Phase-coupled oscillators serve as paradigmatic models of networks of weakly
interacting oscillatory units in physics and biology. The order parameter which
quantifies synchronization was so far found to be chaotic only in systems with
inhomogeneities. Here we show that even symmetric systems of identical
oscillators may not only exhibit chaotic dynamics, but also chaotically
fluctuating order parameters. Our findings imply that neither inhomogeneities
nor amplitude variations are necessary to obtain chaos, i.e., nonlinear
interactions of phases give rise to the necessary instabilities.Comment: 4 pages; Accepted by Physical Review Letter

A. Pikovsky

Christian Bick

Danilo Paulikat

Dirk Rathlev

M. Golubitsky

Marc Timme

P. Ashwin

Peter Ashwin

Y. Kuramoto

English

arXiv

Crossref

Physical Review Letters

Chaos in Symmetric Phase Oscillator Networks

Phase-coupled oscillators serve as paradigmatic models of networks of weakly interacting oscillatory units in physics and biology. The order parameter which quantifies synchronization so far has been found to be chaotic only in systems with inhomogeneities. Here we show that even symmetric systems of identical oscillators may not only exhibit chaotic dynamics, but also chaotically fluctuating order parameters. Our findings imply that neither inhomogeneities nor amplitude variations are necessary to obtain chaos; i e., nonlinear interactions of phases give rise to the necessary instabilities

Bick, Christian

Timme, Marc

Paulikat, Danilo

Rathlev, Dirk

Ashwin, Peter

GRO.publications (Univ. Göttingen)

Max Planck Society eDoc Server

Chaos in Symmetric Oscillator Networks

Bick, C.

Timme, M.

Paulikat, D.

Rathlev, D.

Ashwin, P.

MPG.PuRe

Copyright © 2011 American Physical SocietyThis is the final version. Available from the American Physical Society via the DOI in this recordPhase-coupled oscillators serve as paradigmatic models of networks of weakly interacting oscillatory units in physics and biology. The order parameter which quantifies synchronization so far has been found to be chaotic only in systems with inhomogeneities. Here we show that even symmetric systems of identical oscillators may not only exhibit chaotic dynamics, but also chaotically fluctuating order parameters. Our findings imply that neither inhomogeneities nor amplitude variations are necessary to obtain chaos; i.e., nonlinear interactions of phases give rise to the necessary instabilities

Open Research Exeter

Chaos in Symmetric Phase Oscillator NetworksChristian Bick,1,2 Marc Timme,1,3 Danilo Paulikat,3 Dirk Rathlev,3 and Peter Ashwin41Network Dynamics Group, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37073 Go¨ttingen, Germany2Institute for Mathematics, Georg-August-Universita¨t Go¨ttingen, 37073 Go¨ttingen, Germany3Faculty of Physics, Georg-August-Universita¨t Go¨ttingen, 37077 Go¨ttingen, Germany4Mathematics Research Institute, University of Exeter, Exeter EX4 4QF, United Kingdom(Received 9 May 2011; published 9 December 2011)Phase-coupled oscillators serve as paradigmatic models of networks of weakly interacting oscillatoryunits in physics and biology. The order parameter which quantifies synchronization so far has been foundto be chaotic only in systems with inhomogeneities. Here we show that even symmetric systems ofidentical oscillators may not only exhibit chaotic dynamics, but also chaotically fluctuating orderparameters. Our findings imply that neither inhomogeneities nor amplitude variations are necessary toobtain chaos; i.e., nonlinear interactions of phases give rise to the necessary instabilities.DOI: 10.1103/PhysRevLett.107.244101 PACS numbers: 05.45.a, 02.20.aIntroduction.—Models of coupled oscillators describevarious collective phenomena in natural and artificial sys-tems, including the synchronized flashing of fireflies, thedynamics of superconducting Josephson junctions, oscil-latory neural activity, and oscillations in chemical reactionkinetics [1]. In particular, phase-coupled models serve asparadigmatic approximations for many weakly coupledlimit cycle oscillators [2,3]. The Kuramoto model (andits extensions) provides the gold standard in this fieldbecause it suitably describes the dynamics of a variety ofreal systems, is extensively studied numerically and rea-sonably understood analytically [4,5]. Each oscillator kwith phase ’kðtÞ 2 R=2Z ¼: T on the 1-torus T changeswith time t according tod’kdt¼ !k þ 1NXNj¼1gð’k  ’jÞ (1)for all k 2 f1; . . . ; Ng. For the original Kuramoto model thecoupling function g has a single Fourier mode, g ¼ sin.The dimension of such systems can be reduced to lowdimensions [6–8], implying dynamics that is either peri-odic or quasiperiodic. For coupling functions with two ormore Fourier components the collective dynamics may bemuch more complicated. For example, stable heteroclinicswitching may emerge [9,10]. More irregular, chaoticdynamics of system (1) is observed for nonidentical oscil-lators only [11,12], raising the question whether inhomo-geneities are necessary for the occurrence of suchdynamics. To the best of our knowledge, the only hintthat chaotic dynamics might exist for symmetric phase-oscillator networks are attractors with irregular structurein phase space found recently in a system of N ¼ 5oscillators [9].The complex order parameterRðtÞ ¼ 1NXNj¼1expði’jðtÞÞ 2 C (2)where i ¼ ﬃﬃﬃﬃﬃﬃﬃ1p constitutes an important characteristic forcoupled oscillator systems. In particular, its absolute valuejRðtÞj quantifies their degree of synchrony with jRðtÞj ¼ 1if all oscillators are in phase. For the original Kuramotosystem the full complex order parameter (2) acts as a meanfield variable enabling closed-form analysis [13].In homogeneous, globally coupled systems (1) it re-mains unknown whether or not there exists any couplingfunction g that gives rise to chaotic order parameterfluctuations. Synchronous solutions, antisynchronoussplay states as well as the dynamics of cluster states (where’k ¼ ’j for at least one k  j) all yield a periodic com-plex order parameter RðtÞ. For any invariant solutions ontori, RðtÞ is either periodic or quasiperiodic. The mostirregular dynamics of RðtÞ observed so far is due to hetero-clinic cycles where RðtÞ is nonperiodic as it ‘‘slows down’’each cycle. Even if chaotic dynamics does emerge withinthe system, it may average out due to symmetry, possiblyresulting in a regular dynamics of the order parameter.In this Letter, we answer the question whether chaoticdynamics, and moreover, chaotic order parameter fluctua-tions may arise for some g even in the absence of inhomo-geneities in (1). We show that indeed chaos is notpossible for N < 4. By contrast, for N ¼ 4, chaotic attrac-tors can appear in a specific family of coupling functions g.Interestingly, attractors of all theoretically possible sym-metries exist and we provide further examples of attractingchaos for N ¼ 5 and N ¼ 7. The existence of chaos forinfinite families of N  4 implies that chaos occurs insystems with certain N >N0 for any N0 2 N and suggeststhat chaos is likely to occur in many high-dimensionalsystems with a suitable choice of g.No chaos for N ¼ 2 and N ¼ 3.—Let TN be theN-dimensional torus and let SN denote the group of per-mutations of N symbols. Suppose M is a differentiablemanifold and let  be a group that acts on M. Recall thata vector field X on M is called  equivariant if XPRL 107, 244101 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending9 DECEMBER 20110031-9007=11=107(24)=244101(4) 244101-1  2011 American Physical Society‘‘commutes’’ with the action of , i.e., X   ¼ ^  X forall  2  where ^ denotes the induced action on thetangent space.Equivariance implies restrictions of the dynamics speci-fied by the vector field. We study the dynamical system onTN given by the ordinary differential equations (1). Let ushenceforth assume that the system is homogeneous, i.e.,!k ¼ ! for all k 2 f1; . . . ; Ng. This system is SN  T1equivariant where SN acts by permuting indices and T1through a phase shift. Recall some basic properties of thissystem [2]. Introducing phase differences c j :¼ ’j  ’1for all j 2 f1; . . . ; Ng eliminates the phase-shift symmetry.Write ’ ¼ ð’1; . . . ; ’NÞ and c ¼ ðc 1; . . . ; c NÞ. The re-duced system on TN1 is given by_cj ¼ 1NXNk¼1gðc j  c kÞ XNk¼1gðc kÞ(3)for all j 2 f2; . . . ; Ng.For any partition P ¼ fP1; . . . ; Pmg of f1; . . . ; Ng (that isPr  f1; . . . ; Ng,Smj¼1 Pj ¼ f1; . . . ; Ng, and Pr \ Ps ¼ ;for r  s) the subspacesFP :¼ f’jj; k 2 Pr for any r) ’j ¼ ’kg  TN (4)are flow invariant. The subspaces divide TN1 in ðN  1Þ!invariant (N  1)-dimensional simplices [2]; one of whichC :¼ fc j0 ¼ c 1 < c 2   < c N < 2g  TN1 (5)we refer to as the canonical invariant region. There is aZN :¼ Z=NZ symmetry on the canonical invariant regionand the ‘‘splay state’’ (the phase-locked state with c j ¼2j=N) is the only fixed point of this action at the centroidof this region.The reduction of symmetry has implications for theexistence of chaos in low dimensions. For N ¼ 2 andN ¼ 3 the phase space of the reduced system is a one,resp. two-dimensional torus. This means that by thePoincare´-Bendixon theorem [14] chaos is not possible inthese systems for N < 4.Chaos and symmetry for N ¼ 4.—We choose a parame-trization of the coupling function g in (1) by considering atruncated Fourier seriesgð’Þ ¼ X4k¼1ak cosðk’þ kÞ: (6)In particular, we restrict ourselves to the two parameterfamily given by the parametrizationð1; 2; 3; 4Þ ¼ ð1;1; 1 þ 2; 1 þ 2Þ (7)where 1 and 2 are real valued parameters and a1 ¼ 2,a2 ¼ 2, a3 ¼ 1, and a4 ¼ 0:88 are constants.For N ¼ 4, chaotic attractors do indeed exist. Thedynamics of the absolute value of the order parameterexhibits chaotic fluctuations and exponential divergenceof trajectories, cf. Fig. 1. To explore parameter space, wecalculated the maximal Lyapunov exponent max from thevariational equations [15]. There are regions in (1, 2)-parameter space in which max is greater than zero,cf. Fig. 2. As might be expected, there is fine structure inthis region, for example, islands where the trajectory con-verges to a stable limit cycle. Lines of period doublingcascades [16] bound the chaotic region and end in ahomoclinic flip bifurcation with an inclination flip [17](details not shown). Exploring initial conditions revealedthe coexistence of chaotic attractors and stable limit cyclesin part of the chaotic region.Chaotic attractors in equivariant dynamical system canexhibit symmetries themselves. Let A be a chaotic attractoras defined in [18], i.e., a Lyapunov-stable, closed,and connected set that is the !-limit set of a trajectory,for a dynamical system on a manifold M given bya -equivariant vector field. The subgroup StabðAÞ :¼f 2 jðaÞ ¼ a for all a 2 Ag is the group of instanta-neous symmetries of the attractor; i.e., at any point in timethe action of StabðAÞ keeps every point in A fixed.Furthermore, we define ðAÞ :¼ f 2 jðAÞ ¼ Ag to beFIG. 1. Trajectories of the absolute value of the order parame-ter jRðtÞj fluctuate chaotically (N ¼ 4). Two trajectories withsmall difference in initial condition diverge from each other(coupling function (7) for parameter values 1 ¼ 0:1104 and2 ¼ 0:5586).00.020.040.060.080 0.03 0.06 0.09 0.12 0.15 0.1800.020.040.060.080.10.120.14FIG. 2. Chaos for N ¼ 4. Maximal Lypunov exponents arepositive in a region of parameter space. The initial conditionwas fixed and the coupling function parametrized by (7).PRL 107, 244101 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending9 DECEMBER 2011244101-2the set of symmetries on average, and we haveStabðAÞ  ðAÞ as a subgroup.The subdivision of the phase space by flow-invariantregions restricts the possible symmetries of chaotic attrac-tors. The possible symmetries on average of any chaoticattractor A  C with trivial instantaneous symmetriesStabðAÞ ¼ f1g are limited to subgroups of ZN since theyare contained in one of the invariant simplices (C or one ofits images under the group action) with that symmetry. ForN ¼ 4, any chaotic attractor of this type must have trivialinstantaneous symmetry. Thus, the possible symmetries onaverage are limited to subgroups of Z4, i.e., ðAÞ  Z4. Infact, we have found examples of chaotic attractors for eachpossible symmetry in systems of N ¼ 4 and couplingfunctions given by (7) (Fig. 3). Note that this definitionof attractor is somewhat restrictive—Milnor attractors maydisplay a wider range of symmetries including differentinstantaneous symmetries at the same time.Chaos for N > 4.—Analyzing the same region of pa-rameter space for N > 4 yields attracting chaos in systemsof N ¼ 5 and N ¼ 7 oscillators in large regions. Figure 4shows an overlay of regions for three different N; regionsare shaded where the Lyapunov exponent exceeds 0.01 anddarker areas indicate that several N satisfy this condition.Clearly, there is a single coupling function for whichattracting chaos is present for all N ¼ 4, N ¼ 5, andN ¼ 7. Intriguingly, we did not find chaotic attractors forany N 2 f6; 8; 9; . . . ; 13g in the entire region of parameterspace considered in Fig. 4.The parametrization of the coupling function by a trun-cated Fourier series raises the question of how manyFourier components the coupling function needs to containfor chaos to occur. For N ¼ 5 we also measured positiveLyapunov exponents when the coupling was chosen tobe through the simpler coupling function gð’Þ ¼0:2 cosð’þ 1Þ  0:04 cosð2’ 2Þ as in [9]. Hence,in dimension five, coupling functions with only twoFourier components suffice to generate chaotic dynamicswhereas for N ¼ 4, we did not find an example with lessthan four components.From the above, it is clear that for systems of sizeN ¼ KM with K 2 f4; 5; 7g there are chaotic invariantsets lying in flow-invariant subspaces for coupling func-tions yielding positive max, cf. Fig. 4. For instance, forK ¼ 4, these spaces are given by partitions P¼fP1; . . . ;P4gwith jPjj ¼ M for j 2 f1; . . . ; 4g. For N large we similarlycalculated positive maximal Lyapunov exponents for thesystem reduced to asymmetric 4-cluster states given bypartitions P ¼ fP1; . . . ; P4g with jP1j=N ¼ 1=4þ q andFIG. 3. All possible symmetries of the chaotic attractors for different parameters for N ¼ 4. We have 1 ¼ 0:138 in panel (a),1 ¼ 0:0598 in panel (b), 1 ¼ 0:1104 in panel (c), and 2 ¼ 0:5586 in all panels. The projection is a  ¼ S4 equivariant map,x1 ¼ sinð’2  ’4Þ, x2 ¼ sinð’1  ’3Þ, and x3 ¼ jRj.00.050.10.150.20.250.30 0.05 0.1 0.15 0.2 0.25 0.3FIG. 4. Overlapping regions where the maximal Lyapunovexponent is greater than 0.01 for N 2 f4; 5; 7g. The darker thecolor, the more N for which the condition holds. For parametervalues around (0.115, 0.06) there is a region where there is chaosfor all these three N.00.050.10.15-0.005 -0.0025 0 0.0025 0.005FIG. 5. Positive maximal Lyapunov exponents for asymmetric4-cluster states for large systems, N ¼ KMþ Lðq;MÞ  4.Here q parametrizes the deviation from the symmetric clusterstate and L is the corresponding integer dimension (couplingfunction (7) with 1 ¼ 0:1104 and 2 ¼ 0:5586).PRL 107, 244101 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending9 DECEMBER 2011244101-3jPjj=N ¼ 1=4 q=3 for j 2 f2; 3; 4g as depicted in Fig. 5.However, these chaotic invariant sets in invariant subspa-ces close to the symmetric cluster state may be transverselyrepelling, possibly yielding nonchaotic long-termdynamics.Discussion.—Inhomogeneities or asymmetries are thusnot necessary for collective chaotic dynamics to appear in asystem (1), and even chaotic order parameter fluctuationsemerge in the presence of full SN  T symmetry. We high-lighted that for certain coupling functions chaotic attrac-tors exist for several N. However, the regions in parameterspace for which chaotic attractors exist vary drastically,cf. Fig. 4. For certain coupling functions there are chaoticinvariant sets lying in flow-invariant subspaces that corre-spond to the symmetric and near-symmetric cluster statesbut these may not be transversely attracting. The questionremains whether there are coupling functions giving rise tochaotic sets that are actually attracting for N ¼ 6 andN  8. Moreover, is there a ‘‘universal chaos function’’in the sense that there is a coupling function for which thereis some N0 2 N such that there exists a chaotic attractorsfor all (or at least an infinite number of) N >N0?For coupling functions with only one Fourier compo-nent, finite-dimensional systems and the continuum limitare related; the dynamics for both finite N and in thecontinuum limit reduces to effectively two-dimensionaldynamics [7,8] preventing the occurrence of chaotic tra-jectories. Is chaos possible in the continuum limit for morecomplicated coupling functions? If so, how would such aresult relate to chaos in the finite-dimensional systems wehave studied here?For finite systems, attracting chaos in the system doesnot necessarily imply chaotic dynamics of the order pa-rameter since there could be chaotic fluctuations, for ex-ample, in an ‘‘antiphase state.’’ The converse, however,holds. Additionally, observed chaotic fluctuations of theorder parameter cannot necessarily be traced back to in-homogeneity in the system (possibly through an experi-mental setup, cf. [19]) because, as shown above, even fullysymmetric systems can support such dynamics. Whenconsidering the continuum limit, the problem of theseimplications becomes more subtle and will require furtherinvestigation.The coupling function we considered above is written interms of a truncated Fourier series. As discussed above, thenumber of Fourier components is relevant for the dynam-ics. An alternative approach would be to consider suitablepiecewise linear functions. ForN ¼ 4, we find that systemswith piecewise linear g also exhibit positive maximalLyapunov exponents (not shown). Finding a suitable basisfor the space of coupling functions might be a way toexplain some of the dynamical features that were observed.Coupled phase oscillators are a limit of weakly coupledlimit cycle oscillators [2]. In globally coupled identicalGinzburg-Landau oscillator ensembles, chaotic dynamicscan be observed [20,21]. However, it was thought that theamplitude degree of freedom is crucial for the emergenceof such dynamics. Our results show that this is not the caseand, moreover, suggest that chaotic mean field oscillationsare also present in a large class of higher-dimensionalsymmetrically coupled limit cycle oscillators with a richpossible range of chaotic dynamics.C. B. would like to thank Laurent Bartholdi. Supportedby the Federal Ministry for Education and Research(BMBF) Germany under Grant No. 01GQ1005B, a grantby the NVIDIA Corporation, USA, and a grant of the MaxPlanck Society to M. T.[1] A. Pikovsky, M. Rosenblum, and J. Kurths,Synchronization: A Universal Concept in NonlinearSciences (Cambridge University Press, Cambridge,England, 2003).[2] P. Ashwin and J.W. Swift, J. Nonlinear Sci. 2, 69 (1992).[3] J.W. Swift, S. H. Strogatz, and K. Wiesenfeld, Physica(Amsterdam) 55D, 239 (1992).[4] Y. Kuramoto, Chemical Oscillations, Waves andTurbulence (Springer, New York, 1984).[5] J. Acebro´n, L. Bonilla, C. Pe´rez Vicente, F. Ritort, and R.Spigler, Rev. Mod. Phys. 77, 137 (2005).[6] S. Watanabe and S.H. Strogatz, Phys. Rev. Lett. 70, 2391(1993).[7] E. Ott and T.M. Antonsen, Chaos 18, 037113 (2008).[8] S. A. Marvel, R. E. Mirollo, and S. H. Strogatz, Chaos 19,043104 (2009).[9] P. Ashwin, G. Orosz, J. Wordsworth, and S. Townley,SIAM J. Appl. Dyn. Syst. 6, 728 (2007).[10] P. Ashwin, G. Orosz, and J. Borresen, HeteroclinicSwitching in Coupled Oscillator Networks: Dynamics onOdd Graphs (Springer, Berlin, 2010), pp. 31–50,Understanding Complex Systems.[11] O. V. Popovych, Y. L. Maistrenko, and P. A. Tass, Phys.Rev. E 71, 65201 (2005).[12] S. Luccioli and A. Politi, Phys. Rev. Lett. 105, 158104(2010).[13] S. H. Strogatz, Physica D (Amsterdam) 143, 1 (2000).[14] A. J. Schwartz, Am. J. Math. 85, 453 (1963).[15] The time step of the standard Runge-Kutta scheme wastypically chosen to be t ¼ 0:05 and integration rangedover several thousand time units.[16] Numerical continuation of the bifurcations was donewith AUTO-07p, E. J. Doedel et al., (http://indy.cs.concordia.ca/auto/).[17] A. Homburg and B. Krauskopf, J. Dyn. Differ. Equ. 12,807 (2000).[18] M. Golubitsky and I. Stewart, The Symmetry Perspective(Birkha¨user Verlag, Basel, 2002).[19] H. Kori, C. G. Rusin, I. Z. Kiss, and J. L. Hudson, Chaos18, 026111 (2008).[20] V. Hakim and W.-J. Rappel, Phys. Rev. A 46, R7347(1992).[21] N. Nakagawa and Y. Kuramoto, Physica D (Amsterdam)75, 74 (1994).PRL 107, 244101 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending9 DECEMBER 2011244101-4

Chaos in symmetric phase oscillator networks

Chaos in Symmetric Phase Oscillator Networks

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