Izvestiya of Saratov University.


ISSN 1817-3020 (Print)
ISSN 2542-193X (Online)

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Puzanov A. M., Anishchenko V. S., Strelkova G. I. Chimera Structures in Ensembles of Nonlocally Coupled Sprott Maps. Izvestiya of Sarat. Univ. Physics. , 2019, vol. 19, iss. 4, pp. 246-257. DOI: 10.18500/1817-3020-2019-19-4-246-257

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Chimera Structures in Ensembles of Nonlocally Coupled Sprott Maps

Puzanov Alexey Mikhailovich, Saratov State University
Anishchenko Vadim Semenovich, Saratov State University
Strelkova Galina Ivanovna, Saratov State University

Background and Objectives: Recently, special attention in nonlinear dynamics and related research fields was targeted to the study of chimera states in networks of coupled oscillators. Chimeras were revealed in ensembles of nonlocally coupled identical systems which are described by both discrete- and continuous-time chaotic systems. In the paper we study numerically the dynamics of ring networks of nonlocally coupled chaotic discrete maps in order to find chimera states of different types, namely, phase and amplitude chimeras. The local dynamics of the individual elements is described by 2D and 3D Sprott maps which exhibit a quasi-periodic route to chaos and the regime of hyperchaos. Materials and Methods: The analysis is carried out using a software package which was elaborated for modeling the dynamics of complex networks. This program entitled “Computer program for modeling networks of dynamical elements, which are described by one-dimensional or two-dimensional coupling matrices” got the Certificate on state registration of a computer program. This software enables one to perform a detailed study of the spatio-temporal dynamics of the considered networks as the parameters of the individual elements and of the nonlocal coupling are varied, as well as to construct instantaneous profiles (snapshots) and space-time plots for the ensemble dynamics. Results: The numerical analysis of the dynamics of the ensembles of nonlocally coupled 2D and 3D Sprott maps has shown that the transition from complete chaotic synchronization to the regime of spatio-temporal chaos occurs through the appearance of chimera structures. For certain values of the coupling range and when decreasing coupling strength, the ring of 2D Sprott maps demonstrates amplitude chimera structures, whose elements are characterized by strongly developed chaotic behavior. The regime of coexistence of phase and amplitude chimeras is observed in the network of 3D Sprott maps with nonlocal coupling. Conclusion: The numerical results obtained and described in this paper indicate that the regimes of phase and amplitude chimeras in ensembles of nonlocally coupled chaotic oscillators are typical not only for the case when individual oscillators are characterized by perioddoubling bifurcations, but also for the oscillators which demonstrate the quasi-periodic route to chaos.

  1. Kuramoto Y., Battogtokh D. Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators. Nonlin. Phen. in Complex Syst., 2002, vol. 5, no. 4, pp. 380–385.
  2. Abrams D. M., Strogatz S. H. Chimera States for Coupled Oscillators. Phys. Rev. Lett., 2004, vol. 93, iss. 17, pp. 174102. DOI: https://doi.org/10.1103/PhysRevLett.93.174102
  3. Abrams D. M., Mirollo R., Strogatz S. H., Wiley D. A. Solvable Model for Chimera States of Coupled Oscillators. Phys. Rev. Lett., 2008, vol. 101, no. 8, pp. 084103. DOI: https://doi.org/10.1103/PhysRevLett.101.084103
  4. Panaggio M. J., Abrams D. M. Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity, 2015, vol. 28, no. 3, pp. R67. DOI: https://doi.org/10.1088/0951-7715/28/3/R67
  5. Omelchenko I., Maistrenko Y., Hövel P., Schöll E. Loss of coherence in dynamical networks: spatial chaos and chimera states. Phys. Rev. Lett., 2011, vol. 106, iss. 23, pp. 234102. DOI: https://doi.org/10.1103/PhysRevLett.106.234102
  6. Wolfrum M., Omel’chenko O. E. Chimera States Are Chaotic Transients. Phys. Rev. E, 2011, vol. 84, iss. 1, pp. 015201(R). DOI: https://doi.org/10.1103/PhysRevE.84.015201
  7. Omelchenko I., Riemenschneider B., Hövel P., Maistrenko Y., Schöll E. Transition from spatial coherence to incoherence in coupled chaotic systems. Phys. Rev. E, 2012, vol. 85, iss. 2, pp. 026212. DOI: https://doi.org/211210.1103/PhysRevE.85.026212
  8. Zakharova A., Kapeller M., Schöll E. Chimera Death: Symmetry Breaking in Dynamical Networks. Phys. Rev. Lett., 2014, vol. 112 , iss. 15, pp. 154101. DOI: https://doi.org/10.1103/PhysRevLett.112.154101
  9. Dudkowski D., Maistrenko Y., Kapitaniak T. Different types of chimera states: an interplay between spatial and dynamical chaos. Phys. Rev. E, 2014, vol. 90, iss. 3, pp. 032920. DOI: https://doi.org/10.1103/PhysRevE.90.032920
  10. Bogomolov S. A., Strelkova G. I., Schöll E., Anishchenko V. S. Amplitude and phase chimeras in an ensemble of chaotic oscillators. Technical Physics Letters, 2016, vol. 42, iss. 7, pp. 765–768. DOI: https://doi.org/10.1134/S1063785016070191
  11. Semenova N., Zakharova A., Schöll E., Anishchenko V. Does hyperbolicity impedes emergence of chimera states in networks of nonlocally coupled chaotic oscillators. Europhys. Lett., 2015, vol. 112, no. 4, pp. 40002. DOI: https://doi.org/10.1209/0295-5075/112/40002
  12. Rybalova E., Semenova N., Strelkova G., Anishchenko V. Transition from complete synchronization to spatio-temporal chaos in coupled chaotic systems with nonhyperbolic and hyperbolic attractors. Eur. Phys. J. Spec. Top., 2017, vol. 226, iss. 9, pp. 1857–1866. DOI: https://doi.org/10.1140/epjst/e2017-70023-1
  13. Bogomolov S. A., Slepnev A. V., Strelkova G. I., Schöll E., Anishchenko V. S. 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. DOI: https://doi.org/10.1016/j.cnsns.2016.06.024
  14. Ulonska S., Omelchenko I., Zakharova A., Schöll E. Chimera states in networks of Van der Pol oscillators with hierarchical connectivities. Chaos, 2016, vol. 26, iss. 9, pp. 094825. DOI: https://doi.org/10.1063/1.4962913
  15. Shepelev I. A., Bukh A. V., Strelkova G. I., Vadivasova T. E., Anishchenko V. S. Chimera states in ensembles of bistable elements with regular and chaotic dynamics. Nonlinear Dynamics, 2017, vol. 90, iss. 4, pp. 2317– 2330. DOI: https://doi.org/10.1007/s11071-017-3805-6
  16. Hizanidis J., Kouvaris N. E., Zamora-López G., DíazGuilera A., Antonopoulos C. G. Chimera-like states in modular neural networks. Scientifi c reports, 2016, vol. 6, pp. 19845. DOI: https://doi.org/10.1038/srep19845
  17. Semenova N. I., Zakharova A., Anishchenko V., Schöll E. Coherence-resonance chimeras in a network of excitable elements. Phys. Rev. Lett., 2016, vol. 117, iss. 1, pp. 01410. DOI: https://doi.org/10.1103/PhysRevLett.117.014102
  18. Schmidt A., Kasimatis T., Hizanidis J., Provata A., Hövel P. Chimera patterns in two-dimensional networks of coupled neurons. Phys. Rev. E, 2017, vol. 95, iss. 3, pp. 032224. DOI: https://doi.org/10.1103/PhysRevE.95.032224
  19. Tsigkri-DeSmedt N. D., Hizanidis J., Schöll E., Hövel P., Provata A. Chimeras in Leaky Integrate-and-Fire Neural Networks: Effects of Refl ecting Connectivities. Eur. Phys. J. B, 2017, vol. 90, iss. 7, pp. 139. DOI: https://doi.org/10.1140/epjb/e2017-80162-0
  20. Yeldesbay A., Pikovsky A., Rosenblum M. Chimeralike States in an Ensemble of Globally Coupled Oscillators. Phys. Rev. Lett., 2014, vol. 112, iss. 14, pp. 144103. DOI: https://doi.org/10.1103/PhysRevLett.112.144103
  21. Hizanidis J., Panagakou E., Omelchenko I., Schöll E., Hövel P, Provata A. Chimera States in Population Dynamics: Networks with Fragmented and Hierarchical Connectivities. Phys. Rev. E, 2015, vol. 92, iss. 1, pp. 012915. DOI: https://doi.org/10.1103/PhysRevE.92.012915
  22. Schöll E. Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics. Eur. Phys. J. Spec. Top., 2016, vol. 225, iss. 6–7, pp. 891–919. DOI: https://doi.org/10.1140/epjst/e2016-02646-3
  23. Sawicki J., Omelchenko I., Zakharova A., Schöll E. Chimera states in complex networks: interplay of fractal topology and delay. Eur. Phys. J. Spec. Top., 2017, vol. 226, iss. 9, pp.1883–1892. DOI: https://doi.org/10.1140/epjst/e2017-70036-8
  24. Elhadj Z., Sprott J. C. A minimal 2-D quadratic map with quasi-periodic route to chaos. Int. J. Bifurcation and Chaos, 2008, vol. 18, no. 5, pp. 1567–1577. DOI: https://doi.org/10.1142/S021812740802118X
  25. Elhadj Z., Sprott J. C. Classifi cation of three-dimensional quadratic diffeomorphisms with constant Jacobian. Front. Phys. China, 2009, vol. 4, iss. 1, pp. 111–121. DOI: https://doi.org/10.1007/s11467-009-0005-y
  26. Kuznetsov A. P., Savin A. V., Sedova Y. V., Tyuryukina L. V. Bifurkatsii otobrazhenii [Bifurcations of maps]. Saratov, OOO Izdat. Tsentr “Nauka”, 2012. 196 p. (in Russian).
  27. Henon M. Dvumernoe otobrazhenie so strannym attraktorom [Two-dimensional maps with strange attractor]. In: Strannye attraktory [Strange attractors]. Moscow, Mir Publ., 1981, pp. 152–163 (in Russian).
  28. Anishchenko V. S. Slozhnye kolebaniya v prostih sistemah [Complex oscillations in simple systems]. 2nd ed., ext. Moscow, Knizhnyi Dom “LIBROKOM”, 2009. 320 p. (in Russian).
  29. Gonchenko S. V., Ovsyannikov I. I., Simo C., Turaev D. Three-dimensional Henon-like maps and wild Lorenzlike attractors. Int. J. of Bifurcation and Chaos, 2005, vol.15, no. 11, pp. 3493–3508. DOI: https://doi.org/10.1142/S0218127405014180
  30. Gonchenko S. V., Meiss J. D., Ovsyannikov I. I. Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation. Reg. and Chaot. Dynamics, 2006, vol. 11, no. 2, pp. 191–212. DOI: https://doi.org/10.1070/RD2006v011n02ABEH000345
  31. Gonchenko S. V., Ovsyannikov I. I. On bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a “neutral” saddle fi xed point. J. Math. Sci. (N.Y.), 2005, vol. 128, iss. 2, pp. 2774–2777.
  32. Turaev D. V., Shilnikov L. P. An example of wild strange attractor. Math sb., 1998, vol. 189, no. 2, pp. 137–160 (in Russian).
  33. Shilnikov A. L. Bifurcations and chaos in the MoriokaShimizu system. In: Metody kachestvennoi teorii differenzialnyh uravnenii [Methods of qualitative theory of differential equations]. Gorky, Izd-vo Gorkovskogo universiteta, 1986, pp. 180–193 (in Russian).
  34. Shilnikov A. L. On bifurcations of the Lorenz attractor in the Shimizu–Morioka model. Physica D, 1993, vol. 62, iss. 1–4, pp. 338–346. DOI: https://doi.org/10.1016/0167-2789(93)90292-9
  35. Lozi R. Un Attracteur Entrange du Type Attracteur de Henon. J. de Physique, 1978, vol. 39, no. C5, pp. 9–10.
  36. Bukh A. V., Shepelev I. A. Kompyuternaya programma dlya modelirovaniya setei dinamicheskih elementov opisyvayuzhihsya odnomernymi ili dvumernymi matrizami svyazi [Computer program for modeling networks of dynamical elements, which are described by onedimensional or two-dimensional coupling matrices]. Certificate on state registration of a computer program No. 2017612340 dated 20.02.2017 (in Russian).