvector of A, corresponding to the eigenvalue v. Moreover, pi aij = pj aji for all i, j ∈ U. By Lemmas 4.3 and 4.8, the proportion pi of color i ∈ U is the same as in the balanced k-coloring of a lattice network or in any other ﬁnite network that shares the same k-coloring matrix A. 4.5. Periodic patterns. Suppose that Ψξ is a periodic pattern of a balanced k-coloring on a lattice network. Thus, the fundamental domain of the pattern must have a set of cells whose colors respect the proportion in p, and the number of cells of each color in the fundamental domain are the components of the vector mp for some m ∈ N. 5. Balanced colorings in lattices. In this section we deﬁne a function that projects lattice networks into ﬁnite bidirectional networks under some conditions. Using this projection function, we obtain our main result (Theorem 5.4): we show that any periodic balanced kcoloring of an n-dimensional lattice network with nearest neighbor coupling architecture can be obtained from a ﬁnite bidirectional network admitting the same balanced k-coloring rule. Let A be a k-coloring matrix with ﬁxed valence v, with v even. For V = {1, . . . , v/2}, let Σ = {σi , σi−1 : i ∈ V} ∈ DA . Let L = {l1 , . . . , ln }Z be a Euclidean lattice with nearest neighbors indexed by J = {li , −li : i ∈ V}, where v/2 ≥ n.

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Figure 17. A bidirectional network where each black cell receives four inputs from grey cells and each grey cell receives one input from grey cells and three inputs from black cells.

G2

G1

Figure 18. A decomposition of the bidirectional network in Figure 17. The 2-factors G1 and G2 correspond to the noncommuting permutations σ1 = (c1 c4 c3 c7 c6 c2 c5 ) and σ2 = (c1 c6 c3 c5 c4 c2 c7 ).

Suppose that 1. the permutations commute: σi ◦ σj = σj ◦ σi for all i, j ∈ V, where ◦ denotes the composition, and let φτ be a function φτ : J −→ Σ associated with a permutation τ of the elements in V, such that the following conditions are veriﬁed: 2. related inverse elements: φτ (li ) = στ (i) and φτ (−li ) = στ−1 (i) for all i ∈ V, v/2 3. consistency: for v/2 > n, if li = j=1 mj lj , with m1 , . . . , mv/2 ∈ Z, then στ (i) = mv/2 m1 στ (1) ◦ · · · ◦ στ (v/2) . Remark 5.1. Given Σ ∈ DA , there always exists φτ satisfying condition 2, and it is used only to deﬁne φτ . Condition 3 depends on the set Σ as well as on J , and condition 1 deﬁnes a subset of DA that can be the empty set, as we show in the next example. Consider the matrix A=

0 4 3 1

.

This matrix has left eigenvector 7pT = (3, 4) associated with v = 4. In Figure 17 we represent one bidirectional network G, with three cells of one color and four cells of another color, that has the two-coloring matrix A. Lemma 4.5 ensures the existence of Σ ∈ DA , that is, a decomposition of G. Consider, as an example, the factorization into 2-factors G1 and G2 , represented in Figure 18 and corresponding to Σ = {σ1 , σ1−1 , σ2 , σ2−2 } with the noncommuting permutations σ1 = (c1 c4 c3 c7 c6 c2 c5 ) and σ2 = (c1 c6 c3 c5 c4 c2 c7 ).

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ANA PAULA S. DIAS AND ELIANA MANUEL PINHO

Suppose that there is a factorization such that the permutations in Σ commute. By Theorem 5.4, below, this Σ ensures the existence of a periodic balanced coloring, with the two-coloring matrix A, of the planar square lattice network with nearest neighbor coupling architecture, since for the planar square lattice network we have v/2 = n, and thus 3 is not a restriction. However, Wang and Golubitsky [12, Theorem 1.7] prove that there is not such a coloring, and we conclude that the permutations in Σ do not commute. Definition 5.2. Let J and Σ be such that there exists φ ≡ φτ satisfying the conditions 1–3 above. Then we say that they are identiﬁable, and we use the notation φ(l) = σl

for l ∈ J ,

knowing that, for all l, g ∈ J , • σl ◦ σg = σg ◦ σl ; • σ−l = σl−1 ; • σl+g = σl ◦ σg if l + g ∈ J . Since the lattice is Euclidean, the set J generates the Abelian group (L, +) that we denote by J . Analogously, the set Σ = {σi , σi−1 : i ∈ V}, with commuting permutations (condition 1 above), generates an Abelian group that we denote by Σ. The function φ induces the homomorphism Φ : J −→ Σ such that Φ(l) = φ(l) for all l ∈ J . From the deﬁnition of Φ, it follows that Φ is an epimorphism; that is, Φ(L) = Σ. Therefore, saying that J and Σ are identiﬁable means that there exists a homomorphism between J ∼ = L and Σ. Theorem 5.3. Let L be an n-dimensional Euclidean lattice, A a k-coloring matrix with ﬁxed even valence v, and Σ = Σ(C, E1 , . . . , Ev/2 ) ∈ DA . If J and Σ are identiﬁable by φ, then there is a function Π : L −→ C such that (5.1)

Π(l + g) = σg (Π(l))

for all l ∈ L and for all g ∈ J ,

where σg = Φ(g) is the image of g ∈ J by Φ. ˜ the kernel of Φ, Moreover, Π has set of periods L, ˜ ker (Φ) = {l ∈ L : Φ(l) = } = L, with periods along n linearly independent directions. Proof. Let αL and αC be the actions of J on L and Σ on C, respectively, deﬁned by αL : J × L −→ L, (g, l) −→ g + l,

αC : Σ × C −→ C, (σ, c) −→ σ(c).

If J and Σ are identiﬁable by φ, then we can consider the induced homomorphism Φ : J −→ Σ and deﬁne a function Π : L −→ C such that the next diagram commutes:

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αL J × L −→ L Φ↓ ↓Π ↓Π . Σ × C −→ C αC Thus, for all l ∈ L and g ∈ J , Π(αL (g, l)) = αC (σg , Π(l)) ⇔ Π(l + g) = σg (Π(l)). ˜ then σg = and Π(l + g) = (Π(l)) = Π(l) for all l ∈ L. For each g ∈ J there is If g ∈ L, s ˜ Since the lattice is Euclidean, there an integer sg such that σgg = , and therefore sg g ∈ L. are elements of J along n linearly independent directions, and the result follows. It follows that the function Π, obtained in the above theorem, covers the lattice L with the cells of C such that the coloring ξ of C induces a coloring of L. Moreover, we show in the next theorem that the coloring is balanced for the lattice network GL . Theorem 5.4. Let A be a coloring matrix with even valence v, and let L be a Euclidean lattice with #J = v. The lattice network with nearest neighbor coupling architecture GL has a periodic balanced coloring with the coloring matrix A if and only if DA has an element identiﬁable with J . Proof. 1. From ﬁnite bidirectional networks to lattice networks. If J and Σ are identiﬁable, with Σ = Σ(C, E1 , . . . , Ev/2 ), then by Theorem 5.3 there is a function Π : L −→ C such that Π(l + g) = σg (Π(l)) for all l ∈ L and for all g ∈ J . Beginning with any cell l ∈ L, let Π(l) = c ∈ C. The remaining domain of Π is the orbit of l by group J , having images given by the orbit of c under Σ. Since the cells of C are colored by the balanced k-coloring ξ with k-coloring matrix A, this function Π induces the coloring ξL of GL : ξL (l) = ξ(Π(l))

for all l ∈ L.

Moreover, ξL is also a balanced k-coloring with k-coloring matrix A, as we show now. Recall the matrix formulation (3.1) and (3.2) of balanced colorings of networks and lattice networks. We have AT ξL (l) = =

AT ξ(Π(l)) (d)) ξ(T

since ξ is balanced

d∈I(Π(l))

=

g (Π(l))) ξ(σ

see (3.3)

ξ(Π(l + g))

by (5.1)

g∈J

=

g∈J

=

ξL (l + g)

by the deﬁnition of ξL .

g∈J

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668

ANA PAULA S. DIAS AND ELIANA MANUEL PINHO

Moreover, ξL inherits the periods of Π: if ˜l is a period of Π, then, for all l ∈ L, ξL (l + ˜l) = ξ(Π(l + ˜l)) = ξ(Π(l)) = ξL (l). ˜ with periods along n linearly Therefore, by Theorem 5.3, the coloring ξL has set of periods L, independent directions. 2. From lattice networks to ﬁnite bidirectional networks. Let GL have a periodic balanced k-coloring ξL , with k-coloring matrix A and with set of periods L˜ = {˜l1 , . . . , ˜ln }Z ⊂ L, an n-dimensional lattice, and consider the n-dimensional parallelepiped {˜l1 , . . . , ˜ln }[0,1) . The ˜ intersection of this parallelepiped with L is a set C, isomorphic to the quotient L/L. Let Π : L −→ C be the composition of the isomorphism L/L˜ ∼ = C with the projection ˜ Notice that Π has the set of periods L. ˜ Now consider the permutations of the L −→ L/L. elements in C deﬁned by σg (Π(l)) = Π(l + g)

(5.2)

for g ∈ J .

By deﬁnition, σl+g = σl ◦ σg for all l, g ∈ J . Therefore, we have σ−l = σl−1 and σl ◦ σg = σg ◦ σl for all l, g ∈ J . It follows that J and Σ = {σg : g ∈ J } are identiﬁable. Let G = (C, E) be the ﬁnite identical-edge homogeneous network such that, for all c ∈ C, {T (e) : e ∈ I(c)} = {σg (c) : g ∈ J }.

(5.3)

Since g ∈ J implies −g ∈ J , the network G has all the couplings bidirectional. Consider the coloring ξ of C deﬁned by ξ(Π(l)) = ξL (l)

for all l ∈ L.

We have AT ξ(Π(l))

= AT ξL (l) ξL (l + g) =

since ξL is balanced

g∈J

=

ξ(Π(l + g))

by the deﬁnition of ξL

g (Π(l))) ξ(σ

by (5.2)

g∈J

=

g∈J

=

(d)) ξ(T

see (5.3).

d∈I(Π(l))

Thus ξ is a balanced k-coloring of G = (C, E), with the k-coloring matrix A. Therefore Σ ∈ DA , and the proof is complete. The proof of this theorem is illustrated by the examples in the ﬁrst section. Theorem 5.4 relates the local structure of the colorings, given by the coloring matrix A, to their long-range behavior in lattice networks.

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6. Some consequences of Theorem 5.4. In this section we present some corollaries of Theorem 5.4, as well as some examples, relating balanced colorings of lattices with diﬀerent dimensions. 6.1. Corollaries. In what follows we denote by Zm the m-dimensional lattice generated by the m vectors in the canonical basis of Rm . Note that Zm is a Euclidean lattice. For this lattice network, with nearest neighbor coupling architecture, we have v/2 = m, and one element Σ = {σi , σi−1 : i ∈ V} ∈ DA with V = {1, . . . , v/2} is identiﬁable with J if and only if the permutations commute. Corollary 6.1. Let A be a coloring matrix with even valence v, and let L be an n-dimensional Euclidean lattice, with v/2 > n. If the lattice network with nearest neighbor coupling architecture GL has a periodic balanced coloring with the coloring matrix A, then there is a periodic balanced coloring of a v/2-dimensional Euclidean lattice Zv/2 having the same coloring matrix A. Proof. Under the hypotheses of the corollary and by Theorem 5.4, there is an element Σ ∈ DA identiﬁable with J , the set that indexes the nearest neighbors in L. In particular, the permutations in Σ commute, and so Σ is also identiﬁable with the set that indexes the v nearest neighbors of Zv/2 . Thus, the result follows by Theorem 5.4. Corollary 6.2. Let A be a k-coloring matrix with even valence v, and let L = Zn . Let r ∈ Z+ . If the lattice network with nearest neighbor coupling architecture GL has a periodic balanced coloring with the coloring matrix A, then there is a periodic balanced coloring of the (n + r)-dimensional Euclidean lattice Zn+r having the k-coloring matrix A + 2rIdk . Proof. Under the hypotheses of the corollary and by Theorem 5.4, there is an element Σ ∈ DA identiﬁable with J , the set that indexes the nearest neighbors in L. The set Σ ∪ Σ , where Σ = {, . . . , } with #Σ = 2r, is identiﬁable with the set that indexes the nearest neighbors of Zr+n and, thus, corresponds to a balanced coloring with coloring matrix A . Let A = (aij ) and A = (aij ). In Zn+r , each cell with color i is connected to aij cells of color j if i = j, according to Σ. By Σ , beyond these connections, each cell is coupled 2r times to cells with the same color. That is, aij = aij if i = j, and aij = aij + 2r if i = j. Remark 6.3. By Corollary 6.2 it follows, in particular, that for a given k, the number of balanced colorings of Zn+1 with nearest neighbor coupling architecture is larger than or equal to the number of balanced colorings of Zn with nearest neighbor coupling architecture. We think that a similar result may hold for n-dimensional lattices, with v/2 = n; i.e., given two Euclidean lattices L and L1 with dimensions, respectively, n and n + 1 and such that L = {l1 , . . . , ln }Z and L1 = {l1 , . . . , ln , ln+1 }Z , we make the following conjecture. For a given k, the number of balanced colorings of L1 , with nearest neighbor coupling architecture, is larger than or equal to the number of balanced colorings of L with nearest neighbor coupling architecture. 6.2. Examples. Consider the standard cubic lattice L = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}Z with nearest neighbor coupling architecture J = {±l1 , ±l2 , ±l3 }, where l1 = (1, 0, 0), l2 = (0, 1, 0), and l3 = (0, 0, 1).

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670

ANA PAULA S. DIAS AND ELIANA MANUEL PINHO

Next we present three examples of balanced colorings of L with two colors, black and grey, indexed by the elements in U = {1, 2}. 6.2.1. Case 43. Let A be the matrix

A=

2 4 3 3

.

By Wang and Golubitsky [12], there are balanced colorings of the planar hexagonal lattice with coloring matrix A. Thus, by Corollary 6.1, the existence of a pattern with this coloring matrix in the standard cubic lattice is ensured. For the two-coloring matrix A, the proportion of cells with each color is given by the vector (3/7, 4/7)T . Consider the seven-cell bidirectional network with adjacency matrix ⎞ ⎛ 0 1 1 1 1 1 1 ⎜ 1 0 1 1 1 1 1 ⎟ ⎟ ⎜ ⎜ 1 1 0 1 1 1 1 ⎟ ⎟ ⎜ ⎟ B=⎜ ⎜ 1 1 1 0 1 1 1 ⎟ ⎜ 1 1 1 1 0 1 1 ⎟ ⎟ ⎜ ⎝ 1 1 1 1 1 0 1 ⎠ 1 1 1 1 1 1 0 admitting the ⎛ 0 1 ⎜ 1 0 ⎜ ⎜ 0 1 ⎜ B=⎜ ⎜ 0 0 ⎜ 0 0 ⎜ ⎝ 0 0 1 0

two-coloring rule deﬁned by A. Note that ⎞ ⎛ 0 0 1 0 0 1 0 0 0 0 1 ⎜ 0 0 0 1 0 0 1 0 0 0 0 ⎟ ⎟ ⎜ ⎜ 0 1 0 0 0 ⎟ ⎟ ⎜ 1 0 0 0 1 0 ⎟ 1 0 1 0 0 ⎟+⎜ ⎜ 0 1 0 0 0 1 ⎜ 0 1 0 1 0 ⎟ ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 1 ⎠ ⎝ 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0

we have the decomposition ⎞ ⎛ 0 0 0 1 1 0 0 ⎜ 0 0 0 0 1 1 1 ⎟ ⎟ ⎜ ⎜ 0 ⎟ ⎟ ⎜ 0 0 0 0 0 1 ⎟ 0 ⎟+⎜ ⎜ 1 0 0 0 0 0 ⎜ 1 ⎟ ⎟ ⎜ 1 1 0 0 0 0 0 ⎠ ⎝ 0 1 1 0 0 0 0 0 0 1 1 0 0

0 0 1 1 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

that corresponds to the commuting permutations σ1 = (b1 b2 b3 g1 g2 g3 g4 ), σ2 = (b1 b3 g2 g4 b2 g1 g3 ), and σ3 = (b1 g1 g4 b3 g3 b2 g2 ) deﬁning the balanced coloring in Figure 19. Note that the restriction of this pattern to the planes with constant third coordinate is a two-dimensional pattern (see Figure 20) whose proportion of cells with each color is given by the vector (3/7, 4/7)T . However this is not a balanced coloring. The only coloring matrix with valence 4 that has this left eigenvector corresponding to the eigenvalue 4 is 0 4 . 3 1 By Wang and Golubitsky [12], there is not a balanced pattern in the square lattice with this coloring matrix, and thus this restriction could not be balanced.

6.2.2. Case 41. Let A=

2 4 1 5

.

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(0,1,0)

(0,0,1) (1,0,0) (0,0,0)

Figure 19. A periodic pattern for the standard cubic lattice with nearest neighbor coupling architecture obtained for the case 43.

Figure 20. Restriction to {(x, y, 0) : x, y ∈ R} of the periodic pattern in Figure 19.

The proportion of cells with each color is given by the vector (1/5, 4/5)T . ﬁve-cell bidirectional network with adjacency matrix ⎞ ⎛ ⎞ ⎛ ⎛ 0 1 0 0 1 0 0 1 1 2 1 1 1 1 ⎜ 1 2 1 1 1 ⎟ ⎜ 1 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ ⎟ ⎜ ⎜ ⎟=⎜ 0 1 0 1 0 ⎟+⎜ 1 0 0 0 B=⎜ 1 2 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎜ ⎝ 1 1 1 2 1 ⎠ ⎝ 0 0 1 0 1 ⎠ ⎝ 1 1 0 0 1 1 1 1 2 1 0 0 1 0 0 1 1 0

So we can take the 0 1 1 0 0

⎞ ⎟ ⎟ ⎟ + 2Id5 . ⎟ ⎠

This decomposition corresponds to the commuting permutations σ1 = (b1 g1 g2 g3 g4 ), σ2 = (b1 g2 g4 g1 g3 ), and σ3 = that generate the pattern in Figure 21. Remark 6.4. This coloring matrix does not correspond to a balanced coloring of the planar hexagonal lattice, by the enumeration in [12]. Although the permutations commute, they do not verify the condition σ3 = σ2 ◦ σ1−1 , which is necessary for the identiﬁcation of the permutations and the directions of the nearest neighbors in the planar hexagonal lattice

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ANA PAULA S. DIAS AND ELIANA MANUEL PINHO

(0,1,0)

(0,0,1) (1,0,0) (0,0,0)

Figure 21. A periodic pattern for the standard cubic lattice with nearest neighbor coupling architecture obtained for the case 41.

network. Note that, given A and for nearest neighbor coupling architecture, Corollary 6.1 states that the number of periodic patterns with the k-coloring matrix A in Zv/2 is greater than or equal to the number of periodic patterns with the same coloring matrix in other ndimensional Euclidean lattices with valence v, if v/2 > n. In the example above these numbers are not equal. Since 0 4 A − 2Id2 = 1 3 and this coloring matrix corresponds to a pattern in the planar square lattice (see [12]), the three-dimensional pattern in L was ensured by Corollary 6.2.

6.2.3. Case 33. Let A=

3 3 3 3

.

The proportion of cells with each color is given by the vector (1/2, 1/2)T . We deﬁne a four-cell bidirectional network with adjacency matrix ⎞ ⎛ ⎞ ⎛ 0 1 0 1 0 3 0 3 ⎜ ⎟ ⎜ 3 0 3 0 ⎟ ⎟ = 3⎜ 1 0 1 0 ⎟. B=⎜ ⎝ 0 1 0 1 ⎠ ⎝ 0 3 0 3 ⎠ 3 0 3 0 1 0 1 0 Therefore, we associate each generator of the lattice with the permutation (b1 b2 g1 g2 ), and we obtain the pattern in Figure 22. Remark 6.5. By Wang and Golubitsky [12], there is one balanced coloring of the planar hexagonal lattice with coloring matrix

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673

(0,1,0)

(0,0,1) (1,0,0) (0,0,0)

Figure 22. A periodic pattern for the standard cubic lattice with nearest neighbor coupling architecture obtained for case 33.

A=

3 3 3 3

.

Thus, by Corollary 6.1, the existence of a periodic pattern with this coloring matrix in L = Z3 is ensured. However, in the standard cubic lattice, A corresponds to an inﬁnite family of patterns, as we describe below. The restriction of the pattern in Figure 22 to the planes with constant third coordinate corresponds to case 22 in the planar square lattice, where an inﬁnite family of patterns can be obtained by the diagonal method, interchanging colors along the diagonal where black and grey cells alternate (see [12, section 3.2]). In Figure 23 we show one of these diagonals whose cells have alternating colors. In fact, all the diagonals that are parallel to the one represented are alternating diagonals. Thus, for a given alternating diagonal dr = {(r + x, x, 0) : x ∈ R}, with r ∈ Z, we can deﬁne an inﬁnite set of alternating diagonals Fr = {dr + s(0, 0, 1) : s ∈ Z} (Figure 24) in the plane {(r + x, x, y) : x, y ∈ R}. The lines as = {(r + 2s, 2s, y) : y ∈ R} ∩ L are equally colored for all s ∈ Z (see lines “a” in Figure 25), and the lines bs = {(r + 2s + 1, 2s + 1, y) : y ∈ R} ∩ L are equal for all s ∈ Z (lines “b” in Figure 25). Interchanging lines a and b, we obtain a new balanced coloring with the same coloring matrix A. This happens because along the third coordinate everything remains, and in the planes with constant third coordinate we interchange colors along alternating diagonals. Therefore, this is a three-dimensional version of the diagonal method described in [6]. Acknowledgments. We thank Martin Golubitsky and Yunjiao Wang for fruitful discussions, and Matthias Kriesell and Maria Leonor Moreira for information and suggestions for appropriate literature about graph theory. We also thank the referees for comments that much improved the paper.

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ANA PAULA S. DIAS AND ELIANA MANUEL PINHO

Figure 23. Restriction to {(x, y, 0) : x, y ∈ R} of the periodic pattern in Figure 22. This plane has an alternating diagonal.

(0,1,0)

(0,0,1) (1,0,0) (0,0,0)

Figure 24. The alternating diagonals of the parallel planes with constant third coordinate in the periodic pattern in Figure 22.

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b

a

b

a

b

a

(0,1,0)

(0,0,1)

diagonal

(1,0,0) (0,0,0)

Figure 25. One possible generalization of the diagonal method for three-dimensional patterns. Interchanging the lines a and b along the “diagonal” corresponds to the diagonal method applied in every plane with constant third coordinate. REFERENCES [1] M. A. D. Aguiar, A. P. S. Dias, M. Golubitsky, and M. da C. A. Leite, Bifurcations from regular quotient networks: A ﬁrst insight, Phys. D, 238 (2009), pp. 137–155. [2] F. Antoneli, A. P. S. Dias, M. Golubitsky, and Y. Wang, Patterns of synchrony in lattice dynamical systems, Nonlinearity, 18 (2005), pp. 2193–2209. [3] F. Antoneli, A. P. S. Dias, M. Golubitsky, and Y. Wang, Synchrony in lattice diﬀerential equations, in Some Topics in Industrial and Applied Mathematics, Contemp. Appl. Math. Ser. 8, R. Jeltsch, T. Li, and I. Sloane, eds., World Scientiﬁc Publishing, River Edge, NJ, 2007, pp. 43–56. [4] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing, New York, 1976. [5] R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Encyclopedia Math. Appl. 39, Cambridge University Press, Cambridge, UK, 1992. [6] M. Golubitsky, M. Nicol, and I. Stewart, Some curious phenomena in coupled cell systems, J. Nonlinear Sci., 14 (2004), pp. 119–236. ¨ ro ¨ k, Patterns of synchrony in coupled cell networks with [7] M. Golubitsky, I. Stewart, and A. To multiple arrows, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 78–100. [8] D. L. Isaacson and R. W. Madsen, Markov Chains, Theory and Applications, Wiley, New York, 1976. [9] J. G. Kemeny and J. L. Snell, Finite Markov Chains, Undergrad. Texts Math., Springer-Verlag, New York, 1976. [10] J. Petersen, Die Theorie der regul¨ aren Graphen, Acta Math., 15 (1891), pp. 193–220. [11] I. Stewart, M. Golubitsky, and M. Pivato, Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 609–646. [12] Y. Wang and M. Golubitsky, Two-colour patterns of synchrony in lattice dynamical systems, Nonlinearity, 18 (2005), pp. 631–657.

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