Regular and Chaotic Dynamics

, Volume 23, Issue 7–8, pp 948–960 | Cite as

Synchronization of Chimera States in a Network of Many Unidirectionally Coupled Layers of Discrete Maps

  • Galina I. StrelkovaEmail author
  • Tatiana E. Vadivasova
  • Vadim S. Anishchenko


We study numerically external synchronization of chimera states in a network of many unidirectionally coupled layers, each representing a ring of nonlocally coupled discretetime systems. The dynamics of each element in the network is described by either the logistic map or the bistable cubic map. We consider two cases: when all M unidirectionally coupled layers are identical and when (M − 1) identical layers differ from the first driving layer in their nonlocal coupling parameters. It is shown that the master chimera state in the first layer can be retranslating along the network with small distortions which are defined by a parameter mismatch. We also analyze the dependence of the mean-square deviation of the structure in the ith layer on the nonlocal coupling parameters.


synchronization many layer network chimera states nonlocal coupling unidirectional coupling 

MSC2010 numbers

90B10 34D06 35B36 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Pikovsky, A., Rosenblum, M., and Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences, New York: Cambridge Univ. Press, 2001.CrossRefzbMATHGoogle Scholar
  2. 2.
    Anishchenko, V. S., Astakhov, V. V., Neiman, A. B., Vadivasova, T.E., and Schimansky-Geier, L., Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments, 2nd ed., Berlin: Springer, 2007.zbMATHGoogle Scholar
  3. 3.
    Fujisaka, H. and Yamada, T., Stability Theory of Synchronized Motions in Coupled Oscillatory Systems, Progr. Theor. Phys., 1983, vol. 69, no. 1, pp. 32–47.CrossRefzbMATHGoogle Scholar
  4. 4.
    Pecora, L. and Carroll, T., Synchronization of Chaotic Systems, Phys. Rev. Lett., 1990, vol. 64, no. 8, pp. 821–823.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Anishchenko, V. S., Vadivasova, T. E., Postnov, D. E., and Safonova, M.A., Synchronization of Chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1992, vol. 2, no. 3, pp. 633–644.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Rosenblum, M. G., Pikovsky, A. S., and Kurths, J., Phase Synchronization of Chaotic Oscillators, Phys. Rev. Lett., 1996, vol. 76, no. 11, pp. 1804–1807.CrossRefzbMATHGoogle Scholar
  7. 7.
    Kocarev, L. and Parlitz, U., Generalized Synchronization, Predictability and Equivalence of Unidirectionally Coupled Systems, Phys. Rev. Lett., 1996, vol. 76, no. 11, pp. 1816–1819.Google Scholar
  8. 8.
    Boccaletti, S., Kurths, J., Osipov, G., Valladares, D. L., and Zhou, C. S., The Synchronization of Chaotic Systems, Phys. Rep., 2002, vol. 366, nos. 1–2, pp. 1–101.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Neiman, A.B., Synchronizationlike Phenomena in Coupled Stochastic Bistable Systems, Phys. Rev. E, 1994, vol. 49, no. 4, pp. 3484–3488.CrossRefGoogle Scholar
  10. 10.
    Shulgin, B., Neiman, A., and Anishchenko, V., Mean Switching Frequency Locking in Stochastic Bistable System Driven by a Periodic Force, Phys. Rev. Lett., 1995, vol. 75, no. 23, pp. 4157–4161.CrossRefGoogle Scholar
  11. 11.
    Han, S. K., Postnov, D.E., Sosnovtseva, O.V., and Yim, T.G., Interacting Coherence Resonance Oscillators, Phys. Rev. Lett., 1999, vol. 83, no. 9, pp. 1771–1774.CrossRefGoogle Scholar
  12. 12.
    Kuramoto, Y., Chemical Oscillations, Waves and Turbulence, Berlin: Springer, 1984.CrossRefzbMATHGoogle Scholar
  13. 13.
    Afraimovich, V. S., Nekorkin, V. I., Osipov, G. V., and Shalfeev, V.D., Stability, Structures and Chaos in Nonlinear Synchronization Network, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol. 6, Singapore: World Sci., 1994.Google Scholar
  14. 14.
    Nekorkin, V. I. and Velarde, M. G., Synergetic Phenomena in Active Lattices, Berlin: Springer, 2002.CrossRefzbMATHGoogle Scholar
  15. 15.
    Osipov, G., Kurths, J., and Zhou, C., Synchronization in Oscillatory Networks, Berlin: Springer, 2007.CrossRefzbMATHGoogle Scholar
  16. 16.
    Kuramoto, Y. and Battogtokh, D., Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators, Nonlinear Phenom. Complex Systems, 2002, vol. 5, no. 4, pp. 380–385.Google Scholar
  17. 17.
    Abrams, D.M. and Strogatz, S.H., Chimera States for Coupled Oscillators, Phys. Rev. Lett., 2004, vol. 93, no. 17, 174102, 4 pp.CrossRefGoogle Scholar
  18. 18.
    Omelchenko, I., Maistrenko, Yu., Hövel, P., 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
  19. 19.
    Zakharova, A., Kapeller, M., and Schöll, E., Chimera Death: Symmetry Breaking in Dynamical Networks, Phys. Rev. Lett., 2014, vol. 112, no. 15, 154101, 5 pp.CrossRefGoogle Scholar
  20. 20.
    Panaggio, M. J. and Abrams, D.M., Chimera States: Coexistence of Coherence and Incoherence in Networks of Coupled Oscillators, Nonlinearity, 2014, vol. 28, no. 3, R67–R87.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Maistrenko, Y., Sudakov, O., Osir, O., and Maistrenko, V., Chimera States in Three Dimensions, New J. Phys., 2015, vol. 17, no. 7, 073037.CrossRefGoogle Scholar
  22. 22.
    Schöll, E., Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics, Eur. Phys. J. Spec. Top., 2016, vol. 225, nos. 6–7, pp. 891–919.CrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Komarov, M. and Pikovsky, A., Effects of Nonresonant Interaction in Ensembles of Phase Oscillators, Phys. Rev. E, 2011, vol. 84, no. 1, 016210, 12 pp.CrossRefGoogle Scholar
  25. 25.
    Komarov, M. and Pikovsky, A., Dynamics of Multifrequency Oscillator Communities, Phys. Rev. Lett., 2013, vol. 110, no. 13, 134101, 5 pp.CrossRefGoogle Scholar
  26. 26.
    Amengual, A., Hernández-García, E., Montagne, R., and San Miguel, M., Synchronization of Spatiotemporal Chaos: The Regime of Coupled Spatiotemporal Intermittency, Phys. Rev. Lett., 1997, vol. 78, no. 23, pp. 4379–4382.CrossRefGoogle Scholar
  27. 27.
    Kocarev, L., Tasev, Z., and Parlitz, U., Synchronizing Spatiotemporal Chaos of Partial Differential Equations, Phys. Rev. Lett., 1997, vol. 79, no. 1, pp. 51–54.CrossRefzbMATHGoogle Scholar
  28. 28.
    Boccaletti, S., Bragard, J., Arecchi, F.T., and Mancini, H., Synchronization in Nonidentical Extended Systems, Phys. Rev. Lett., 1999, vol. 83, no. 3, pp. 536–539.CrossRefGoogle Scholar
  29. 29.
    Boccaletti, S., Bragard, J., and Arecchi, F.T., Controlling and Synchronizing Space Time Chaos, Phys. Rev. E, 1999, vol. 59, no. 6, pp. 6574–6578.CrossRefGoogle Scholar
  30. 30.
    Bragard, J. and Boccaletti, S., Integral Behavior for Localized Synchronization in Nonidentical Extended Systems, Phys. Rev. E, 2000, vol. 62, no. 5, pp. 6346–6351.CrossRefGoogle Scholar
  31. 31.
    Kazantsev, V.B., Nekorkin, V. I., Artyuhin, D.V., and Velarde, M.G., Synchronization, Re-Entry, and Failure of Spiral Waves in a Two-Layer Discrete Excitable System, Phys. Rev. E, 2001, vol. 63, no. 1, 016212, 9 pp.Google Scholar
  32. 32.
    Akopov, A., Astakhov, V., Vadivasova, T., Shabunin, A., and Kapitaniak, T., Frequency Synchronization of Clusters in Coupled Extended Systems, Phys. Lett. A, 2005, vol. 334, nos. 2–3, pp. 169–172.CrossRefzbMATHGoogle Scholar
  33. 33.
    Shepelev, I. A., Slepnev, A. V., and Vadivasova, T.E., Different Synchronization Characteristics of Distinct Types of Traveling Waves in a Model of Active Medium with Periodic Boundary Conditions, Commun. Nonlinear Sci. Numer. Simul., 2016, vol. 38, pp. 206–217.MathSciNetCrossRefGoogle Scholar
  34. 34.
    Watts, D. J. and Strogatz, S.H., Collective Dynamics of Small-World Networks, Nature, 1998, vol. 393, no. 6684, pp. 440–442.CrossRefzbMATHGoogle Scholar
  35. 35.
    De Domenico, M., Solé-Ribalta, A., Cozzo, E., Kivelä, M., Moreno, Y., Porter, M.A., Gómez, S., and Arenas, A., Mathematical Formulation of Multilayer Networks, Phys. Rev. X, 2013, vol. 3, no. 4, 041022, 15 pp.Google Scholar
  36. 36.
    Boccaletti, S., Bianconi, G., Criado, R., del Genio, Ch., Gómez-Garde˜nes, J., Romance, M., Sendi˜na-Nadal, I., Wang, Z., and Zanin, M., The Structure and Dynamics of Multilayer Networks, Phys. Rep., 2014, vol. 544, no. 1, pp. 1–122.MathSciNetCrossRefGoogle Scholar
  37. 37.
    Sevilla-Escoboza, R., Sendi˜na-Nadal, I., Leyva, I., Gutiérrez, R., Buldú, J. M., and Boccaletti, S., On the Inter-Layer Synchronization in Multiplex Networks of Identical Layers, Chaos, 2016, vol. 26, no. 6, 065304, 6 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    del Genio, Ch., Gómez-Garde˜nes, J., Bonamassa, I., and Boccaletti, S., Synchronization in Networks with Multiple Interaction Layers, Sci. Adv., 2016, vol. 2, no. 11, e1601679, 10 pp.CrossRefGoogle Scholar
  39. 39.
    Maksimenko, V.A., Makarov, V. V., Bera, B.K., Ghosh, D., Dana, S.K., Goremyko, M.V., Frolov, N. S., Koronovskii, A. A., and Hramov, A.E., Excitation and Suppression of Chimera States by Multiplexing, Phys. Rev. E, 2016, vol. 94, no. 5, 052205, 9 pp.CrossRefGoogle Scholar
  40. 40.
    Majhi, S., Perc, M., and Ghosh, D., Chimera States in a Multilayer Network of Coupled and Uncoupled Neurons, Chaos, 2017, vol. 27, no. 7, 073109, 15 pp.MathSciNetCrossRefGoogle Scholar
  41. 41.
    Majhi, S., Perc, M., and Ghosh, D., Chimera States in Uncoupled Neurons Induced by a Multilayer Structure, Sci. Rep., 2016, vol. 6, 39033, 11 pp.CrossRefGoogle Scholar
  42. 42.
    Andrzejak, R.G., Ruzzene, R., and Malvestio, I., Generalized Synchronization between Chimera States, Chaos, 2017, vol. 27, no. 5, 053114, 6. pp.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Ghosh, D., Zakharova, A., and Jalan, S., Non-Identical Multiplexing Promotes Chimera States, Chaos Solitons Fractals, 2018, vol. 106, pp. 56–60.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Bukh, A., Rybalova, E., Semenova, N., Strelkova, G., and Anishchenko, V., New Type of Chimera and Mutual Synchronization of Spatiotemporal Structures in Two Coupled Ensembles of Nonlocally Coupled Interacting Chaotic Maps, Chaos, 2017, vol. 27, no. 11, 111102, 7 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Shepelev, I. A., Bukh, A. V., Strelkova, G. I., Vadivasova, T.E., and Anishchenko, V. S., Chimera States in Ensembles of Bistable Elements with Regular and Chaotic Dynamics, Nonlinear Dynam., 2017, vol. 90, no. 4, pp. 2317–2330.MathSciNetCrossRefGoogle Scholar
  46. 46.
    Shepelev, I. A., Bukh, A. V., Vadivasova, T.E., Anishchenko, V. S., and Zakharova, A., Double-Well Chimeras in 2D Lattice of Chaotic Bistable Elements, Commun. Nonlinear Sci. Numer. Simul., 2018, vol. 54, pp. 50–61.MathSciNetCrossRefGoogle Scholar
  47. 47.
    Kholuianova, I.A., Bogomolov, S.A., and Anishchenko, V. S., Synchronization of Chimera States in Ensembles of Nonlocally Coupled Cubic Maps, Izv. SGU. Novaya Seriya. Fizika, 2018, vol. 18, no. 2, pp. 103–111 (Russian).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Galina I. Strelkova
    • 1
    Email author
  • Tatiana E. Vadivasova
    • 1
  • Vadim S. Anishchenko
    • 1
  1. 1.Department of PhysicsSaratov State UniversitySaratovRussia

Personalised recommendations