11 and 21 phase entrainment in a system of two coupled limit cycle oscillators

J. Math. Biology (1984) 20:133-152

Journal of

Mathematical Biology 9 Springer-Verlag 1984

1:1 and 2:1 phase entrainment in a system of two coupled limit cycle oscillators W. L. Keith and R. H. Rand Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA

Abstract. A model of a pair of coupled limit cycle oscillators is presented in order to investigate the extent of, and the transition between, 1 : 1 and 2:1 phase entrainment, a phenomenon exhibited by numerous diverse biological systems. The mathematical form of the model involves a flow on a phase torus given by two coupled first order nonlinear ordinary differential equations which govern the oscillators' phase angles (i.e. their respective positions around their limit cycles). The regions corresponding to 1 : 1 and 2 : 1 phase entrainment in an appropriate parameter space are determined analytically and numerically. The bifurcations occurring during the transition from 1 : 1 to 2:1 phase entrainment are discussed. Key words: Oscillators - - limit cycles - - phase entrainment - - bifurcations

Introduction Recent research on problems in bio-oscillation theory have utilized simplified models of limit cycle oscillators (Cohen et al. [1], Guevara and Glass [7], Ermentrout and Kopell [2], Hoppensteadt and Keener [8], Rand et al. [12]). In this paper we extend previous work by proposing a model of two limit cycle oscillators which takes into account both 1 : 1 and 2:1 phase entrainment. We briefly cite the following examples of biological systems which have exhibited 2:1 phase entrainment: In the vertebrate heart disorder known as cardiac arrhythmias, every other pulse generated by the sinuatrial node pacemaker is ineffective in driving the ventricular rhythm. The process by which a normal heart becomes diseased may be viewed as a transition from 1 : 1 to 2 : 1 phase entrainment (Guevara and Glass [7]). In experiments on the neurobiology of swimming in the fish Icthyomyzon unicuspis (the lamprey), Cohen et al. [1] reported a transition between 1 : 1 and 2:1 phase entrainment as follows: An isolated spinal cord undergoing phase locked "fictive swimming" was lesioned, resulting in the observation that the two

134

W.L. Keith and R. H. Rand

portions of the cord on either side of the lesion became 2:1 phase entrained (whereas before the lesion the two portions were 1 : 1 phase entrained). In this p a p e r we shall be interested in certain aspects of the general problem of 1 : 1 versus 2:1 phase entrainment in a system of two oscillators. In particular we wish to address the question of the extent of 1 : 1 versus 2 : 1 entrainment as a function of the uncoupled frequencies and the coupling between the oscillators. In addition we shall discuss the nature of the transition between these two entrained states, and will show that in our model a complicated sequence of bifurcations accompanies this transition. Model

We follow Cohen et al. [1] and postulate a general biological oscillator which exhibits a unique stable limit cycle oscillation. Since we wish to make the model as general as possible, we make no assumptions about the containing space, except that the number of dimensions (and hence the number of physical variables needed to describe the system) be greater than one. In order to describe the oscillation mathematically, we parameterize the limit cycle by its phase 0 chosen to run from 0 to 2 ~ radians over the course of a complete cycle. Moreover we assume that the phase O(t) flows uniformly in time t: 0=~o

implies O ( t ) = t o t + O ( O )

(mod2~-).

(1)

Here m o d 2~r means that we do not distinguish between values of 0 which differ by multiples of 2~- (since the limit cycle is topologically a circle). In order to model the interaction between two such oscillators, we assume that the limit cycle in an individual oscillator continues to persist when perturbed by feedback from another oscillator (i.e. we assume that the limit cycle is structurally stable). If 01 and 02 represent the respective phases of the two oscillators, we write: 01 -=0) 1 +h1(Ol, 02) (2) 02 = w2 +h2(01, 02)

(3)

where hi(01, 02) represents the influence of oscillator 2 on oscillator l, and similarly for h2(Ol, 02). Since each 0~ is defined m o d 2~r, we further assume that the functions h~(01, 02) are 21r periodic in their arguments. We may envision the resulting flow as taking place on a two dimensional torus which may be represented as a square with opposite sides identified (see Fig. 1). In Cohen et al. the functions hi were chosen thus: 01 = ~ol - a sin(01 - 02)

(4)

02 = o)2+a s i n ( 0 1 - 02)

(5)

where a is a coupling coefficient. By subtracting these equations it is easy to show that if Itol-WEI < 12al this system exhibits two l : l phase locked motions, one stable and one unstable: 9 [wl-w2~ Ol(t) - 02(t) = a r c s l n ~ , ] .

(6)

135

1 : 1 and 2 : 1 phase entrainment in two coupled oscillators

(o) Oz

Oi

(b) O, Fig. 1. The 0i, 02 phase torus (c) can be represented by a square with opposite sides identified, as in (a). Fig. l(b) shows the intermediate step in which the left and right sides of (a) have been joined, giving a cylinder. Fig. l(c) results from joining the top and bottom ends of (b). The curve L represents a possible limit cycle

2rr

2~~ '

/c)

These motions represent limit cycles on the phase torus, see Fig. 2. N o t e that when plotted on a square with identified sides as in Fig. 2, these phase locked motions consist o f straight line segments with slope unity. N o t e also that these limit cycles on the phase torus in the c o u p l e d system are distinct from and should not be c o n f u s e d with, the limit cycles in the space of physical variables in the d y n a m i c s o f the original oscillators which were parameterized by 01 and 02. In the case that [Wl-w21> 12=1, no such 1:1 phase locked limit cycles exist, and the m o t i o n is said to "drift". In order to m o d e l 2 : 1 phase locking, we m o d i f y Eqs. (4), (5) as follows: 0~ = W l - / 3 s i n ( 0 1 - 2 0 2 )

(7)

02 = w2+/3 sin(01-202).

(8)

Here /3 is a coupling coefficient. In the case that ]wl-2w21 <]2/31, this system exhibits two 2:1 phase locked motions, one stable and one unstable: 01 (t) - 202(t) = a r c s i n ( t~ - 2oJ2]. 2/3 ] \

(9)

2rr

01 Fig. 2. A 1 : 1 phase

locked motion. For clarity, only one of the two solutions in Eq. (6) has been shown

0

0z

2rr

136

W.L. Keith and R. H. Rand

2-n-

Oi

0

02

Fig. 3. A 2 : l phase locked motion. For clarity, only one of the two solutions in Eq. (9) has been shown

See Fig. 3. In the case that Ito1-2~o21> 12/31, Eqs. (7), (8) do not admit 2:1 phase locked limit cycle solutions, but rather exhibit drift. In Appendix E both sets of Eqs. (4), (5) and (7), (8) are derived from a perturbation method applied to two coupled van der Pol oscillators. In order to obtain a model which exhibits both I : 1 and 2:1 phase locked motions, we combine the systems previously discussed in the following form: 01 = (.01 --OL

sin(01- 0 2 ) - f l s i n ( 0 , - 2 0 2 )

02 = o92 + a sin(01 - 02) +/3 sin(01-202).

(10) (11)

For the special cases/3 = 0 and a = 0 , Eqs. (10), (11) reduce to Eqs. (4), (5) and (7), (8) respectively. Although these limiting cases can be derived from coupled van der Pol oscillators (see Appendix E), we propose the present model of coupled oscillators as independent of any derivation based on van der Pol oscillators, but rather as being of interest in its own right without a more elementary derivation. In a similar spirit, Hoppensteadt and Keener [8] presented a study of phase locking !n radial isochron clocks based on the system: 01=l 02 = ~o + e

~ tn, n~

am,, sin(m01 - nO2) co

(with am., = a . . . . V m ~ n). Note the similarity in the coupling term between their model and our posited Eqs. (10), (11). In order to reduce the number of parameters involved in this study, we set /3 = 1 - a in what follows: 01 = w l - a sin(01 - 02) - ( 1 - a ) sin(01-202)

(12)

02 = ~o2+a sin(01- 02) +(1 - a ) sin(0~- 202).

(13)

We shall be interested in the dynamics of Eqs. (12), (13) as a is tuned from 0 to 1 for various values of Wl and w2. In particular we wish to investigate for which values of a the system behaves in a 1 : 1 or 2:1 manner. Before proceeding with the analysis of this model, we wish to distinguish between the kind of 1 : 1 (or 2: 1) behavior exhibited by Eqs. (12), (13) in the case that a = 1 (or cr = 0), called phase locking, and a more general kind of

1: 1 and 2: l phase entrainment in two coupled oscillators

137

277-

Ol

/ Fig. 4. A l : 1 phase entrained motion

0

27r

01

Fig. 5. A 2:1 phase entrained motion

0

02

:// 0a

277"

interaction called phase entrainment. While a 1 : 1 phase locked m o t i o n involves a linear relationship between 01 and 02 (Eq. (6)), and plots as a straight line (Fig. 2), a 1 : 1 phase entrained m o t i o n will be defined to be a periodic m o t i o n for which 01 advances by 2~- whenever 02 advances by 2zr. That is, a m o t i o n will be said to be 1 : 1 phase entrained if

011o2=0.-011o2-0.+2= 27r for all 0". A 1 : 1 phase entrained m o t i o n is s h o w n in Fig. 4. Note that 1 : 1 phase locked motions are special cases of 1 : 1 phase entrained motions in which a linear relationship exists between 01 and 02 for all time. Similarly we define a 2 : 1 phase entrained motion to be one for which

011o~_o*-011o2-o*+2~=4~ for all 0". That is, 01 advances by 4~- w h e n e v e r 02 advances 2~r. See Fig. 5. The importance o f phase entrained motions lies in the fact that Eqs. (12), (13) admit 1 : 1 and 2 : 1 phase entrained motions for a wide range o f parameter values, while these equations only exhibit 1 : 1 or 2:1 phase locked motions for a = 1 or 0, respectively.

Analysis of the model In the previous section we showed that the two oscillator model given by Eqs. (12), (13) exhibits 1 : 1 and 2:1 phase locking when a = 1 and 0 respectively. In A p p e n d i x A we show that this model also exhibits a 3 : 2 phase locked motion

138

w . L . Keith and R. H. R a n d 091

211 2

a

092 Fig. 6. Phase locked regions in 0)1, 0)2, o~ parameter space. The planes c~ = 0, 1 contain regions of 2:1 and 1 : 1 locking, respectively, s h o w n shaded. The plane a = 89contains the indicated line on which 3 : 2 locking occurs

8I

a--O.O~/ e2

a=.5/~ 91 a=.594~ ~

/a =.590

/~ ~~a=.400 a=.593

Fig. 7. Flow on the 01, 02 phase torus during transition from 2:1 to 1 : 1 phase entrained motions. As c~ increases from 0.39 to 0.4, a complicated sequence of bifurcations occurs which is conjectured to involve p: q phase entrained motions for all rational p/q between 1 and 2. Here, we display the 2: 1, 3 : 2, 4 : 3, 5 : 4, and 1 : 1 cases. Note that w h e n t~ = 0 we have a 2: 1 phase locked motion

1 : 1 and 2:1 phase entrainment in two coupled oscillators

139

to I

57 -2

I 0

~

,

I:1

la

Z 2

Fig. 8. 1 : 1 and 2:1 phase entrainment regions in Wl, c~ space with w2 = 1. The dashed line represents the boundary of the 1 : 1 region and intersects the line a = 1 at w1= 3 (see Appendix A)

w h e n a = 89 a n d t h a t no o t h e r p h a s e l o c k e d m o t i o n s are possible. See Fig. 6 w h e r e the r e g i o n s o f p h a s e l o c k i n g are d i s p l a y e d in Wl, w2, a p a r a m e t e r space. W e p a s s n o w f r o m c o n s i d e r a t i o n o f p h a s e l o c k e d m o t i o n s to p h a s e e n t r a i n e d m o t i o n s . To r e d u c e the n u m b e r o f p a r a m e t e r s , we set w2 = 1 in all t h a t follows. W e b e g i n b y p r e s e n t i n g the results f r o m t y p i c a l n u m e r i c a l i n t e g r a t i o n s o f Eqs. (12), (13). W e set w~ = oJ2 = 1 a n d vary c~ f r o m 0 to i. F o r 0 < a ~ 0.39, the system d i s p l a y s 2 : 1 p h a s e e n t r a i n m e n t , while for 0.40 ~ ~ < 1, 1 : i p h a s e e n t r a i n m e n t is o b s e r v e d . A c o m p l i c a t e d s e q u e n c e o f b i f u r c a t i o n s occurs in the t r a n s i t i o n region, 0.39 ~ a ~ 0.40, see Fig. 7. W e will r e t u r n to a d i s c u s s i o n o f these b i f u r c a t i o n s later. Since the a t t r a c t i o n onto a stable limit cycle b e c o m e s very w e a k as a a p p r o a c h e s a critical ( b i f u r c a t i o n ) value, the n u m e r i c a l i n t e g r a t i o n s m u s t be c a r r i e d o u t over very long time intervals. In o r d e r to a v o i d this difficulty, we d e v e l o p e d an a n a l y t i c a l - n u m e r i c a l t e c h n i q u e w h i c h m o r e efficiently yields the critical v a l u e o f c~ at w h i c h a p: q p h a s e e n t r a i n e d s o l u t i o n ceases to exist. T h e details o f this m e t h o d are given in A p p e n d i x B. The resulting 1:1 a n d 2 : 1 e n t r a i n m e n t r e g i o n s in the w~, a p l a n e are s h o w n in Fig. 8. W h e n a = 0 a n d 1, the t r a n s i t i o n curves for 2 : 1 a n d 1 : 1 e n t r a i n m e n t r e s p e c t i v e l y yield the critical v a l u e o f o9~ o b t a i n e d f r o m the p r e v i o u s p h a s e l o c k e d analysis.

2"0-

|

"

S t j

01

ir .."

I

Fig. 9. A 2 : 1 drift flow (3 distinct trajectories are shown)

0

02

Jt

2v"

140

w.L. Keith and R. H. Rand

In p a r t i c u l a r , for a = 1, the 2 : 1 e n t r a i n m e n t t r a n s i t i o n curves intersect at Wl =7, f o r m i n g a w e d g e s h a p e d region (Fig. 8). The p o i n t ~ol =7, a = 1 m a y b e a n a l y t i c a l l y s h o w n to c o r r e s p o n d to a 2 : 1 drift s o l u t i o n ( K e i t h [9]). (By a 2 : 1 drift s o l u t i o n we m e a n a situation in w h i c h the p h a s e torus is filled with 2 : 1 p h a s e e n t r a i n e d m o t i o n s , n o n e o f w h i c h are attracting, see Fig. 9.) W e s u p p l e m e n t e d this n u m e r i c a l w o r k in the n e i g h b o r h o o d o f this w e d g e s h a p e d r e g i o n with a p e r t u r b a t i o n m e t h o d . At the p o i n t Wl =~,7 o~ = 1, an exact s o l u t i o n o f the 2 : 1 drifting m o t i o n is k n o w n ( C o h e n et al. [1]). W e set e = 1 - a a n d ~o~ = 7 + he + O(62), a n d p e r t u r b e d off the e x a c t s o l u t i o n for small e. W e offer a b r i e f s u m m a r y o f the m e t h o d in A p p e n d i x C, a n d here note j u s t that the resulting values o f s l o p e A turn out to be 0 , - 3 . The resulting w e d g e s h a p e d region agrees closely with the p r e v i o u s n u m e r i c a l results, Fig. t0. N e x t we offer a p r o o f o f the existence o f 1 : 1 a n d 2 : 1 p h a s e e n t r a i n e d m o t i o n s b a s e d o n the P o i n c a r e - B e n d i x s o n t h e o r e m . T h e exact n a t u r e o f this p r o o f c o m p l e ments the a p p r o x i m a t e n a t u r e o f the p r e v i o u s n u m e r i c a l a n d p e r t u r b a t i o n results. A p p e n d i x D c o n t a i n s a s u m m a r y o f this work, w h i c h results in the regions o f 1 : t a n d 2 : 1 p h a s e e n t r a i n m e n t s h o w n in Fig. 11. N o t e t h a t the m e t h o d involves OAI

7/2

L~

0.w

Fig. 10. The wedge shaped region of 2:1 phase entrainment near a = 1. The dotted lines are the results from perturbation theory. The solid lines are the results from the numerical integration of the F equation

(OI

5

3 2:1 I 0

O.5

Ct

1.0

Fig. 11. The regions in the o)a, a plane (with o)2 = 1) in which the existence of limit cycles (corresponding to 1: 1 and 2 : 1 phase entrained motions) may be proven from the Poincare-Bendixson theorem. See Appendix D. The boundaries of the displayed regions are obtained analytically, and found to be straight line segments

1 : 1 and 2 : 1 phase entrainment in two coupled oscillators

141

only sufficient conditions on the existence o f phase entrained motions, so the associated regions in Fig. 11 lie inside the c o r r e s p o n d i n g numerically obtained regions s h o w n in Fig. 8. Let us n o w consider the nature o f the bifurcations which occur during the transition f r o m 1 : 1 to 2 : 1 phase entrainment. The analytical-numerical technique discussed in A p p e n d i x B and utilized to obtain the limiting parameter values for 1 : 1 a n d 2 : 1 entrainment in Fig. 8, was used again here. For a given ratio p/q, a differential equation was derived (see A p p e n d i x B) which exhibits periodic solutions only for those parameter values for which there is p: q phase entrainment. We numerically integrated this equation for nine different ratios p/q between 1 and 2, fixing wl = w2 = 1 and varying a (cf. Fig. 7). The results are shown in Fig. 12. E a c h value o f p/q corresponds to a plateau of phase entrainment, the size of the plateau being larger, the smaller the value o f q is. Since this flow on the torus T 2 has no singular points, a Poincare m a p on a cross section can be

2:1 7:4 *~5:3 3:2

9:8 4:5

5:4

,, 6:5,7:

I ~

,

i

F

0.595

0.5925

i

0.5955

w

0.594

0.5945

Fig. 12. Plateaus of p: q phase entrainment in the t~,p/q plane. (Note the breaks in the horizontal scale)

I

§

I

I §

I 2

......

I §

+ I

I +

o Fig. 13. The Cantor function

i

i--

g(o~) with 15 intervals shown

142

W.L. Keith and R. H. Rand

constructed. This m a p will be a diffeomorphism on S ~, and therefore one can apply the results of Denjoy [4] (cf. Guckenheimer and Holmes [6]) which show that the rotation number depends continuously on parameters (here a and ~ol). We conclude that all rational p/q between 1 and 2 are exhibited by this system. We also expect generically to find frequency plateaus for all rational rotation numbers. An example of function which exhibits this behavior is provided by the Cantor function, defined as follows: 1 OL< 2 we set g ( a ) = 8 9 for We consider the closed interval 0 <~ a <~ 1. For ~< ~ < a < 3 and 3 < a < ~ we set g(a)=a1 and 43 respectively, and so forth. At the nth step, the function g(a) is given values over the 2 n-~ open intervals of length (1), midway between the values of g ( a ) on the neighboring 2 "-2 intervals of length (1)n-l. In the limit as n ~ oo, we have a continuous function g(a), whose slope is zero almost everywhere. The Cantor function g(a) is shown in Fig. 13 with 15 intervals plotted. S u m m a r y and conclusions

A system of two coupled first order nonlinear ordinary differential equations has been proposed to govern the behavior of 2 generalized biological oscillators. Each uncoupled oscillator is assumed to exhibit a limit cycle which persists after coupling and which is modeled by its phase Oi(t). The system exhibits "phase-locked" motions, for which (in the 1 : 1 case, for example) the difference O~(t)-02(t) is constant, as well as "phase entrained" motions, for which (again in the 1:1 case) O~(t)-O2(t) varies periodically. Although phase-locked motions represent very special behavior in the model, phase entrained motions (1 : 1 and 2 : 1) occur in a large region of parameter space. In the neighborhood of a = 1, where the 1 : 1 coupling term predominates, the 1 : 1 phase-entrainment region is centered around a natural frequency ratio of w~/w2= 1. Similarly, in the neighborhood of a = 0, where the 2"1 coupling term predominates, the 2:1 phase-entrainment region is centered around a frequency ratio of w~/w2=2. See Fig. 8. For a given value of ~o~, the two regions are separated by a narrow band which becomes vanishingly small as w~->0. See Fig. 8. Across this band, all rational p: q phase-entrained motions are conjectured to occur, for 1

1 : 1 and 2:1 phase entrainment in two coupled oscillators

143

The SA node and AV node have been modeled as two coupled oscillators as early as 1928 by van der Pol and van der Mark [13], and more recently by Guevara and Glass [7]. (Note, however, that these previous models are mathematically very different from the approach taken in this paper.) Using our model, the onset of 2:1 AV block could be explained as being caused either by a change in frequency ratio or in coupling coefficient leading to transition from 1 : 1 to 2 : 1 phase entrainment. Although the model presented in this work offers a very simplified picture of the dynamics of a two oscillator system, it is hoped that it will be useful to biologists in thinking about the qualitative behavior associated with 1 : 1 and 2 : 1 entrainment.

Acknowledgments. The authors wish to thank Philip Holmes for numerous contributions to this work. This work was partially supported by Air Force Grant ~AFOSR-84-0051.

A p p e n d i x A. P h a s e l o c k e d m o t i o n s In this section we show that Eqs. (12), (13) exhibit 1 : 1, 2: 1, and 3 : 2 phase locked motions, and no others. For p: q phase locking we require that 01(t)=P-02(t)+01o q

for all t,

(A1)

where p and q are relatively prime (that is, are integers with no c o m m o n factor besides unity), and where we take without loss of generality 02(0 ) = 0, 01(0)= 010. Differentiating (A1) with respect to time, we obtain the condition @1 = pO2.

(A2)

Substituting Eqs. (12), (t3) into (A2), we find q(wl-e~sin(Ol-O2)-(1-a)sin(Oi-2Oz))=p(wz+asin(Ol-Oz)+(l-a)sin(01-202)).

(A3)

Using (A1) this may be written sin(A02(t) + 01o) + ( 1 - a ) sin(B02(t) + 010) + C = 0

(A4)

where A = - P - 1, q

B=P-2, q

Now Eq. (A4) is of the general form kl s i n f ( t ) + k2 sin g(t) + k 3 = 0

(A5)

where kl=oL ,

k2= 1 - a ,

k3=C , f(t)=AO2(t)+Olo ,

g(t) = BO2(t ) +Oio.

In order for (A5) to be satisfied on an open interval in t, in the general case in which the functions f ( t ) , g(t) and the constant function 1 are linearly independent, we require that k 1 = k 2 = k 3 = 0.

(A6)

If, however, f ( t ) =- +g(t) ~ constant, then (A5) is satisfied by k I • k 2 = 0,

k 3 = 0.

(A7)

144

W . L . Keith and R. H. Rand

On the other hand, if f (t) -= constant, while g(t) ~ constant, then (A5) gives k 1 sin f + k 3 = 0,

k 2 = 0.

(A8)

kl=0.

(A9)

If g(t) =-constant, while f ( t ) ~ constant, then (A5) gives

k2sing+k3=O,

Finally, if both f ( t ) and g(t) are constants, then (A5) simply becomes k 1 sin f + k 2 sin g + k 3 = 0 .

(AI0)

Thus (A4) may possibly be satisfied in one of the 5 ways (A6)-(A10). However, it m a y be seen immediately that (A6) and (A10) are impossible. Equation (A7) requires either f ( t ) ~ - g ( t ) or f ( t ) - = -g(t). The former condition requires that A = B, which is impossible, while the latter condition requires A = - B and 01o = 0. The condition A = - B is satisfied by taking p / q = 3, in which case (A7) requires that k 1- k 2 = O , i.e. a =89 as well as k 3 = C = 0 , i.e. 0)2/0)1 = 23-" Thus we find that when a - 1 and 0)2/0)1 = 3, a 3 : 2 phase locked motion exists. Returning to conditions (A6)-(A10), we find that (A8) is satisfied if f ( t ) - = constant, i.e. if A = 0 which requires that p = q = 1, in which case (A8) gives c~sin01o+

tO 1 -- 0)2

2

=0

and

a=l

i.e. sin 01o-

0 ) 2 - - 0) I

2

which requires [0)i- o921<~2. Thus when a = 1, a 1:1 phase locked motion exists only if Io92- o)11~< 2. Similarly, (A9) requires g(t) =- constant, i.e. B = 0 which requires p = 2q, in which case (A9) gives 2 0 ) 1 -- 0) 2

( 1 - cQ sin 010 -~ - 3

0

and

a=0

i.e. sin 01o = which requires

0)2--2o)1

-

I0)=-20)tl ~<3. Thus when c~ = 0, a 2:1

3

phase locked motion exists only if

I0)=-20),1 ~<3.

Appendix B. Derivation of the F equation In this section we develop an analytical-numerical method in order to find the regions of p: q phase entrainment more efficiently than by simply numerically integrating Eqs. (12), (13). We derive an equation, Eq. (B6) below, which exhibits periodic solutions if and only if Eqs. (12), (13) exhibit p: q phase entrainment. Setting 0)2 = 1 and adding Eqs. (12) and (13) gives (B1)

O1 -~- 02 = 0)1 -{- 1.

Integrating (B1) with respect to time we obtain

Ol(t)+Oz(t)=(0)t +l)t+01(O)+O2(O).

(B2)

Equation (BI) may be satisfied by solutions of the form 01(0 =

t+F(t)=-rlt+F(t),

rl=--T-

(B3)

02(t)=

t-F(t)-~r2t-F(t)'

r2- T

(B4)

1 : 1 and 2:1 phase entrainment in two coupled oscillators

145

where to= 0 and 01(0 ) = - 0 2 ( 0 ) = F(0). Adding (B3) and (B4) and comparing the result to (B2) we observe rl + r2 = o91 + 1.

(B5)

Substituting (B3) and (B4) into (12) we obtain P = w 1- r I + a sin((r 2 -

rot - 2 F ) +(1 - a) sin((2r2 - q)t - 3 F )

which is a first-order, nonautonomous, nonlinear differential equation for we have

A(qO1-p02)l,-r = (q +p)AF(t)I,=r.

(B6)

F(t). From (B3) and (B4) (B7)

For p: q phase entrainment to exist at time t = T, we require AF(t)],= r = 0. This implies that a periodic solution must exist for F(t). Equation (B6) must then yield a periodic solution for F(t), where the period is governed by the p and q of interest, which determine r 1 and r2. For p = 2 and q = 1, we obtain r I = 2r 2.

(B8)

r 1 =2(w, + 1)

(B9)

Combining (B5) and (BS) we obtain

and r2 =

o91+1

(too)

3

Substituting (BI0) and (B9) into (B6) gives /~= t o l - 2 [ [ o9, +1"~ 3 -a sin~-)t+2F)

\

-(l-cOsin3E

(Bll)

Values of tom and a for which periodic solutions of period 6z'/(w I + l) exist for Eq. (Bl l) correspond to regions of 2:1 phase entrainment in the to1, a plane. Equation (B1 l) was integrated numerically. For a fixed value of (Ol, a was varied until loss of periodicity occurred at or before 5 periods. For l : 1 entrainment, an equation analogous to (B1 l) was derived and regions of 1 : 1 entrainment were similarly obtained by numerical integration.

Appendix

C. Perturbation

method

In this section, we present a perturbation method which serves to supplement information about the phase entrainment plot in the neighborhood of the point in parameter space where ro1= 7, o~2 = I, and c~ = 1. We approximate the upper and lower boundaries of the 2" I phase entrainment region in the neighborhood of a = 1 by straight lines which intersect at ~o1 =7 and a = 1 forming a wedge. In the neighborhood of the wedge, a = 1- e

(C1)

w I = 7 + A e + O(e2).

(C2)

When ~ = 0 , the exact solution to Eqs. (12) and (13) is known (Cohen et al. [1]). When e is not identically zero, 01(t) and O2(t) will depend on e. For 2:1 phase entrainment, at some time t = t*, Ol(t*) = 4rr and O2(t*) = 2~r + 0o. Since t* will depend on e, it is convenient to stretch time such that r = t(1 + Ue +O(e2)), where U is an undetermined constant. We then transform the independent variable from t to z and look for a solution to (12) and (13) of the form

01(r) = O~

+ e(l(r )

(C3)

02(7) = 0~162+ ~2(~).

(c4)

146

W . L . Keith and R. H. Rand

We then substitute (C3) and (C4) into (12) a n d (13), expand the sine terms in a Taylor series, and neglect terms of O(e2). Equating terms of O(e) we obtain 7U ~] = - ~ - + ( U

+ 1) sin(0 ~ 0 ~ +)t

- cos(0 ~ - 0~ (~:1 - ~:2)- sin(0 ~ - 20~

(C5)

~:~= - U - ( U + 1 ) s i n ( 0 ~ - 0 ~ + s i n ( 0 ~ - 2 0 ~ )

0~)(~1- ~2).

+ cos(0 ~

(c6)

For 2 : 1 phase entrainment, it may be shown (Keith [9]) that ~-*= 4~r/3

(C7)

U=2A/9.

(C8)

Subtracting (C6) from (C5), using (C8), and defining Tt = ~:1- ~:2, we obtain -q' +2• cos(0 ~ - 0 ~ = 2(2A/9 + ( 2 h / 9 + 1) sin(0 ~ - 0 ~ - s i n ( 0 ~ - 20~

(C9)

Equation (C9) is of the form

tt+P(r)V = Q(~').

(C10)

This first-order linear o.d.e, has the solution "q(7") = e -l~p(~) d~

io

Q(1")el;P(')U'd'r+~e -l~p('~)a'~

(ell)

where ~(0) = El(0) - ~2(0) = ~. Without loss of generality we choose ~:1(0) = ~::(0), i.e. ~ = 0. Equation (C 11) evaluated at ~"= ~'* = 4~r/3 becomes 0=

f

4~/3

Q ( r ) eg e(~) ~ dr

(C12)

*/0

which is o f the form H(Oo, )t) = 0 where 0o = 02(0). Equation (C12) may be shown to be 27r periodic in 0o. Treating 0o as a variable and h as a parameter, values of h for which zeroes of H(Oo, h) exist correspond to the region of 2:1 phase entrainment in the ~o1- a plane. The limiting values of h for which zeroes exist correspond to the boundaries of the 2:1 entrainment region. Equation (Cl2) was integrated numerically for -4<~ h ~< 1. The results of this analysis are shown in Fig. 10. The limiting values of )t are found to be approximately ;t = - 3 and h = 0.

Appendix

D.

Poincar~,--Bendixson

theorem

In this section we offer a proof of the existence of the limit cycles in 01, 02 phase space. We begin by reviewing the theory of Poincarr's proofs of existence of limit cycles in the plane. The Poincarr-Bendixson theorem for flow on a plane states that it is sufficient for the existence of at least one closed trajectory in a domain D that (Fig. D1) a) Trajectories enter (or leave) D through every point on the bounding curves C~ and C2, and b) There are no singular points in D or on C~ or C 2. Let the vector field in the x, y phase plane be given by A

A

V=~i +~j where ~ and f are orthogonal unit vectors in the x and y directions respectively. If a curve C exists such that at no point on C is the vector field V zero or tangent to C, then C is called a curve without contact. For a continuous vector field V, a curve with contacts cannot generally serve as a bounding curve C~ or C2 since the vector field along C enters D at some points and leaves at others, as shown in Fig. D2. An exception occurs when the contact is of higher order (Minorsky [14]). For a "topographic family" consisting of closed curves given by f(x, y) = k, and a given vector field V, let k~ < k < k 2 correspond to members of this family which are curves with contacts, and

1 : 1 and 2:1 phase entrainment in two coupled oscillators

147

yJ

Cz Fig. D1. The region D in which a stable limit cycle exists

k < k 1 and k > k 2 correspond to members without contacts. The locus of points at which contact exists forms a continuous curve which may or may not be closed. The case where such a curve is closed (and the topographic family consists of circles centered at the origin) is shown in Fig. D3. Poincare refers to this curve as the curve o f contacts. When the curve of contacts is closed, the domain D defined by the topographic family for kl < k < k a contains a closed trajectory that is a limit cycle. The curves f ( x , y) = kl, and f ( x , y) = k 2 serve as the bounding curves along which all trajectories enter or leave /9. If all trajectories enter D, the limit cycle is stable, and if all trajectories leave D, the limit cycle is unstable. We now proceed to apply this theory to our system. We showed that for p: q entrainment, a periodic solution of period T exists for the corresponding differential equation for F(t). The phase space for this flow may be viewed as a cylinder whose circumference is the period T and whose axis is F ( t ) , as shown in Fig. D4. For this system we choose the topographic family to be f ( F , t) = F = constant. We then proceed to find a range of parameter values over which a domain D such as that shown in Fig. D5 exists.

Y

D

"• Fig. D2. C3 is a curve with contact (since the vector field is tangent to C 3 at point P) which cannot serve as a bounding curve such as C~ or C 2 in Fig. DI

5

D

Fig. D3. COC is the curve of contacts, with the topographic system consisting of circles centered at the origin

148

W. L. Keith and R. H. Rand

, F(t)

_~~nT

Fig. D4. The cylindrical phase space

F(t)

.D

",

",

~

",

F(t)=CI

I

7i

l

.... a = .700

-7/"

Fig. DS. The topographical system

2rr I. t

F(t)= constant

Fig. D6. The solids lines are curves of contacts. The dotted lines are members of the topographic family which serve as boundaries of the regions in which limit cycles exist. The region D, (D2) lies between the upper (lower) pair of dotted lines. Here c~ = 0.7 and to~ = 1

a : .550 -7/"

Fig. D7. The curves of contacts with c~ =0.55 and to 1 = 1

=. 5 0 5 -7/'-t

Fig. D8. The curves of contacts with a =0.505 and w 1 = 1

1 : 1 and 2 : 1 phase entrainment in two coupled oscillators

149

F(t) 7/'-

~

t a =.501

Fig. D9. The curves of contacts with ~ = 0.501 and to 1 = 1 -TJ

The vector field in the F - t plane is given by

v=;+tZ The normal to the topographic family f ( F , t) = F is given by

Vf(F,t)=ff+fFf=f Then for curves of contacts, Vf. V=/~=0. From Eq. (B6) derived in Appendix B for/~(t) we obtain the equation for the curve of contacts for 1 : 1 phase entrainment,

O=(~-) -ct sin2F+(l-a) sin((W~ l)t-3F)

(DI)

and for 2:1 phase entrainment,

O=(~)-olsin((~-)t+2F)-(1-a)sin3F

(D2)

where we has been set equal to 1, and to 1 and a are retained as parameters. Figure D6 shows two of the curves of contacts given by Eq. (D1) as solid lines (where to I = 1 and a =0.7). The dotted lines are members of the topographic family which serve as the boundaries of the region in which the limit cycles exist. Here the region D 1 contains a stable limit cycle and D 2 contains an unstable limit cycle. It should be noted that in general the limit cycles (which are the periodic solutions for and not shown in Fig. D6) do not coincide with the curves of contacts and may or may not touch the bounding curves of a region D. As cr is varied, the diagram corresponding to Fig. D6 (for a = 0.7) changes, as shown, for example, in Figs. D 7 - D 9 (for c~ =0.55, 0.505, and 0.501, respectively). While the previous argument for the existence of limit cycles applies to Figs. D7 and D8, it does not hold for Fig. D9, since the intersection of the two curves of contacts eliminates the associated regions D of Fig. D6. By analytically requiring the two curves of contacts to just touch, we obtain the boundaries for 1 : 1 and 2 : 1 phase entrainment shown in Fig. 11 (see Keith [9] for further details).

F(t),

Appendix

E. C o u p l e d van der P o l o s c i l l a t o r s

In this section we outline the derivation of Eqs. (4), (5) and (7), (8) from coupled van der Pol oscillators. We consider two van der Pol oscillators with weak coupling given by: ii + W u - e ( I - U2)ti = eg where

u=X[j W:[o

(El)

150

W.L. Keith and R. H. Rand

Here 0)2 has been taken to be unity without loss of generality. We take the coupling to be a general quadratic function of the velocities and displacements:

1=I k=l

This extends the study of Rand and Holmes [1 I], who took the g~ to be linear functions of Xj and Xs. Using the two variable expansion perturbation method (cf. Cole [3]), the independent variable t is replaced by two new independent variables, ~ (stretched time) and 7/ (slow time): ~:= Ot,

O = 1 + O l e +O(e2),

*/=et.

Under this transformation, Eq. (El) becomes ue~ +2eu~, + W u - e ( I - U2)u~ = eg

(E3)

where subscripts denote partial differentiation. Next we expand the dependent variables XI(~, */) and X2(~, */) in power series in e:

Xi(.~, */)= Xio(/~, */)+eXil(.~, */)+O(e2),

i=1,2.

Substituting this expression into (E3) and equating coefficients of e ~ we obtain: X10 = AI(*/) cos tol~: + BI(*/) sin 0)1~:

(E4)

X20 = A2(~7) cos ~ +B2(*/) sin ~.

(E5)

Equating the coefficients o f e 1 we obtain:

X l l ~ + 0)2X H = ( 1 - X2o)X~o~ - 2Xlo~ + gl

(E6)

X21~r +X21 = (1 - X22o)X20r -2X2oen +g2.

(E7)

Equations (E4) and (E5) are substituted into (E6) and (E7), and the coefficients of the secular terms sin tol~ and cos tOl~ in (E6) and sin ~ and cos ~ in (E7) are set equal to zero. Due to the large number of terms in gl and g2, this calculation was performed using the symbolic manipulation program MACSYMA (cf. Rand [10]). We now have four differential equations for AI(*/), A2(*/), BI(*/), and B2(~). We transform to polar coordinates such that Xlo = RI(*/) cos(Ol(~, 7/)) (E8) X2o = 82(*/) cos(02(~, */)). We wish to consider the 1 : 1 and 2 : 1 cases where co~ = 1 and to, = 2 respectively. For the 1 : 1 case, after manipulation (cf. Keith [9]) the differential equations for 01 and 02 become:

eh,, eR~ dO I __-~z +7-~-_"(h,: cos(01 - 02)- 112 sin(01 - 02)) --~-= 1 2 2R I dO2 dt

l

eh22 eRl (h21 cos(01 - 02) -/2, sin(01 - 02)). 2 2R 2

(E9) (El0)

For the 2: I case, the differential equations for 01 and 02 become:

dot --=dt

2

ehll 4

eR2 ~11 ((a122-c122) cOs(O1-202)+b122sin(OI-202))

(Ell)

dO2 eh22 R1 1 - 2 - e-4-((2(c212 +c221) +(a212 5-a221)) cos(01-202) d-T = - (26212 - b221) sin(01 - 202)).

(El2)

Note that R 1 and R 2 in Eqs. (E9) through (El2) also satisfy differential equations which we omit in our derivation of Eqs. (4), (5), (7), and (8).

1 : I and 2:1 phase entrainment in two coupled oscillators

151

For the 1 : 1 case, we can obtain Eqs. (4) and (5) by making the following substitutions: Set h~2 = h21 = hl~ = h22 = 0 and obtain:

01 = 1 - ~( R2la2~ sin(Ol- 02)

(El3)

\ 2R1 ] 02 = 1

+e(Rll21)

\-'~2"-2 / s i n ( 0 , - 02).

(El4)

The presence of e in each of the coefficients of the sin terms restricts the phase coupling to be of O(e), which we expect since the van der Pol oscillators are weakly coupled. In the absence of coupling, each van der Pol oscillator moves on a limit cycle of radius 2. In the presence of weak coupling ( 0 < e << 1), the perturbed limi~t cycles may be assumed to be close to the unperturbed ones (cf. Rand and Holmes [11]), leading to the assumption R 1= R 2 = 2. In order to get (El3) and (El4) to be of the form (4) and (5), we further take 112= 121 = 2a* and obtain:

01 = 1 - ca* sin(01 - 02)

(El5)

02 = 1 +ca* sin(01 - 02).

(El6)

For the 2:1 case, we similarly require 2(c212 + c221) = -(a212 + a221) to eliminate the cos terms from Eqs. (El 1) and (El2). The coefficients of ffflX2 and XIJf 2 in g2 are b212 and b22 ~ respectively. For convenience we set b221 = 0, and b~22 = 4b212 = 4/3* to obtain: 01 = 2 - e/3* sin(01-202)

(El7)

02 = 1 +e/3* sin(01 -,202).

(El8)

References 1. Cohen, A. H., Holmes, P. J., Rand, R. H.: The nature o f the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model. J. Math. Biol. 13, 345-369 (1982) 2. Ermentrout, G. B., Kopell, N.: Frequency plateaus in a chain of weakly coupled oscillators, I. SIAM J. Math. Anal. 15, 215-237 (1984) 3. Cole, J. D.: Perturbation Methods in Applied Mathematics. Waltham: Blaisdell 1968 4. Denjoy, A.: Sur les courbes d~finies par les ~quations differentielles a la surface du tore. J. Math. 17 (IV), 333-375 (1932) 5. Glass, L., Mackey, M. C.: Pathological conditions resulting from instabilities in physiological control systems. Ann. N.Y. Acad. Sci. 316, 214--235 (1979) 6. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer 1983 7. Guevara, M. R., Glass, L.: Phase locking, period doubling bifurcations, and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation o f cardiac dysrhythmias. J. Math. Biol. 14, 1-23 (1982) 8. Hoppensteadt, F. C., Keener, J. P.: Phase locking of biological clocks. J. Math. Biol. 15, 339-349 (1982) 9. Keith, W. L.: Phase Locking and Entrainment in Two Coupled Limit Cycle Oscillators. Ph.D. Thesis, Cornell University, pp. 17-32, 1983 10. Rand, R. H.: Computer Algebra in Applied Mathematics: An Introduction to MACSYMA. Boston: Pitman 1984 11. Rand, R. H., Holmes, P. J.: Bifurcation o f periodic motions in two weakly coupled van der Pol oscillators. Int. J. Nonlinear Mech. 3, 387-399 (1980)

152

W.L. Keith and R. H. Rand

12. Rand, R. H., Storti, D. W., Upadhyaya, S. K., Cooke, J. R.: Dynamics of coupled stomatal oscillators. J. Math. Biol. 15, 131-149 (1982) 13. van der Pol, B., van der Mark, J.: The heart beat considered as a relaxation oscillation and electrical model of the heart. Phil. Mag. 6, 763-775 (1928) 14. Minorsky, N.: Nonlinear oscillations. New York: Kreiger 1974 Received January 27/April 14, 1984