Regular and Chaotic Dynamics

, Volume 22, Issue 2, pp 148–162 | Cite as

“Coherence–incoherence” transition in ensembles of nonlocally coupled chaotic oscillators with nonhyperbolic and hyperbolic attractors

  • Nadezhda I. Semenova
  • Elena V. Rybalova
  • Galina I. Strelkova
  • Vadim S. Anishchenko


We consider in detail similarities and differences of the “coherence–incoherence” transition in ensembles of nonlocally coupled chaotic discrete-time systems with nonhyperbolic and hyperbolic attractors. As basic models we employ the Hénon map and the Lozi map. We show that phase and amplitude chimera states appear in a ring of coupled Hénon maps, while no chimeras are observed in an ensemble of coupled Lozi maps. In the latter, the transition to spatio-temporal chaos occurs via solitary states. We present numerical results for the coupling function which describes the impact of neighboring oscillators on each partial element of an ensemble with nonlocal coupling. Varying the coupling strength we analyze the evolution of the coupling function and discuss in detail its role in the “coherence–incoherence” transition in the ensembles of Hénon and Lozi maps.


ensemble of nonlocally coupled oscillators chimera states solitary states hyperbolic and nonhyperbolic attractors coupling function 

MSC2010 numbers

90B10 34D06 35B36 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kuramoto, Y. and Battogtokh, D., Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators, Nonlin. Phen. Compl. Sys., 2002, vol. 5, no. 4, pp. 380–385.Google Scholar
  2. 2.
    Vadivasova, T.E., Strelkova, G. I., Bogomolov, S.A., and Anishchenko, V. S., Correlation Analysis of the Coherence-Incoherence Transition in a Ring of Nonlocally Coupled Logistic Maps, Chaos, 2016, vol. 26, no. 9, 093108, 9 pp.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Omelchenko, I., Maistrenko, Yu., Hövel, Ph., and Schöll, E., Loss of Coherence in Dynamical Networks: Spatial Chaos and Chimera States, Phys. Rev. Lett., 2011, vol. 106, no. 23, 234102, 4 pp.CrossRefGoogle Scholar
  4. 4.
    Omelchenko, I., Riemenschneider, B., Hövel, Ph., and Schöll, E., Transition from Spatial Coherence to Incoherence in Coupled Chaotic Systems, Phys. Rev. E, 2012, vol. 85, no. 2, 026212, 9 pp.CrossRefGoogle Scholar
  5. 5.
    Omelchenko, I., Provata, A., Hizanidis, J., Schöll, E., and Hövel, Ph., Robustness of Chimera States for Coupled FitzHugh–Nagumo Oscillators, Phys. Rev. E, 2015, vol. 91, no. 2, 022917, 13 pp.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bastidas, V.M., Omelchenko, I., Zakharova, A., Schöll, E., and Brandes, T., Quantum Signatures of Chimera States, Phys. Rev. E, 2015, vol. 92, no. 6, 062924, 5 pp.CrossRefGoogle Scholar
  7. 7.
    Hizanidis, J., Panagakou, E., Omelchenko, I., Schöll, E., Hövel, Ph., and Provata, A., Chimera States in Population Dynamics: Networks with Fragmented and Hierarchical Connectivities, Phys. Rev. E, 2015, vol. 92, no. 1, 012915, 11 pp.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Rosin, D.P., Rontani, D., and Gauthier, D. J., Synchronization of Coupled Boolean Phase Oscillators, Phys. Rev. E, 2014, vol. 89, no. 4, 042907, 7 pp.CrossRefGoogle Scholar
  9. 9.
    Vanag, V. K. and Epstein, I.R., Pattern Formation in a Tunable Medium: The Belousov–Zhabotinsky Reaction in an Aerosol OTMicroemulsion, Phys. Rev. Lett., 2001, vol. 87, no. 22, 228301, 4 pp.CrossRefGoogle Scholar
  10. 10.
    Tinsley, M. R., Nkomo, S., and Showalter, K., Chimera and Phase Cluster States in Populations of Coupled Chemical Oscillators, Nature Phys., 2012, vol. 8, pp. 662–665.CrossRefGoogle Scholar
  11. 11.
    Rogister, F. and Roy, R., Localized Excitations in Arrays of Synchronized Laser Oscillators, Phys. Rev. Lett., 2007, vol. 98, no. 10, 104101, 4 pp.CrossRefGoogle Scholar
  12. 12.
    Böhm, F., Zakharova, A., Schöll, E., and Ludge, K., Amplitude-Phase Coupling Drives Chimera States in Globally Coupled Laser Networks, Phys. Rev. E, 2015, vol. 91, no. 4, 040901, 6 pp.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Levy, R., Hutchison, W.D., Lozano, A.M., and Dostrovsky, J.O., High-Frequency Synchronization of Neuronal Activity in the Subthalamic Nucleus of Parkinsonian Patients with Limb Tremor, J. Neurosci., 2000, vol. 20, no. 20, pp. 7766–7775.Google Scholar
  14. 14.
    Rattenborg, N.C., Amlaner, C. J., and Lima, S. L., Behavioral,Neurophysiological and Evolutionary Perspectives on Unihemispheric Sleep, Neurosci. Biobehav. Rev., 2000, vol. 24, no. 8, pp. 817–842.CrossRefGoogle Scholar
  15. 15.
    Funahashi, S., Bruce, C. J., and Goldman-Rakic, P. S., Neuronal Activity Related to Saccadic Eye Movements in the Monkey’s Dorsolateral Prefrontal Cortex, J. Neurophysiol., 1991, vol. 65, no. 6, pp. 1464–1483.Google Scholar
  16. 16.
    Swindale, N.V., A Model for the Formation of Ocular Dominance Stripes, Proc. R. Soc. Lond. B Biol. Sci., 1980, vol. 208, no. 1171, pp. 243–264.CrossRefGoogle Scholar
  17. 17.
    Omel’chenko, O., Wolfrum, M., and Maistrenko, Yu., Chimera States as Chaotic Spatiotemporal Patterns, Phys. Rev. E, 2010, vol. 81, no. 6, 065201(R), 4 pp.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wolfrum, M., Omel’chenko, O., Yanchuk, S., and Maistrenko, Yu., Spectral Properties of Chimera States, Chaos, 2011, vol. 21, no. 1, 013112.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Semenova, N., Zakharova, A., Schöll, E., and Anishchenko, V., Does Hyperbolicity Impede Emergence of Chimera States in Networks of Nonlocally Coupled Chaotic Oscillators?, Europhys. Lett., 2015, vol. 112, no. 4, 40002, 6 pp.CrossRefGoogle Scholar
  20. 20.
    Bogomolov, S., Slepnev, A., Strelkova, G., Schöll, E., and Anishchenko, V., Mechanisms of Appearance of Amplitude and Phase Chimera States in Ensembles of Nonlocally Coupled Chaotic Systems, Commun. Nonlinear Sci. Numer. Simul., 2017, vol. 43, pp. 25–36.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Dziubak, V., Maistrenko, Yu., and Schöll, E., Coherent Traveling Waves in Nonlocally Coupled Chaotic Systems, Phys. Rev. E, 2013, vol. 87, no. 3, 032907, 5 pp.CrossRefGoogle Scholar
  22. 22.
    Hénon, M., A Two-Dimensional Mapping with a Strange Attractor, Commun. Math. Phys., 1976, vol. 50, no. 1, pp. 69–77.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lozi, R., Un attracteur étrange (?) du type attracteur de Hénon, J. Phys. Colloques, 1978, vol. 39, no.C5, pp.C5-9–C5-10.CrossRefGoogle Scholar
  24. 24.
    Shil’nikov, L.P., A Contribution to the Problem of the Structure of an Extended Neighbourhood of a Rough Equilibrium State of Saddle-Focus Type, Math. USSR-Sb., 1970, vol. 10, no. 1, pp. 91–102; see also: Mat. Sb. (N. S.), 1970, vol. 81(123), no. 1, pp. 92–103.CrossRefMATHGoogle Scholar
  25. 25.
    Anishchenko, V. S., Complex Oscillations in Simple Systems: Mechanisms of the Structure and Properties of Dynamical Chaos in the Radio-Physical Systems, Moscow: Nauka, 1990 (Russian).Google Scholar
  26. 26.
    Afraimovich, V. S. and Shil’nikov, L.P., Strange Attractors and Quasiattractors, in Nonlinear Dynamics and Turbulence, G. I. Barenblatt, G. Iooss, D. D. Joseph (Eds.), Interaction Mech. Math. Ser., Boston,Mass.: Pitman, 1983, pp. 1–34.Google Scholar
  27. 27.
    Kuznetsov, S.P. and Pikovsky, A. S., Universality and Scaling of Period-Doubling Bifurcations in a Dissipative Distributed Medium, Phys. D, 1986, vol. 19, no. 3, pp. 384–396.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kuznetsov, A.P. and Kuznetsov, S.P., Critical Dynamics of Coupled Map Lattices at the Onset of Chaos, Radiophys. Quantum El., 1991, vol. 34, nos. 10–12, pp. 845–868; see also: Izv. Vyssh. Uchebn. Zaved. Radiofizika, 1991, vol. 34, nos. 10–12, pp. 1079–1115.MathSciNetGoogle Scholar
  29. 29.
    Kuznetsov, S.P., Renormalization Group, Universality and Scaling in Dynamics of Coupled Map Lattices, in Theory and Applications of Coupled Map Lattices, K. Kaneko (Ed.), Nonlinear Science: Theory and Applications, vol. 12, Chichester: Wiley, 1993, pp. 51–94.Google Scholar
  30. 30.
    Bunimovich, L.A. and Sinai, Ya.G., Spacetime Chaos in Coupled Map Lattices, Nonlinearity, 1988, vol. 1, no. 4, pp. 491–516.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Bunimovich, L.A. and Sinai, Ya.G., Statistical Mechanics of Coupled Map Lattices in Theory and Applications of Coupled Map Lattices, K. Kaneko (Ed.), Nonlinear Science: Theory and Applications, vol. 12, Chichester: Wiley, 1993, pp. 169–189.Google Scholar
  32. 32.
    Kuptsov, P. V. and Kuznetsov, S.P., Transition to a Synchronous Chaps Regime in a System of Coupled Nonautonomous Oscillators Presented in Terms of Amplitude Equations, Nelin. Dinam., 2006, vol. 2, no. 3, pp. 307–331 (Russian).CrossRefGoogle Scholar
  33. 33.
    Kuptsov, P. V. and Kuznetsov, S.P., Violation of Hyperbolicity in a Diffusive Medium with Local Hyperbolic Attractor, Phys. Rev. E, 2009, vol. 80, no. 1, 016205, 11 pp.MathSciNetCrossRefGoogle Scholar
  34. 34.
    Maistrenko, Yu., Penkovsky, B., and Rosenblum, M., Solitary State at the Edge of Synchrony in Ensembles with Attractive and Repulsive Interactions, Phys. Rev. E, 2014, vol. 89, no. 6, 060901, 5 pp.CrossRefGoogle Scholar
  35. 35.
    Jaros, P., Maistrenko, Yu., and Kapitaniak, T., Chimera States on the Route from Coherence to Rotating Waves, Phys. Rev. E, 2015, vol. 91, no. 2, 022907, 5 pp.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • Nadezhda I. Semenova
    • 1
  • Elena V. Rybalova
    • 1
  • Galina I. Strelkova
    • 1
  • Vadim S. Anishchenko
    • 1
  1. 1.Department of PhysicsSaratov National Research State UniversitySaratovRussia

Personalised recommendations