Rotating waves in rings of coupledoscillatorsCarlo R. Laing,Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Silver Street, Cambridge CB3 9EW, U.K.October 19, 1998Current address: Department of Mathematics and Statistics,University of Surrey, Guildford GU2 5XH, United Kingdom.Telephone: (01483) 300800 extn 2620Fax: (01483) 259385Email: mas1cl@ee.surrey.ac.uk 1

AbstractIn this paper we discuss the types of stable oscillation created via Hopf bifurcations fora ring of identical nonlinear oscillators, each of which is di�usively and symmetricallycoupled to both its neighbours, and which, when uncoupled, undergo a supercriticalHopf bifurcation creating a stable periodic orbit as a parameter, �, is increased.We show that for small enough coupling, the only stable rotating waves producedare either one or a conjugate pair, depending on the parity of the number of oscillatorsin the ring and the sign of the coupling constant, and that the magnitude of thephase di�erence between neighbouring oscillators for these rotating waves is eitherzero (i.e. the oscillators are synchronised) or the maximum possible, depending onthe sign of the coupling constant. These branches of rotating waves are producedsupercritically.

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1 IntroductionThe behaviour of rings of identical oscillators has been studied by a number of au-thors [1, 2, 3, 4, 7, 11]. Such rings are used in the biological world to model suchstructures as the cross{section through a plant stem [9] or an intestine, the ring oftissue from which the petals of a ower emanate [11], or a ring of neuronal oscillators,each of which controls the motion of one leg of an animal during locomotion [3]. Inthe world of physics, they have been used to model rings of semiconductor lasers [10].Various approaches have been taken to try and understand the types of possiblebehaviour in such rings. One approach, taken by Ashwin and Swift [2], is to assumethat the coupling between oscillators is weak relative to the attractiveness of theperiodic orbit that exists in the phase space of each oscillator. This assumptionallows a decoupling of the dynamics for the phases of each of the oscillators fromthe dynamics of their amplitudes. Since each oscillator has only one phase variableassociated with it, this reduces the dynamical system from one in RNn to one on TN ,where N is the number of oscillators in the ring, the phase space for each uncoupledoscillator is Rn, and TN is the N{torus. The symmetry group of N oscillators coupledin a ring with no preferred direction around the ring is the dihedral group DN , i.e. thegroup of symmetries of a regular N{gon. The ow on TN inherits this symmetry, sothe problem of �nding the possible dynamics for the ring of identical oscillators hasreduced to that of �nding possible types of DN{equivariant ows on TN . (Under theassumption that the di�erences in phase between oscillators vary on a slower timescale than the oscillatory motion of each oscillator, the DN{equivariant ow on TNcan be averaged, resulting in a DN{equivariant ow on TN�1.)Grasman and Jansen [7] use a similar weak coupling assumption, but considerperturbing away from a ring of relaxation oscillators that have an in�nite rate ofrelaxation.The other approach that has had some success is assume that each oscillator isparametrised in such a way that it undergoes a Hopf bifurcation as a parameter isvaried, and to use the theory of Hopf bifurcations, with or without symmetry, to3

determine the types of oscillation produced. This was the approach used by Collinsand Stewart [3] in relation to the modelling of animal gaits. These authors usedresults from the theory of Hopf bifurcation with either DN or ZN symmetry [5, 6] todetermine the possible types of oscillation that could exist, purely as a result of thesymmetry of the network. As they mention, the question of the stability of any ofthese types of oscillation depends on the exact form of the oscillators and couplingused. In this paper we extend their results to show that for quite general oscillators,one speci�c branch of oscillations is stable.When the linearisation of the coupled system about a �xed point has a simple pairof purely imaginary eigenvalues, the theory of simple Hopf bifurcation (see e.g. [8, 12])can be used to determine the type of oscillation produced in that bifurcation, and byappropriate centre manifold reduction calculations, the direction of branching and theconsequent stability of the orbit can be determined. This is the approach taken byAlexander and Auchmuty [1], among others. However, for systems with DN symme-try, most of the eigenvalues are forced to appear as pairs, and simple Hopf bifurcationtheory can no longer be used. This problem can sometimes be resolved in particularcases. For example, in Silber et al. [10], the equations governing the behaviour of aring of semiconductor lasers are such that periodic solutions can be found explicitly,and these can be linearised about to give their stabilities. Ermentrout [4] studied thecase of weak coupling near a Hopf bifurcation and found families of rotating wavesand necessary conditions for their stabilities in the limit of large N . His results are aspecial case of the results we derive.The tools we use are centre manifold and normal form theory. For a general cou-pled system (1) we determine which Hopf bifurcation may produce a stable periodicorbit, perform a centre manifold reduction at this bifurcation point, and then do thenormal form calculations necessary to determine the sub{ or supercriticality of thebranches produced here and their stabilities. For more details on centre manifold andnormal form theory, see [8, 12]. 4

2 Presentation of systemWe assume that our system of N coupled oscillators in a ring is governed by equationsof the form_zj � dzjdt = (�+ i)zj + F2(zj; �zj) + F3(zj; �zj) + (�r + i�i)(2zj � zj+1 � zj�1) (1)where zj 2 C; �;; �r; �i 2 R, i2 = �1, the subscripts (which label the oscillators) aretaken mod N , and F2 and F3 contain the second and third order terms, respectively,in the Taylor series expansion of the vector �eld of an uncoupled oscillator. To bemore explicit, we write F2(zj; �zj) = �1z2j + �2zj�zj + �3�z2jand F3(zj; �zj) = �1z3j + �2z2j �zj + �3zj�z2j + �4�z3jwhere the �'s and �'s are complex constants. We take � as our bifurcation parameter,which we assume to be close to zero. The �'s and �'s will typically depend on �, butbecause we are only concerned with the case j�j � 1, we �x them at the value theyhave when � = 0.We assume that each uncoupled oscillator undergoes a supercritical Hopf bifurca-tion as � increases through zero. This means that the normal form coe�cient, a, foreach oscillator is negative. For a system0B@ _x_y 1CA = 0B@ 0 �!! 0 1CA0B@ xy 1CA + 0B@ f(x; y)g(x; y) 1CA x; y 2 Rwith f(0; 0) = g(0; 0) = 0 and Df(0; 0) = Dg(0; 0) = 0 at a simple Hopf bifurcation,the expression for a is [8]a = 116 [fxxx + fxyy + gxxy + gyyy ] + 116! [fxy(fxx + fyy)� gxy(gxx + gyy)� fxxgxx + fyygyy]where subscripts indicate partial di�erentiation with respect to that variable. It isstraight{forward to calculate this quantity in terms of the �'s and �'s in (1):a = Ref�2g � Imf�1gRef�2g+Ref�1gImf�2g = Ref�2g � Imf�1�2g (2)5

Note that this expression is linear in coe�cients of cubic terms in the Taylor seriesand quadratic in coe�cients of second order terms | this fact will be used later.The Jacobian of (1) evaluated at the origin is circulant, so we can easily �nd itseigenvalues and eigenvectors. We use the eigenvectors to diagonalise the linear partof the system, de�ning a new coordinate system w by w = Az, wherew = 0BBBBBBBBBBBB@ w1w2...wN�1wN 1CCCCCCCCCCCCA A = 0BBBBBBBBBBBB@ 1 �(N�1)(N�1) : : : �2(N�1) �N�1... ... ... ...1 �2(N�1) � � � �4 �21 �N�1 � � � �2 �1 1 � � � 1 1 1CCCCCCCCCCCCA z = 0BBBBBBBBBBBB@ z1z2...zN�1zN 1CCCCCCCCCCCCA(3)and � = e2�i=N , i.e. Ajk = �(N�j)(N�k+1). After this transformation the linear partof the _w equation is diagonal with entries of � + i + 2(�r + i�i) h1� cos �2�jN �i forj = 0; : : : ; N�1 with the j = 0 entry at bottom right and the j = N �1 entry at topleft. From the form of these diagonal terms we see that there is a Hopf bifurcation(which may be simple or double) from the origin when �+2�r h1� cos �2�jN �i = 0 forsome 0 � j � N � 1.For � < �4j�rj all of the eigenvalues of the Jacobian at the origin have negativereal parts and the origin is stable, so we look for the �rst Hopf bifurcation to occur as� is increased, which we hope will produce a stable branch of oscillations. Once thisHopf bifurcation has occurred the origin is unstable, and all of the orbits subsequentlycreated from it in Hopf bifurcations will necessarily be unstable. First we take thecase �r < 0; the analysis of the case �r > 0 depends on whether N is even or odd, andwe deal with these cases in subsequent sections.3 �r < 0When �r < 0 the �rst Hopf bifurcation to occur as � is increased is the simple onewith j = 0 at � = 0, irrespective of the value of N . At this bifurcation the centremanifold in the w coordinate system is in the wN direction and since the Jacobian isdiagonal and all of its other eigenvalues have negative real part, the dynamics for all6

the other components of w are dominated by exponential contraction onto the centremanifold. We know from (3) that wN = NXj=1 zj;so using this and (1) we have_wN = NXj=1 _zj= NXj=1f(�+ i)zj + F2(zj; �zj) + F3(zj; �zj) + (�r + i�i)(2zj � zj�1 � zj+1)g= (� + i)wN + NXj=1fF2(zj; �zj) + F3(zj; �zj)g (4)= (� + i)wN + F2(wN ; �wN)N + [: : : 2 : : : ] + F3(wN ; �wN)N2 + [: : : 3 : : : ]where [: : : 2 : : : ] represents second order terms inw1; �w1; : : : ; wN�1; �wN�1 and [: : :3 : : : ]represents all cubic terms that include at least one of w1; �w1; : : : ; wN�1; �wN�1. Thesecond and fourth terms in the last line of (4) were obtained by using the inverse ofA, A�1jk = � 1N � �(j�1)(N�k);so that zj = NXk=1A�1jk wk = 1N NXk=1 �(j�1)(N�k)wk (5)and substituting into the expressions for F2(zj; �zj) and F3(zj; �zj).There is a subtle point here regarding the di�erence between [: : :2 : : : ] and [: : :3 : : : ].We might expect there to be second order terms in (4) of the form wNwk, wN �wk, �wNwkor �wN �wk for some k 6= N , and when we then perform a centre manifold reduction (inwhich we write wk and �wk as a sum of second{order terms in wN and �wN) these termswill be third order in wN ; �wN . However, it is possible to calculate the coe�cients ofthe second order terms in wNwk, wN �wk, �wNwk and �wN �wk for any k 6= N , and they areall zero. Hence, when we perform the centre manifold reduction all terms in [: : :2 : : : ]are fourth order in wN ; �wN and will therefore be ignored. For a similar reason, all7

terms in [: : :3 : : : ] will be of order at least 4 (possibly up to 6) and will similarly beignored. Thus, after performing the centre manifold reduction we can write (4) as_wN = (� + i)wN + F2(wN ; �wN)N + F3(wN ; �wN)N2 +O(jwN j4) (6)Comparing this with (1) at (�r; �i) = (0; 0) we see that to third order they are identicalexcept that the coe�cients of the second order terms in (6) have been divided by Nand the third order ones by N2. Going back to the expression for a (equation (2)),we see that the value of a for (6) is equal to that for an uncoupled oscillator dividedby N2. Since only the sign of a is important, we see that in the dynamics restrictedto the centre manifold, (6), there is a supercritical Hopf bifurcation as � increasesthrough 0 when �r < 0. To see how this manifests itself in the oscillators, we use theinverse of A, (5). Since wN is small at the onset of oscillation and the wk for k 6= Nare second order in wN , we can ignore their contribution to the zj. From (5) we seethat zj � wN for all j, i.e. this branch of orbits manifests itself as the completelysynchronised state, zj = zk for all j; k. The next case we do is �r > 0, N even.4 �r > 0, N evenIn this case the �rst Hopf bifurcation to occur as � is increased is the simple one forj = N=2 at � = �4�r. The centre manifold is in the wN=2 direction and as above, thedynamics in the other directions of w are dominated by exponential contraction ontothe centre manifold. From (3) we havewN=2 = NXj=1 �AN2 j� zj = NXj=1(�1)N�j+1zj8

so using (1) we obtain_wN=2 = NXj=1(�1)N�j+1 _zj= NXj=1(�1)N�j+1f(� + i)zj + F2(zj; �zj) + F3(zj; �zj) + (�r + i�i)(2zj � zj+1 � zj�1)g= [�+ i + 4(�r + i�i)]wN=2 + NXj=1(�1)N�j+1fF2(zj; �zj) + F3(zj; �zj)g= [�+ i + 4(�r + i�i)]wN=2 (7)+ 1N h2�1wN=2wN + �2(wN=2 �wN + �wN=2wN ) + 2�3 �wN=2 �wNi+ [: : :2 : : : ] + F3(wN=2; �wN=2)N2 + [: : :3 : : : ]where [: : :2 : : : ] now represents second order terms in w1; �w1; : : : ; wN ; �wN that have nofactors of wN=2 or �wN=2, and [: : :3 : : : ] represents cubic terms not composed exclusivelyof wN=2 and �wN=2. After performing the centre manifold reduction, terms in [: : :2 : : : ]and [: : :3 : : : ] will be of order at least 4 in wN=2 and �wN=2, and will therefore be ignoredfrom now on. The last step is to actually perform (part of) the centre manifoldreduction in order to write wN (and hence �wN) as a function of wN=2 and �wN=2 sothat we can substitute these expressions into (7) and obtain an equation for motionon the centre manifold.We start by writingwN = g(wN=2; �wN=2) = �1w2N=2 + �2wN=2 �wN=2 + �3 �w2N=2 (8)as an approximation to the centre manifold, where �1;2;3 2 C are (as yet) unknown.We �nd them by equating two equivalent expressions for _wN :_wN = @g@wN=2 _wN=2 + @g@ �wN=2 _�wN=2 = (�+ i)g(wN=2; �wN=2) + f2(wN=2; �wN=2) (9)where f2(wN=2; �wN=2) are the terms in the [: : :2 : : : ] of (4) involving only wN=2 and�wN=2. It is straight{forward to show that f2(wN=2; �wN=2) is in fact F2(wN=2; �wN=2)=N .9

Thus (9) becomes[2�1wN=2 + �2 �wN=2]� [�+ i + 4(�r + i�i)]wN=2+ [�2wN=2 + 2�3 �wN=2]� [�� i + 4(�r � i�i)] �wN=2= (� + i)� [�1w2N=2 + �2wN=2 �wN=2 + �3 �w2N=2] + F2(wN=2; �wN=2)=NWe equate coe�cients of second order terms in wN=2 and �wN=2 in this expression toget�1 = �1N [�+ i + 8(�r + i�i)]; �2 = �2N [�� i + 8�r]; �3 = �3N [�� 3i + 8(�r � i�i)]Now that we have found �1; �2 and �3, we can substitute the expression for wN , (8),into (7) to obtain the equation for motion on the centre manifold. It is_wN=2 = [�+ i + 4(�r + i�i)]wN=2 + Cw2N=2 �wN=2 + [: : : iii : : : ]where C = �2N2 + 2�1�2N + �2N (��2 + �1) + 2�3��3Nand [: : : iii : : : ] represents all terms of order 3 or more in the Taylor series excludingthe term in w2N=2 �wN=2. If the real part of C is negative at the bifurcation value(� = �4�r), we will have a supercritical Hopf bifurcation in the wN=2 direction as �increases through �4�r.For small �r; �i we can expand C at � = �4�r in powers of �r and �i asC = 1N2 ��2 + i�1�2 � i�2��2 � i2�3��33 �+ �rN22 �12�1�2 + 4�2��2 + 8�3��39 �+ �iN22 �8i�1�2 + 16i�3��39 �+O(j�r; �ij2)and thusRefCg = 1N2 �a+ �r2 �12Ref�1�2g+ 4�2��2 + 8�3��39 �� �i2 (8Imf�1�2g)�+ O(j�r; �ij2)where a is as given in (2). Thus for small enough �r and �i, RefCg will be negative,and we will have a supercritical Hopf bifurcation in the wN=2 direction. As before, we10

use (5) to see how this branch of oscillations appears in the oscillators. We �nd thatzj � (�1)j�1wN=2, i.e. zj+1 = �zj for all j and thus neighbouring oscillators are halfa period out of phase with one another. We call this the exact antiphase state. Thelast case to do is �r > 0, N odd.5 �r > 0, N oddIn this case the �rst Hopf bifurcation to occur as � increases is double, correspondingto both j = (N + 1)=2 and j = (N � 1)=2, at � = �2�r[1 + cos ( �N )]. For notationalconvenience, we de�ne p � (N�1)=2 and q � (N+1)=2. At this bifurcation the centremanifold is four{dimensional, in the directions of wp and wq, and there are genericallythree branches of periodic orbits (labelled eZN ;Z2(�) and Z2(�; �) by Golubitsky etal. [6]) that emanate from the double Hopf bifurcation. The sub{ or supercriticalityof these branches and their stability or otherwise depends on the coe�cients of thecubic terms in the normal form of the equations on the centre manifold. We derivethese coe�cients below for (1) in terms of the equation for an uncoupled oscillator.Once we have these coe�cients, we can compare them with those in the normalform of the DN symmetric double Hopf bifurcation (see x3, Ch. XVIII of [6]) whichwe choose to write as follows:_u1 = �u1 +Bju1j2u1 + Cju2j2u1 +O(ju1; u2j5) (10)_u2 = �u2 +Bju2j2u2 + Cju1j2u2 +O(ju1; u2j5)where all parameters and variables (except time) are complex and Ref�g is the bifur-cation parameter. The di�erence between this presentation of the normal form andthat in [6] is that we have collected all terms of order 5 or more in the \O(ju1; u2j5)"term. We have done this because, although higher order terms are necessary to de-termine the stability of the Z2(�) and Z2(�; �) branches, we show below that for j�rjand j�ij small enough there is a bifurcation to a stable eZN branch, the direction of bi-furcation and stability of this branch being completely determined by the coe�cientsof the cubic terms, B and C; thus the exact form of the higher order terms is not11

relevant.The normal form has three types of nontrivial solution:1. u1 = u2, which we associate with the Z2(�) orbit,2. u1 = �u2, which we associate with the Z2(�; �) orbit, and3. either (u1; u2) = (u1; 0) or (u1; u2) = (0; u2), both of which we associate witheZN orbits.The bifurcation set for the normal form (10) is shown in Figure 1, which is reparametrisedversion of Figure 3.1 in Ch. XVIII of [6] showing how the bifurcation diagrams for (10)depend on the real parts of B and C. See [6] for more details.De�ning �� � �(N�1)=2 and �+ � �(N+1)=2, where � = e2�i=N , we have, using (3)wq = NXk=1 �N�k+1� zk and wp = NXk=1 �N�k+1+ zkDi�erentiating the �rst of these with respect to time and using (1) we have_wq = NXk=1 �N�k+1� f(�+ i)zj + F2(zj; �zj) + F3(zj; �zj) + (�r + i�i)(2zj � zj+1 � zj+1)g= f�+ i + 2(�r + i�i)[1 + cos (�=N)]gwq + NXk=1 �N�k+1� fF2(zk; �zk) + F3(zk; �zk)g= f�+ i + 2(�r + i�i)[1 + cos (�=N)]gwq (11)+ [2�1wqwN + �2(wq �wN + �wqw1) + 2�3 �wq �wN�1]=N+ [2�1wpw1 + �2(wp �wN�1 + �wpwN) + 2�3 �wp �wN ]=N + [: : : ii : : : ]+ �2N2 [w2q �wq + 2wqwp �wp] + [: : : iii : : : ]where [: : : ii : : : ] represents second order terms with no factors of wq; �wq; wp or �wp,and [: : : iii : : : ] represents cubic terms excluding those of the form w2q �wq and wqwp �wp.When the centre manifold reduction is performed terms in [: : : ii : : : ] and [: : : iii : : : ]will be of order at least 4 in jwpj and jwqj, or if not, can be removed with normalform transformations, and will thus be ignored from now on. We obtain an expressionanalogous to (11) for _wp, with wp and wq exchanged, as expected from the symmetryof the problem. 12

The next step is to perform the centre manifold reduction in order to get expres-sions for w1; wN and wN�1 in terms of wq; �wq; wp and �wp so that we can substitutethem into (11). We writew1 = f(wq; �wq; wp; �wp) = 1w2p + 2wp �wp + 3wpwq + 4wp �wq + 5 �w2p+ 6 �wpwq + 7 �wp �wq + 8w2q + 9wq �wq + 10 �w2qwN = g(wq; �wq; wp; �wp) = �1w2p + �2wp �wp + �3wpwq + �4wp �wq + �5 �w2p (12)+ �6 �wpwq + �7 �wp �wq + �8w2q + �9wq �wq + �10 �w2qwN�1 = h(wq; �wq; wp; �wp) = �1w2p + �2wp �wp + �3wpwq + �4wp �wq + �5 �w2p+ �6 �wpwq + �7 �wp �wq + �8w2q + �9wq �wq + �10 �w2qwhere 1; : : : ; �10 2 C are unknown coe�cients. We �nd them in the usual way| writing equivalent expressions for each of _w1, _wN and _wN�1 and then equatingcoe�cients of like powers of wq; �wq; wp and �wp. By substituting the expressions (12)into (11), we can see that after the centre manifold reduction has been performed thecoe�cient of the term in w2q �wq in (11) will beB � �2N2 + 2�1�9 + �2(��9 + 8) + 2�3��10N (13)while that of the term in wqwp �wp will beC � 2�2N2 + 2�1�2 + �2��2 + 2�1 6 + �2(��4 + �3) + 2�3��7N (14)(We have made the correspondence u1 = wq and u2 = wp for comparison between (11)and (10).) Actually doing the centre manifold reduction, i.e. �nding 1; : : : ; �10, we13

see that at the double Hopf bifurcation (i.e. when � = �2�r[1 + cos (�=N)])�9 = i�2N + �r � 2�22N (1 + cos (�=N))�+O(j�r; �ij2) 8 = � i�1N + �r � 2�12N (cos (�=N) + cos (2�=N))�+ �i � 2i�1N2 (1 + 2 cos (�=N) + cos (2�=N))�+O(j�r; �ij2)�10 = i�33N + �r � 2�392N (cos (�=N) + cos (2�=N))�+ �i � 2i�39N2 (cos (2�=N)� 2 cos (�=N) � 3)�+O(j�r; �ij2)�2 = i�2N + �r � 2�22N (1 + cos (�=N))�+O(j�r; �ij2) 6 = i�2N + �r � 2�22N (cos (�=N) + cos (2�=N))�+ �i � 2i�22N (cos (2�=N) � 1)�+O(j�r; �ij2)�4 = i�2N + �r � 2�22N (cos (�=N) + cos (2�=N))�+ �i � 2i�2N2 (cos (2�=N) � 1)�+O(j�r; �ij2)�3 = �2i�1N + �r � 4�12N (1 + cos (�=N))�+ �i � 8i�12N (1 + cos (�=N))�+O(j�r; �ij2)�7 = 2i�33N + �r � 4�392N (1 + cos (�=N))�+ �i ��8i�39N2 (1 + cos (�=N))�+O(j�r; �ij2)for small j�rj; j�ij. Substituting these expansions into the expressions for B and C (13{14) we obtainRefBg = 1N2 "Ref�2g � Imf�1�2g # (15)+ �rN22 �2Ref�1�2g�2 + 3 cos � �N �+ cos�2�N ��+ 2j�2j2 �1 + cos� �N ��+ 4j�3j29 �cos� �N �+ cos�2�N ��#� �iN22 �2Imf�1�2g�1 + 2 cos � �N �+ cos�2�N ���+O(j�r; �ij2)14

and RefCg = 2N2 "Ref�2g � Imf�1�2g # (16)+ �rN22 �4Ref�1�2g�2 + 3 cos� �N �+ cos�2�N ��+ 2j�2j2 �1 + 2 cos � �N �+ cos�2�N ��+ 8j�3j29 �1 + cos� �N ��#� �iN22 �4Imf�1�2g�1 + 2 cos� �N �+ cos�2�N ���+O(j�r; �ij2)Looking at these expressions when �r = �i = 0, we see that for �r; �i small enough, weare in the region RefCg < RefBg < 0 of Figure 1 (asRef�2g � Imf�1�2g = a < 0)and thus there is a supercritical Hopf bifurcation to the eZN oscillation as � increasesthrough �2�r[1+ cos (�=N)]. The eZN branch corresponds to two types of oscillation,depending on whether solutions of (11) and its symmetric counterpart are of the form(wp; wq) = (wp; 0) or (wp; wq) = (0; wq). Using (5) we see that if wp = 0 then zj ��j�1� wq, i.e. zj+1 = (��)zj, so the phase di�erence between neighbouring oscillators is�1 + 1N ��. Similarly, if wq = 0, then zj � �j�1+ wp, i.e. zj+1 = (�+)zj, so the phasedi�erence between neighbouring oscillators is �1 � 1N ��. Thus we have shown thatas � increases through �2�r [1 + cos (�=N)] we have a supercritical bifurcation to twoconjugate stable rotating waves, one rotating in each direction, both of which have themaximum possible magnitude of phase di�erence between neighbouring oscillators.6 An asideLooking at (15{16) we see that for �r; �i small we have RefCg � 2RefBg, andthese are either both positive or both negative, depending on whether a is positive ornegative, respectively. Looking at Figure 1, we see that the case a > 0 (correspondingto a subcritical Hopf bifurcation in an uncoupled oscillator) is uninteresting, in thatthe DN{symmetric Hopf bifurcation will not create any stable orbits. The only sectorof interest in this respect is the one whereRefBg < RefCg < �RefBg; for RefBg < 015

because here we have the creation of either a stable Z2(�) orbit or a stable Z2(�; �)orbit in the double Hopf bifurcation. A simple rearrangement of the above expressionsfor RefBg (15) and RefCg (16) givesRefCg = 2RefBg (17)+ �rN22 "2j�2j2�cos�2�N �� 1�+ 8j�3j29 �1� cos�2�N ��#+O(j�r; �ij2);so by choosing various parameters correctly, it may be possible to push the normalform for the oscillators from the line RefCg = 2RefBg < 0 that we know we areon for �r = �i = 0 across the boundary RefCg = RefBg < 0 by increasing �r, asshown schematically in Figure 2 (compare with Figure 1, which shows the bifurcationdiagrams in each sector). (Note that as N increases, the coe�cient of the term in�r in (17) decreases, all other things being equal.) We demonstrate this transitionbelow in an example for N = 3, the smallest number of oscillators to have the threedi�erent types of orbit created in a double Hopf bifurcation.We are in the region �r > 0, and we want to increase RefCg, so we set �2 = 0.For simplicity we also set �i; �1; �3 and �4 to be zero. The example we use is_zj = (� + 1:5i)zj � 0:7z2j + 2z2j � jzjj2zj + �(2zj � zj+1 � zj�1) (18)for j = 1; 2; 3 and the subscripts are taken mod 3, which corresponds to equation (1)with = 1:5; �1 = �0:7; �3 = 2; �2 = �1 and �r = �. For this system,RefBg = �19 + �9� 1:52 "4� 229 �cos��3�+ cos�2�3 ��# +O(�2) = �19 +O(�2)so using (17), to �rst order in � we should get a transition from Hopf bifurcation toa stable eZ3 orbit to Hopf bifurcation to a stable Z2 orbit of some kind whenRefCg � 2RefBg = �RefBgi.e. �N22 "8j�3j29 �1 � cos�2�N ��# = �9 � 1:52 "8 � 229 �1 � cos�2�3 ��# = 19 ;i.e. � = 27=64 � 0:42. That this transition does occur is demonstrated in Figure 3,where we show the type of orbit that is stable near the double Hopf bifurcation for (18)16

as a function of �. This clearly shows that for � <� 0:45, there is a Hopf bifurcationto a stable eZ3 orbit, while for � >� 0:45, the bifurcation is to a stable Z2(�; �) orbit,in good agreement with the predicted behaviour.7 Conclusion and further workWe have shown that if we take N identical oscillators, each of which undergoes asupercritical Hopf bifurcation as a parameter say, �, increases, and couple them dif-fusively in a ring geometry with complex coupling constant �r + i�i, then (for smallenough j�rj and j�ij) the system as a whole undergoes a supercritical Hopf bifurcationto a stable rotating wave state as � is increased, and that the magnitude of the phasedi�erence between neighbouring oscillators is a maximum for �r > 0 (i.e. � for Neven and (1 � 1=N)� for N odd) and a minimum (i.e. zero, corresponding to thesynchronised state) for �r < 0.Further extensions could include coupling that is not restricted to nearest{neighbour,while still preserving the DN symmetry of the system, or using nonlinear coupling,although we would need the coupling to have some linear component in order to splitthe eigenvalues of the Jacobian into single and double pairs.AcknowledgementsThe author is supported by the Cambridge Commonwealth Trust.17

References[1] J. C. Alexander and G. Auchmuty.Global bifurcations of phase{locked oscillators.Arch. Rat. Mech. Anal. 93. No. 3. p. 253. 1986.[2] P. Ashwin and J. W. Swift. The dynamics of N weakly coupled identical oscilla-tors. J. Nonlinear Sci. 2, No. 1. p. 69. 1992.[3] J. J. Collins and I. Stewart. A group{theoretic approach to rings of coupled bio-logical oscillators. Biol. Cybern. 71. p. 95. 1994.[4] G. B. Ermentrout. The behavior of rings of coupled oscillators. J. Math. Biol. 23p. 55. 1985.[5] M. Golubitsky and I. N. Stewart. Hopf bifurcation with dihedral group symmetry:coupled nonlinear oscillators. In Multiparameter Bifurcation Theory. Contempo-rary Mathematics 56 p. 131. 1986.[6] M. Golubitsky, I. Stewart and D. Schae�er. Singularities and Groups in Bifurca-tion Theory, Vol II. Applied Mathematical Sciences 69. Springer{Verlag 1988.[7] J. Grasman and M. J. W. Jansen. Mutually Synchronized Relaxation Oscillatorsas Prototypes of Oscillating Systems in Biology. J. Math. Biology. 7. p. 171. 1979.[8] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems,and Bifurcations of Vector Fields. Applied Mathematical Sciences 42. Springer{Verlag 1990. (Corrected third printing.)[9] S. Lubkin. Unidirectional waves on rings: models for chiral preference of circum-nutating plants. Bull. Math. Biol. 56, No. 5. p. 795. 1994.[10] M. Silber, L. Fabiny and K. Wiesenfeld. Stability results for in{phase and splay{phase states of solid{state laser arrays. J. Opt. Soc. Am. B 10, No. 6. p. 1121.1993. 18

[11] A. M. Turing. The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lon-don. B237. p.37. 1952.[12] S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos.Springer{Verlag Texts in Applied Mathematics. 2. 1990.

19

List of Figures1 Generic bifurcation set for the normal form of the DN symmetric Hopfbifurcation (10) for N � 3, N 6= 4, after Figure 3.1, Ch. XVIII of [6].Within each sector of the (RefCg; RefBg) plane is a schematic bi-furcation diagram with Ref�g horizontally and some measure of theorbit vertically. Solid lines refer to stable solutions and dotted to un-stable. Note that for any branch to be stable all must be supercritical,and then at most one branch is stable. Fifth{order terms in the normalform may interchange the Z2(�) and Z2(�; �) orbits. We have assumedthat the origin is stable for Ref�g < 0. (We use \supercritical" and\subcritical" to refer to the direction in which a branch of orbits iscreated as Ref�g is increased: supercritical branches are created asRef�g is increased, while subcritical are created as Ref�g is decreased.) 212 Schematic diagram showing how it might be possible to move from theline RefCg = 2RefBg < 0, which we know we are on at �r = �i = 0,across the line RefBg = RefCg < 0 by increasing �r. Compare withFigure 1, which shows bifurcation diagrams for the relevant sectors. . 223 Transition from Hopf bifurcation to a stable eZ3 orbit to Hopf bifur-cation to a stable Z2(�; �) orbit as � is varied in equation (18). Inregion F the eZ3 orbit is stable, and in region G, the Z2(�; �) orbit isstable. There is non{periodic behaviour in the wedge H. The verticalcoordinate is the distance in � from the Hopf bifurcation. . . . . . . . 2320

���������������@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Z2(�)Z2(�; �)eZN eZNZ2(�)Z2(�; �)

Z2(�; �)Z2(�)eZNZ2(�; �)Z2(�)eZN eZNZ2(�)Z2(�; �)6 - RefCgRefBgRefB + Cg = 0RefBg = RefCg

RefBg = 0Figure 1: Generic bifurcation set for the normal form of the DN symmetric Hopfbifurcation (10) for N � 3, N 6= 4, after Figure 3.1, Ch. XVIII of [6]. Withineach sector of the (RefCg; RefBg) plane is a schematic bifurcation diagram withRef�g horizontally and some measure of the orbit vertically. Solid lines refer tostable solutions and dotted to unstable. Note that for any branch to be stable allmust be supercritical, and then at most one branch is stable. Fifth{order terms inthe normal form may interchange the Z2(�) and Z2(�; �) orbits. We have assumedthat the origin is stable for Ref�g < 0. (We use \supercritical" and \subcritical" torefer to the direction in which a branch of orbits is created as Ref�g is increased:supercritical branches are created as Ref�g is increased, while subcritical are createdas Ref�g is decreased.) 21

Re{B}=Re{C}/2

Re{C}

Re{B}Re{B}=Re{C}

eps=0

Figure 2: Schematic diagram showing how it might be possible to move from theline RefCg = 2RefBg < 0, which we know we are on at �r = �i = 0, across theline RefBg = RefCg < 0 by increasing �r. Compare with Figure 1, which showsbifurcation diagrams for the relevant sectors.22

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

ε

λ+3ε

F

G

H

Figure 3: Transition from Hopf bifurcation to a stable eZ3 orbit to Hopf bifurcationto a stable Z2(�; �) orbit as � is varied in equation (18). In region F the eZ3 orbit isstable, and in region G, the Z2(�; �) orbit is stable. There is non{periodic behaviourin the wedge H. The vertical coordinate is the distance in � from the Hopf bifurcation.23