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Weak coupling of strongly nonlinear, weakly dissipative identicaloscillatorsAshwin Peter

To cite this Article Peter, Ashwin(1989) 'Weak coupling of strongly nonlinear, weakly dissipative identical oscillators',Dynamical Systems, 10: 3, 2471 — 2474To link to this Article: DOI: 10.1080/02681119508806204URL: http://dx.doi.org/10.1080/02681119508806204

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Dynamics and Stability of Systems, Vol. 10, No. 3, 1995

Weak coupling of strongly nonlinear, weakly dissipative identical oscillators

Peter Ashwin Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

(Received March 1995 I

Abstract. This paper considers a network of coupled oscillators with (a) weak coupling and (bj weak dissipation (but strongly nonlinear response). By changing to energy- angle coordinates, equations for the slow evolution of the amplitudes and phases of the individual oscillators are obtained. Restricting to problems with identical oscillators and coupling weaker than dissipation, an approximation to an attracting invariant torus for two osciiiators is consiructed uiid mei-agiiig ij p e ~ f o m e d ox this t~ ;educe tc a s k g b equation for the phase difference. For two van der Pol-Duffing type oscillators coupled by a cubic function this method is used to predict bifurcation from in-phase stabilit-Y (througn a region where nei~her in-phuse rrvr uni@hase sijiiiiiiii~ are : t ~ b k ) to aiit$hase stability as a coupling parameter is changed. Furthermore, the method is used to investigate analytically the destabilization of the in-phase solution when two oscillators are dissipative& coupled through one state variable. For a large class of systems the stability of the in-phase solution depends on the sign of the shear or dependence of frequency on amplitude of the individual oscillators.

1 Introduction

Groups of weakly coupled dissipative oscillators can give useful models for many physical processes. The method of averaging (Bogoliubov, 1961; Sanders & Verhulst, 1985) has often been applied to systems of coupled oscillators, especially for the case of weakly nonlinear perturbations of linear oscillators (see, for example, Bogoliubov, 1961; Kevorkian, 1987; Aronson et al., 1990; Poliashenko & McKay, 1992); but also for more arbitrary weakly coupled oscillators as a method (Ermentrout & Kopell, 1984, 1991; Baesens et al., 1991; Ashwin & Swift, 1992) to reduce the equations to a flow on an invariant torus.

Present address: CNRS-INLN, 1361 Route des Lucioles, F-06560 Valbonne, France. E-mail: ashwinQecu.unice.fr.

0268-1110/95/030203-16 01995 Journals Oxford Ltd

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Luke (1965) jser I(ev-orkiaii & Cole, 198 1; Kevorkiaii, 198:) developed a method for systems with weak dissipation and slowly varying coefficients allowing one to work with strongly nonlinear oscillators. This limit of weak dissipation is relevant in many mechanical, electronic and biological systems, for example the motion of celestial bodies with tidal dissipation (Kevorkian, 1987) (see also references in Bourland et al. (1991)). Bourland et al. have extended this method to study phase shifts for slowly varying oscillators using multiple scales (1988) and averaging (1991) for individual oscillators. Dangelmayr and Kirby (1992) consider diffusively coupled oscillators in a continuum limit of many oscillators, both weakly and strongly anharmonic, and examine fluctuations of phase and energy about an in-phase (uniform) oscillating state using second-order averaging.

This paper uses a weak dissipation assumption to investigate weakly coupled oscillators. Weakly dissipative oscillators give models for, for example, Josephson junction oscillators (Sanders & Cushman, 1986) or electronic oscillator systems (Ashwin et al., 1993). A class of systems is discussed where it is possible to construct an approximation to the invariant torus. Using the assumption that the coupling is weaker than the dissipation allows us to use the normal hyperbolicity (Hirsch et

---- b--* *I. KLU u r coupling going to zero faster to e ..'...ro tho ..oro;.-tn"r!a the ;n.,".-;"nt

113UI L L 1 1 L pLI JIJLLIILC VI L I 1 L A l l V U l l U l l L

torus in the singular limit; the coupling is assumed to be of the order of the square of the dissipation. The fact that the oscillators are perturbations of Hamiltonian oscillators allows approximation of the limit cycles and thus the invariant torus. This enables discussion of the stabilities of periodic orbits with arbitrary phase shifts between the oscillators, not just the in-phase orbits whose stability is &cussed i n Dangelmayr and_ Kirby (1992):

Networks of n coupled cells are often modelled by smooth ordinary differential equations (ODES) of the form:

where xi is in some Rk and i = 1, . . . , n. The variable E governs the strength of the coupling and for K = 0, the equation x= f(x) has a hyperbolic stable limit cycle y with period, say, 2 n (see, for example, Ashwin & Swift, 1992). The asymptotic dynamics of such weakly coupled systems can be reduced from the full phase space Rkn to a torus 8" with only as many degrees of freedom as there are oscillators, i.e.

with 0 = (8,, . . . , 8,) and each of the B i € [ O , 2 ~ ) . Ermentrout and Kopell (1991) give a procedure to calculate G, given either weak coupling or strongly attracting limit cycles. For small enough coupling, each of the oscillators can be ascribed a phase for each asymptotic state of the system and the dynamics is governed by the dynamics of these phases.

Because (2) is a small perturbation of the linear flow (8, = 1) it is possible to use averaging to simplify the asymptotic dynamics even further. Define the average

and to write

d, = 1 + K(G,(O)) + 0 ( x 2 )


WEAK COUPI.ING OF DISSII'ATIVE OSCILLATOKS 205

Averaging takes advantage of the splitting of the time-scales between the fast oscillations and the slow drift of phase shifts relative to each other. The averaged system (3) (for small enough K) is an accurate predictor of periodic orbits and other hyperbolic structures in the original system (2), but fails at non-hyperbolic stiiiciures, e.g. tangencies or bifurcarinr? points.

The rest of the paper is organized in the following way: in Section 2, a system of two degree of freedom weakly dissipative oscillators (modelled by small dissipative perturbations of planar Hamiltonian oscillators) is considered. Using a standard procedure of introducing a normalized period, the equations for the slow variation of amplitude and phase are found. Under the assumptions that the oscillators are identical and the coupling is weaker than the dissipation, one can average these equations, get an approximation to the invariant torus, and hence obtain an expression for the evolution of the phases. This approximation is valid for coupling weak enough that the invariant torus persists.

Section 3 examines how this applies to a system of two identical symmetrically coupled oscillators, by finding the averaged ODE governing their phase difference.

ic ij.wlie? . . :n 4 svst~m of two van der Pn!-Dcff?.n~ oscillators weakly coupled by a cubic function. Varying the linear par: of the coupling produces branches of solutions with trivial symmetry (i.e. periodic solutions that are neither in-phase nor a n t i ~ h ~ s e ) and one can detect symmetry-breaking pitchfork bifurcations of the in- phase and antiphase solutions.

As another application, Section 4 considers a pair of diffusively coupled oscillators

x, = f(xl, x,, E) + K ~ ( x ~ - xl)

with d(x) = - d(-x) and the equation for x, obtained by interchanging i and 2. The coupling is dissipative if df(0) > O. The averaged equu im wi!! be of the form

with G(8) diffusive in the sense of Kopell and Ermentrout (i98bj, i.e. G(Oj = 9. in- phase solution stability is shown to be determined by the product of d'(0) with the shear or dependence of frequency on amplitude, and thus can be unstable even for dissipative coupling. Recent numerical work of Sherman and Rinzel (1992) has shown an example of dissipative coupling between two Hodgkin-Huxley type oscillators where the in-phase solution is unstable. In conclusion, this behaviour should occur in a large class of systems with weak coupling and a numerical example of this is given.

2 Groups of oscillators with weak dissipation and coupling

2.1 Weak coupling and dissipation; slow variables

Consider the following class of systems.

Definition 1. A weakly coupled weakly dissipative (WCWD) network of n nonlinear two-dimensional oscillators is a dynamical system governed by ODES of the form

v, + Ui.(v,) = EF~(v,, . . . , vn, vl, . . . , vn) (4)


where O < E < < ~ and i = l ... n, U,, F, are smooth and U,(a)-a as Iv I - a . (We write U1(v) = d U/dv.)

For this it is possible to derive the equations for the slow variation of energy and phase angles of each oscillator (cf. Bourland et al. (1991)), i.e. equations (12) and (i3j. By assumption, each osciiiator is a perturbation of a conservative osciiiator with energy

constant for E = 0. Suppose that the contours of H are smooth closed curves for all values of a except possibly isolated values corresponding to homoclinic or heteroclinic orbits.

In order to define a useful notion of phase, the period on the level curves is normalized in the following manner. Define w(a)>O such that all solutions z(a , ly) of the following equation (excluding those of infinite period)

have period 2 n in ly and such that

Note that 2n/u is the period of the corresponding solution of equation ( 5 ) . The LI.. .:. 1- 11 ... I .I-.. r I:-..A -I- ..-.. 7 / , \ ..-.. -.,c .I--.I. vd~iav~c yi 1s callcu u1c 1wIuLauLcu p u a x , jii, i j j j ale 1clulJ i" zs '~ii~rg4.-~iip!i

coordinates'. One can look for solutions of equation (4) (setting n = 1 for simplicity) of the form

Since the number of independent variables has been increased from one (v(t)) to two (a ( t ) , ~ ( t ) ) , one can apply a constraint relating a and ly implicitly

d v az - = w(a) --- ( a , q ) d t av

This is equivalent to defining

see, for example, Bourland et al. (1991). For compactness of notation (and to allow subscripts indexing the oscillators) m rite the derivatives

Differentiating (7) with respect to i gives

bz,, + (11,- w)z,,, = 0


W E A K COUPLING OF DISSIPATIVE OSCILLATORS 207

The normalized periodic orbits z ( a , v ) are defined such that they are differentiable in both a and q, and all derivatives are periodic in ly. Assuming U, and therefore z is sufficiently smooth, their derivatives satisfy the identity

4 ~ z , y r z , < l y > + W,<~(Z,J-- ~ ? ~ ~ ? ~ l = 1 (10)

obtained by differentiation of (6). Moreover, z has a time reversal symmetry given by

z(a , q) = z(a , - q )

This follows directly from equation (6).

Remark. Note that (10) reduces to cos2 q + sin2 11, = 1 for the case of U(z) = z2/2 and the oscillators being simple harmonic.

2.2 The slow time-scale evolution equations

where w, = w(cr,). For the rest of this section the subscript i has been dropped for ease of reading. Using (6) to eliminate U the previous expression reduces to

~ ( w , , , z , ~ , + ~ 3 , ~ , , ~ ) + ($I- / t ) \ l ~ t ) ~ - 1 \--,~yi \ = EF (11)

Taking the combination (1 1) iiiz,, - (9) w , vv gives

whereas taking the combination (11) wz,, - (9) ~ (wz , , , + w,,z,,) gives

and again using (10) gives the equation for the variation of the angles =

q = W - EOZ,,F (13)

The two sets of equations (12) and (13) are exact equations for the evolution of the system. They are in standard form to apply averaging over t to separate the time- scales fuily; note that n second-order ODEs have been transformed into 2n first- order ODEs.

2.3 Identical pairwise coupled networks

The dynamics of a network of n coupled oscillators can have a very complicated dependence on the frequency ratios (Baesens et al., 1991). To simplify matters, consider the case where the uncoupled oscillators are identical and therefore have the same frequency. Furthermore, restrict to identical painvise coupled oscillators


208 P. A S H W I N

with full permutation symmetry, where

and

However, the subsequent analysis is applicable to identical oscillator networks with any set of smooth coupling functions {gi].

One can apply an assumption of very weak coupling ( W C ) by requiring that K = O ( E ) as E-0. This ensures that in the singular limit of the dissipation E-0, the dynamics takes place on a normally hyperbolic invariant torus, as discussed in Section 3.1. It is of course possible that the dissipation function f will permit many stable limit cycles, but we shall only consider perturbations around a state where all the oscillators are on identical limit cycles.

Physically, this double limit is reasonable as E is associated with the internal . P . . .. . . . .., dpdfil ics U i ific iiiJ1LiCiial U X L I L ~ ; ~ ~ ~ ~ v i l ~ i ~ a b fi ;a cix&&i;d iFik ;:I; &i.&:iii~

strength; the two are often independent parameters. The equations for (a,, ly,) can be written

where

Consider {ai . . . di) to be functions of {a,, vi}. Write ai = a(u ;, qi), ci= c(ai, ~1;) and wi= w(ai).

The uncoupled equations. For K = 0 the ith oscillator (a , ly) = ( a ;, qi) satisfies

I a = ~ a ( a , ly)

y = w(u) - x ( a , v)

This can be written

where the right-hand side is periodic in y, with period 2n. One can average this equation to obtain periodic solutions for the single oscillators. Define


WEAK COL!PLING OF DISSIPA-TIVE OSCII.I.ATOKS

and so averaging (17) gives

independent of 41 to lowest order. Assume one is given an energy a* such that there is a linearly stable steady solution of the truncated equation, i.e. such that

First-order averaging (Sanders & Verhulst, 1985) predicts that for sufficiently small E > 0 there will be an exact periodic solution (a , (t), v,(t)) of (l7), with period 2n/w(a*) + O(E) and energy a,(t) = a* + O(E). The attracting Floquet exponent of this limit cycle is given by

Vety weak coupling. This section considers perturbations caused by taking small K > 0; more specifically, consider the case

K'E

The following analysis should be applicable to the more general case of K = o(1) in E, at the expense of introducing two slow time-scales. Change coordinates to (yi, $;) by the transformation

and look for solutions with yi of O(1). The equations for the evolution of the perturbed eiiergks y , and i;hases Gi "re

(a:, q!~: are the derivatives with respect to their arguments). Only arguments with respect to energy are shown; the dependence on 4, has been suppressed (note that a , = a,(#,)). The equation for 4, can be rewritten

and expanding this in powers of E gives

The equation for j i can be rewritten


210 P . A S H W I N

Substituting in the equation for 4; and expanding in powers of E gives

where all the functions are evaluated at a = a , . Cancelling one factor of E gives the equations for very weak coupling of the weakly dissipative system to be

Note that a , = a* + O ( E ) means that

and therefore

so that the only qhi dependence is in the O ( t 2 ) error terms and in the functions bi.

3 Two identical weakly coupled oscillators

For i'ir'o oscillators, it is of great interest to fd:oii; the djiiia~iiic~ of ihs phase difference. Therefore, defining

A = y l + ~ 2

1 = Y , - Y2

0 = 4 , + 4 2

e= 41 - 4 2

and transforming (25) into these variables, one obtains

A = e(A1(a*)A + b,(a*) + b2(a*)) + 0 ( c 2 )

= & ( A t ( a * ) l + b,(a*) - b2(a*)) + 0 ( e 2 )

At leading order, the equations for (8, A) decouple from those for ( 0 , A).


WEAK COUPLING OF DISSIPATIVE OSCILLATORS 211

3.1 The invariant torus

Up to error terms of O(E*), the equations for the evolution of the amplitude and phase differences can be written

As O evolves on a time-scale of 0(1), one can average P= b, - b, to give P(8 ,0 ) = p(8) + O(E) where

that is

evaluated on a = a*. The symmetry of interchanging 1 and 2 is inherited by p(8)= -p(-8) and by the error terms. The fixed points for the ODE (26) are given by solutions of

to lowest order, and the symmetry of p implies that the following solutions always exist

[in-phase 8 = 0

[antiphase 8 = n

The stability of the fixed point (0, 8,) is determined by the eigenvalues of

Since trace(M) = &Ar < 0 and det (M) = - E * W , , ~ ' ( O ~ ) / ~ , the eigenvalues of M are given to first order in E as EA' and -~w,,p'(8,)/(wA'), and the steady solution is linearly stable if

is negative; linearly unstable if it is positive. If w,, = O then higher-order terms need be considered in order to obtain the dynamics and second-order averaging (Bourland et al., 1991) is necessary.

Thus, if all fixed points are hyperbolic and o,, # 0, the dynamics is topologically equivalent to the dynamics of the ODE


212 P. A S H W I N

(Two systems are topologically equivalent if their orbits can be mapped to each other, preserving their orientation (Guckenheimer & Holmes, 1983).) Equation (27) is obtained in Ermentrout and Kopell (1991, Appendix A2) by first solving 1 = 0 in (26), sometimes called 'adiabatic elimination'. Equation (27) will correctly predict f ~ e d points of (26) and their stability. However, if the phase space of the averaged equations has dimension higher than one (i.e. more than two oscillators), the stabilities may not be predicted correctly (as in the Lyapunov-Schmidt method (Kielhofer, 1976)).

Although there is a global invariant one-dimensional manifold for (26) on which the dynamics can be rigorously reduced to an ODE topologically equivalent to (27), this manifold is not smooth, in general it is only finitely differentiable (Hirsch et al., 1977). The invariant manifold can be constructed by continuing the unique unstable manifolds of the saddle points until they meet the stable fixed points. Figure 1 shows an example of an invariant manifold constructed in this way. There is an O(E) neighbourhood of 1 = 0 that is attracting. The manifold is only finitely differentiable at the stable fixed points but is smooth elsewhere. The degree of differentiability is typically given by the integer part of the ratio of eigenvalues of

- . lhr hxr(; p&l~ ;iacarira~;oa~, a& is of order GIi ,-,. 'I - - I I O W ~ I I ~ A"- 1 K - L ~ Z ' J CWIU CL?USC

the invariant manifold to be smoother. Note that the attractivity of the O(F) neighbourhood means that linearization

with respect to 1 is justified as long as E is small. The assumption K = o j l j as E + O is vital for our discussion and corresponds to assuming the coupling is weaker than the dissipation. If ~c = 0(1), quadratic and higher terms in 1 are important, and the invariant torus will in general no longer exist.

For two symmetrically coupled identical oscillators with very weak coupling we now apply the results of Section 2. This is followed by a specific example calculation for two coupled van der Pol-Duffing oscillators. For a pair of coupled, two degree of freedom oscillators

(the other equation is obtained by interchanging 1 and 2) set K = 0, define

and assume that a* satisfies A(a*) =O, A1(a*) < O . Then there is an attracting invariant torus

Fig. 1. Equation (26) has an invariant manifold obtained by continuing the unstable manifolds of fixed points. This is shown by the dashed line; the strong stable manifolds at the fixed points are shown by double arrows. The differentiability of the invariant manifold is determined by the ratio of the linearization eigenvalueg at the stable fixed points. On the invariant manifold, the dynamics is

topologically equivalent to that of equation (27).


W E A K COUPLING OF DISSIPATIVE OSCILLATORS 213

with u,(4) - a* + O ( E ) and t$;€ [O, 2nj. In the following discussion assume that w,, # 0. If the phase-shifted z is defined

zO(a *, $J) = z(a *, q~ + 8)

then the results of the subsection on 'very weak coupling' imply that the dynamics of the phases are topologically conjugate to the dynamics of (27) with

3.2 An example: van der Pol-Duffing oscillators

Combining the nonlinear dissipation term of the van der Pol oscillator with the nonlinear response of the DufEng oscillator gives a van der Pol-Duffing oscillator (Holmes & Rand, 1980; Guckenheimer & Holmes, 1983)

Such oscillators arise in generic codimension two interactions between Hopf and pitchfork bifurcations (Carr, 1981; Guckenheimer & Holmes, 1983) in symmetric systems, e.g. in oscillators that have a steady-state bifurcation of a trivial solution as well as a Hopf bifurcation giving rise to limit cycles.

There have been many investigations of weak coupling of weakly nonlinear oscillators (Bogoliubov, 1961; Poliashenko & McKay, 1992). This work however considers strongly nonlinear response. Duffing oscillators (for which the response term is v 3 - v) have dynamics corresponding to a Hamiltonian oscillator with a twin-well potential. The effect of the dissipation term (that is assumed to be weak) is to select certain energy levels to be stable limit cycles.

Consider two identical van der Pol-Duffing oscillators coupled diffusively through one component

The coupling function G is chosen to be a cubic polynomial

with the parameter y giving the linear part of the coupling. The limit E+O (WDWC) gives the integrable system

for i= 1, 2. As before, define the normalized periodic orbits z(a, 1/1) using equation (6 ) and

seek solutions of equation (28) of the form


214 P. ASHWIN

The averaged system. Proceeding as in Section 2, the uncoupled system has a periodic solution (a,, ly,) = a * + O(E) and a satisfies

0 = ((v- z ' ) ~ ~ z ; ~ ; )

for z evaluated on (a*, q). Thus, one defines

with R a ratio of Abelian integrals evaluated on level curves of U(v). A complete analysis of the function R(a) is discussed in detail in (Carr, 1981; Guckenheimer & Holmes, 1983); this work is only concerned with a >O. The dependence of R and o on a is illustrated in Figs 2(a) and (b). For values of a > O not shown, both functions increase monotonically. Note that o,, > 0 for all values of a > 0.

Thus the limit cycles of individual oscillators can be found close to a * satisfying (18). This means that

where the integrals are over one period in q or equivalently

where the integrals are over the loop a = a*. Using the explicit form of the coupling function G(v) = v(q+ v2) allows us to

write p ( 8 ) in (27) as

Fig. 2. (a) R(a) for van der Pol-Duffing oscillators. The minimum of the curve R(a) corresponds to a saddle node of periodic orbits. (b) The function o ( a ) for the same potential. Note that it is mono-

tonically increasing in a > 0.


WEAK COUPLING OF DISSIPATIVE OSCILLATORS 215

3.3 Numerical method and results

Having reduced the problem from a flow in R4 to the calculation of p(0; a, r), a Runge-Kutta method was used to approximate solutions w(a) and z(a, ly) of equation (29). These functions were used to approximate numerically the integrals R and p.

A typical result is displayed in Fig. 3, showing O1=p(O; a, v ) for a = 0.241. The contour p = 0 gives steady states of the phase difference 0 and therefore periodic orbits of the original equations. Note that on varying 17 there is a branch of periodic orbits connecting the in-phase and antiphase solutions, terminating at pitchfork bifurcations at either end. The contours of p give not only the periodic orbits on the invariant torus, but also their stabilities and transient solutions in the limit IC= t 2 and E-0.

4 Synchronization with dissipative coupling

Fnr twn rwp!~d nscillatnrs with diffusive couplin~ through one comoonent. i.e.

g, = g(z27 z 1 7 z,,,., z1.J = 4 2 2 - 2,)

and d: R - R an odd function, the phase difference 0 is governed by (27)

0 '= W , , P ( ~ ) = w,,(d(z-z&,,- d(zo-z)zo,,)

Recall that the stability of the in-phase solution is given by the sign of

Pig. 3. Contours of p(& a, r ] ) with a =0.241 for weak symmetric cubic coupling of two van der Pol-Duffing oscillators (the phase difference between the two oscillators is governed by 8' =p(8; a, 17)). Note how, for r] > 0, all phases except antiphase are attracted to the in-phase solution and as r] passes through zero a supercritical pitchfork bifurcation gives rise to stable limit cycle solutions without symmetry. These stabilize the antiphase solution at another supercritical pitchfork

bifurcation.


....+ w;Lh n + n ~ ' ~ ; t . r a ~ a u i i l ~ j fix r t , A ;nr to~;~; t , , I l l i ) L L I V I I I L J fnr IV I S > 0. The cmsequence of this is that the in-phase solution is stable if

in the limit. This agrees with a result of Aronson et al. (1990) for systems near Hopf bifurcation and extends it to a strongiy noniinear setting. They examine two coupled oscillators of the form

near Hopf bifurcation with XI E R2 and D a constant 2 x 2 coupling matrix,

They find that the in-phase solutions are stable at the weak coupling limit if , .. ,,

"y ' I " V

(to first order in j i , y), where q gives the shear, or twist i.e. the dependence of the frequency of the individual oscillators on the sqmre ef the amp!itude. .!'he !inear part of our coupling is of the form

and as in Aronson et al. (1990), it is possible to have unstable in-phase solutions wih d ' (0 ) > 0 even if, for example, the limit cycles are nc !mger cmvex er are near homoclinic.

This change in sign of the shear seems to be a common mechanism for de- siaxiiziiig ;he iii-phasc s o l u ~ o n s of wcak!j; couI;!cd osci!!a:ors :ha: are fZr from

Hopf bifurcation. Numerical results of Sherman and Rinzel (1992) display this behaviour for two Hodgkin-Huxley type oscillators coupled by one state variable. The van der Pol-Duffing oscillators discussed in Section 3 do not display this behaviour for a > O because the frequency w is monotonic. However, for the potential

there are homoclinic and heteroclinic curves at cr = 0 and a = 6.06 respectively (Fig. 4(a) shows the phase portrait in the Hamiltonian limit). Thus w(a) has zeros at a =O and a = 6.06 and so there must be a reversal in the sign of o,,, at the turning point of the graph in Fig. 4(b). Numerical experiments using the interactive dynamical systems packaged stool on the system

for y = 0.1 show that in-phase solutions are stable for 0 < a < 4.40 and unstable for 4.40 < u < 6.06. The critical value of a = 4.40 is attained at /3 = 0.62 for the dissipation function used and is close to the maximum of w, in confirmation of the theory.


WEAK COLJPLING O F DISSIPATIVE OSCILLATORS 217

Fig. 4. (a) Phase portrait for the Hamiltonian oscillator x+ U'(x)=O with potential U ( x ) = x2!1 -x2! !4 -x2! . Note that there are two energy levels for which the level curves are homoclinic or heteroclinic orbits. (bj The dependence of the angular f iewci iq oi on :he energy !or a range ot ( I

between the heteroclinic and homoclinic orbits. There is a maximum at which the stability of weakly diffusively coupled oscillators changes.

Acknowledgements

The author thanks British Gas for a Research Scholarship and the SERC for support during the completion of this work. Thanks to Gerhard Dangelmayr, Martin Krupa, Mark Muldoon and Ralph Sebastian for some very helpful sugges- tions and discussions.

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