Izvestiya of Saratov University.


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

For citation:

Bogomolov S. A., Rybalova E. V., Strelkova G. I., Anishchenko V. S. Spatiotemporal Structures in an Ensemble of Nonlocally Coupled Nekorkin Maps. Izvestiya of Sarat. Univ. Physics. , 2019, vol. 19, iss. 2, pp. 86-94. DOI: 10.18500/1817-3020-2019-19-2-86-94

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 72)

Spatiotemporal Structures in an Ensemble of Nonlocally Coupled Nekorkin Maps

Bogomolov Sergei Alekseevich, Saratov State University
Rybalova Elena Vladislavovna, Saratov State University
Strelkova Galina Ivanovna, Saratov State University
Anishchenko Vadim Semenovich, Saratov State University

Background and Objectives: Studying chimera states is a subject of special attention among specialists in nonlinear dynamics, and the issue on the mechanisms for implementing chimeras is today one of the topic directions. In the paper we consider the mechanism for realizing a chimera state regime which is based on the so-called “solitary states” (SSC) and is actively discussed by experts. The problem is solved by analyzing the dynamics of a one-dimensional ring of nonlocally coupled discrete-time systems. The individual element of the ring is given by the discrete model describing neuronal activity. Materials and Methods: Numerical simulation of large and multicomponent ensembles (networks) is the basic method of our research. Software systems specially developed by the authors are used that allow to carry out numerical experiments under variations in the system parameters, initial conditions and provide a graphic illustration of the obtained results. Results: We have established that the ensemble of nonlocally coupled Nekorkin maps can exhibit the SSC regime which is resulted from the appearance of bistability in the individual elements of the ensemble. This fact has been corroborated by the numerical results for the phase portraits of the coexisting attractors and their basins of attraction. It has been shown that randomly distributed initial conditions lead to the situation when one part of the ensemble oscillators fall into one attractor, while the other one belongs to the second attracting set. This mechanism was presented by the authors in the previously published papers for ensembles of different discrete-time systems. The results of this work confirm the commonality of the mechanism for implementing the SSC mode established earlier by the authors. Conclusion: The results of this article make a significant contribution to the understanding of the peculiarities in the formation of complex spatiotemporal structures of the SSC type.


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. DOI: https://doi.org/10.1063/1.4858996

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. 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

4. 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

5. 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

6. 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

7. 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

8. Semenova N., Zakharova A., Schöll E., Anischenko 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

9. Bogomolov S. A., Strelkova G. I., Schöll E., Anischenko 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

10. Vadivasova T. E., Strelkova G. I., Bogomolov S. A., Anishchenko V. S. Correlation analysis of the coherenceincoherence transition in a ring of nonlocally coupled logistic maps. Chaos, 2016, vol. 26, iss. 9, pp. 093108. DOI: https://doi.org/10.1063/1.4962647

11. Kemeth F. P., Haugland S. W., Schmidt L., Kevrekidis I. G., Krischer K. A classifi cation scheme for chimera states. Chaos, 2016, vol. 26, iss. 9, pp. 094815. DOI: https://doi.org/10.1063/1.4959804

12. 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

13. 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

14. 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. 89–919. DOI: https://doi.org/10.1140/epjst/e2016-02646-3

15. Hizanidis J., Kouvaris N. E., Zamora-López G., Díaz- Guilera 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

16. 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

17. Vadivasova T. E., Strelkova G. I., Bogomolov S. A., Anischenko V. S. Correlation characteristics of phase and amplitude chimeras in an ensemble of nonlocally coupled maps. Technical Physics Letters, 2017, vol. 43, iss. 1, pp. 118–121. DOI: https://doi.org/10.1134/S1063785017010278

18. Rybalova E., Semenova N., Strelkova G., Anischenko V. Transition from complete synchronization to spatiotemporal 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

19. Semenova N. I., Strelkova G. I., Anischenko V. S., Zakharova A. Temporal intermittency and the lifetime of chimera states in ensembles of nonlocally coupled chaotic oscillators. Chaos, 2017, vol. 27, iss. 6, pp. 061102. DOI: https://doi.org/10.1063/1.4985143

20. 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

21. Shepelev I. A., Bukh A. V., Vadivasova T. E., Anischenko V. S., Zakharova A. Double-well chimeras in 2D lattice of chaotic bistable elements. Commun. Nonlinear Sci. Numer. Simul., 2018, vol. 54, pp. 50–61. DOI: https://doi.org/10.1016/j.cnsns.2017.05.017

22. 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

23. Kholuianova I. A., Bogomolov S. A., Anishchenko V. S. Synchronization of Chimera States in Ensembles of Nonlocally Coupled Cubic Maps. Izv. Saratov Univ. (N. S.), Ser. Physics, 2018, vol. 18, iss. 2, pp. 103–111 (in Russian). DOI: https://doi.org/10.18500/1817-3020-2018-18-2-103-111

24. 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

25. Maistrenko Y., Penkovsky B., Rosenblum M. Solitary state at the edge of synchrony in ensembles with attractive and repulsive interactions. Phys. Rev. E, 2014, vol. 89, iss. 6, pp. 060901. DOI: https://doi.org/10.1103/PhysRevE.89.060901

26. Jaros P., Maistrenko Y., Kapitaniak T. Chimera states on the route from coherence to rotating waves. Phys. Rev. E, 2015, vol. 91, iss. 2, pp. 022907. DOI: https://doi.org/10.1103/PhysRevE.91.022907

27. Premalatha K., Chandrasekar V. K., Senthilvelan M., Lakshmanan M. Imperfectly synchronized states and chimera states in two interacting populations of nonlocally coupled Stuart-Landau oscillators. Phys. Rev. E, 2016, vol. 94, iss. 1, pp. 012311. DOI: https://doi.org/10.1103/PhysRevE.94.012311

28. Jaros P., Brezetsky S., Levchenko R., Dudkowski D., Kapitaniak T., Maistrenko Y. Solitary states for coupled oscillators with inertia. Chaos, 2018, vol. 28, iss. 1, pp. 011103. DOI: https://doi.org/10.1063/1.5019792

29. Bukh A., Rybalova E., Semenova N., Strelkova G., Anischenko 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, iss. 11, pp. 111102. DOI: https://doi.org/10.1063/1.5009375

30. Semenova N., Vadivasova T. E., Anischenko V. S. Mechanism of solitary state appearance in an ensemble of nonlocally coupled Lozi maps. Eur. Phys. J. Spec. Top., 2018, vol. 227, iss. 10–11, pp. 1173–1183. DOI: https://doi.org/10.1140/epjst/e2018-800035-y

31. Rybalova E. V., Strelkova G. I., Anischenko V. S. Mechanism of realizing a solitary state chimera in a ring of nonlocally coupled chaotic maps. Chaos, Solitons & Fractals, 2018, vol. 115, pp. 300–305. DOI: https://doi.org/10.1016/j.chaos.2018.09.003

32. Nekorkin V. I., Vdovin L. V. Map-based model of the neural activity. Izvestiya VUZ, Applied nonlinear dynamics, 2007, vol. 15, no. 5, pp. 36–60 (in Russian). DOI: https://doi.org/10.18500/0869-6632-2007-15-5-36-60