Regular and Chaotic Dynamics

, Volume 23, Issue 7–8, pp 974–982 | Cite as

Heteroclinic and Homoclinic Structures in the System of Four Identical Globally Coupled Phase Oscillators with Nonpairwise Interactions

  • Evgeny A. GrinesEmail author
  • Grigory V. Osipov


Systems of N identical globally coupled phase oscillators can demonstrate a multitude of complex behaviors. Such systems can have chaotic dynamics for N > 4 when a coupling function is biharmonic. The case N = 4 does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies have shown that some of chaotic attractors in this system are organized by heteroclinic networks. In the present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also discuss some of the cases regarding homoclinic trajectories to saddle-foci equilibria.


phase oscillators heteroclinic networks 

MSC2010 numbers

34C15 37C29 37C80 37E99 


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  1. 1.
    Winfree, A. T., Biological Rhythms and the Behavior of Populations of Coupled Oscillators, J. Theor. Biol., 1967, vol. 16, no. 1, pp. 15–42.CrossRefGoogle Scholar
  2. 2.
    Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics, vol. 19, Berlin: Springer, 1984.Google Scholar
  3. 3.
    Sakaguchi, H. and Kuramoto, Y., A Soluble Active Rotater Model Showing Phase Transitions via Mutual Entertainment, Progr. Theor. Phys., 1986, vol. 76, no. 3, pp. 576–581.CrossRefGoogle Scholar
  4. 4.
    Popovych, O., Maistrenko, Yu., and Tass, P., Phase Chaos in Coupled Oscillators, Phys. Rev. E, 2005, vol. 71, no. 6, 065201, 4 pp.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Watanabe, S. and Strogatz, S., Integrability of a Globally Coupled Oscillator Array, Phys. Rev. Lett., 1993, vol. 70, no. 16, pp. 2391–2394.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Marvel, S., Mirollo, R., and Strogatz, S., Identical Phase Oscillators with Global Sinusoidal Coupling Evolve by Möbius Group Action, Chaos, 2009, vol. 19, no. 4, 043104, 11 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bick, Ch., Ashwin, P., and Rodrigues, A., Chaos in Generically Coupled Phase Oscillator Networks with Nonpairwise Interactions, Chaos, 2016, vol. 26, no. 9, 094814, 8 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ashwin, P., Orosz, G., Wordsworth, J., and Townley, S., Dynamics on Networks of Cluster States for Globally Coupled Phase Oscillators, SIAM J. Appl. Dyn. Syst., 2007, vol. 6, no. 4, pp. 728–758.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bick, Ch., Timme, M., Paulikat, D., Rathlev, D., and Ashwin, P., Chaos in Symmetric Phase Oscillator Networks, Phys. Rev. Lett., 2011, vol. 107, no. 24, 244101, 4 pp.CrossRefGoogle Scholar
  10. 10.
    Ashwin, P. and Rodrigues, A., Hopf Normal Form with SN Symmetry and Reduction to Systems of Nonlinearly Coupled Phase Oscillators, Phys. D, 2016, vol. 325, pp. 14–24.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ashwin, P., and Swift, J. W., The dynamics of n weakly coupled identical oscillators, J. Nonlinear Sci., 1992, vol. 2, no. 1, pp. 69–108.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Tresser, C., About Some Theorems by L.P. Shil’nikov, Ann. Inst. H. Poincaré Physique théorique, 1984, vol. 40, no. 4, pp. 441–461.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Shilnikov, L.P., A Case of the Existence of a Denumerable Set of Periodic Motions, Soviet Math. Dokl., 1965, vol. 6, pp. 163–166; see also: Dokl. Akad. Nauk SSSR, 1965, vol. 160, pp. 558–561.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Lobachevsky State University of Nizhni NovgorodNizhni NovgorodRussia

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