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Vadivasova T. E., Arinushkin P. A., Anishchenko V. S. Mutual synchronization of complex structures in interacting ensembles of non-locally coupled rotators. Izvestiya of Saratov University. Physics , 2021, vol. 21, iss. 1, pp. 4-20. DOI: 10.18500/1817-3020-2021-21-1-4-20, EDN: RMLJUY

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Mutual synchronization of complex structures in interacting ensembles of non-locally coupled rotators

Vadivasova Tatyana Evgen'evna, Saratov State University
Arinushkin Pavel A., Saratov State University
Anishchenko Vadim Semenovich, Saratov State University

Background and Objectives: One of the actual problems in nonlinear dynamics is the formation and interaction of complex spatial structures such as chimeras and solitary states arising in multicomponent systems. Chimera states are typical for ensembles of identical oscillators with regular, chaotic, and even stochastic behavior in a case of nonlocal interaction of the elements. They represent cluster structures, including groups of elements with synchronous and non-synchronous oscillations. Chimeras were discovered and investigated in real experiments, that indicates the possibility of observing such regimes in complex systems in living nature and in technology. Solitary states are less studied today. The regime of solitary states is characterized by the synchronous behavior of most elements of the ensemble, while individual oscillators behave in a “special state”. In the present work, an ensemble of phase oscillators with inertia (rotators) is chosen as the basic model for investigation. Such ensembles with a specific coupling topology are widely used in modeling the operation of energy networks. Ensembles of rotators with nonlocal coupling are characterized by both chimera states and solitary state regimes. The problem of interaction of ensembles of rotators with nonlocal coupling and synchronization of complex spatial structures (chimeras and solitary states) formed in them has not been studied yet. Materials and Methods: A two-layer multiplex network of rotators with a nonlocal character of intralayer interactions is considered. Each layer consists of 100 elements with the same value of the coupling coefficient and coupling phase shift for each element within one layer. The interlayer coupling is symmetric. At the initial stage, with a random choice of initial conditions, steady regimes (chimeras or solitary states) in non-interacting layers were found. Next, the interlayer coupling was introduced and the evolution of the layer dynamics in the selected initial regimes was studied. Four cases of interaction with various initial states of the layers were considered. In the first case, the two layers are completely identical and demonstrate slightly different chimera structures without interlayer coupling. Their evolution with the introduction and growth of the interlayer coupling is considered for two values of the coupling phase shift. It is shown that, starting from a certain threshold value of the interlayer coupling coefficient, the complete synchronization regime is established in the layers, and the coupling phase shift significantly affects the value of the synchronization threshold. In the second case, the previous experiment is reproduced for the two layers with a frequency mismatch. Chimera states established without interlayer interaction are characterized by significantly different average frequencies of the elements in the two layers. In the presence of non-identity of the layers (in this case, frequency mismatch), the regime of complete synchronization of spatial structures is impossible. However, with an increase in the interlayer coupling coefficient, effective synchronization can be obtained which corresponds to a slight difference in the phases of rotators in the interacting layers with full frequency synchronization. In the third case, we consider the interaction between the layers in the solitary state regimes with different spatial structures. In this case, a frequency mismatch is also introduced for the elements of the two layers. For solitary states, the effective synchronization regime with an increase in the interlayer coupling is also established. In both layers the same configurations of solitary states are realized and frequency synchronization is observed. In the fourth case, a heterogeneous multiplex network is considered, in which one layer is in the chimera state, the second layer shows the solitary state mode. With a certain strength of the interlayer coupling the complex structures are destroyed in both layers of the network and a spatially uniform regimes are established. In this case, all the rotators of the two layers rotate at the same frequency, and the difference in the regimes in the layers reduces to a small phase shift, the same for all pairs of coupled rotators of the two layers. Conclusion: The effects of synchronization in the multiplex network were established for two layers in the regimes of complex spatio-temporal dynamics, such as chimera states and solitary states. The influence of the frequency mismatch of the network elements and the phase shift in the interlayer coupling on the synchronization phenomena was studied.

  1. Rosenblum M., Pikovsky A., Kurths J. Synchronization ‒ a universal concept in nonlinear sciences. Cambridge, New York, Cambridge University Press, 2001, Cambridge Nonlinear Science Series 12. 411 p. DOI:  https://doi.org/10.1119/1.1475332
  2. Anishchenko V. S., Astakhov V. V., Neiman A. B., Vadivasova T. E., Shimansky-Geier L. Nonlinear dynamics of chaotic and stochastic systems. Tutorial and modern development. Ed. 2. Berlin, Heidelberg, Springer-Verlag, 2007, Springer Series in Synergetics. 446 p. DOI:  https://doi.org/10.1007/978-3-540-38168-6
  3. Kuramoto Y. Chemical oscillations, waves, and turbulence. Berlin, Heidelberg, Springer-Verlag, 1984, Springer Series in Synergetics. 158 p. DOI:  https://doi.org/10.1007/978-3-642-69689-3
  4. Nekorkin V. I., Velarde M. G. Synergetic phenomena in active lattices. Berlin, Heidelberg, Springer-Verlag, 2002, Springer Series in Synergetics. 359 p. DOI:  https://doi.org/10.1007/978-3-642-56053-8
  5. Osipov G. V., Kurths J., Zhou C. Synchronization in oscillatory networks. Berlin, Heidelberg, SpringerVerlag, 2007, Springer Series in Synergetics. 370 p. DOI:  https://doi.org/10.1007/978-3-540-71269-5
  6. Barreto E., Hunt B., Ott E., So P. Synchronization in networks of networks: the onset of coherent behavior in systems of interacting populations of heterogeneous oscillators. Phys. Rev. E, 2008, vol. 77, 036107. DOI:  https://doi.org/10.1103/PhysRevE.77.036107
  7. Louzada V. H., Araújo N. A., Andrade J. S. Jr., Herrmann H. J. Breathing synchronization in interconnected networks. Sci. Rep., 2013, vol. 3, 03289. DOI:  https://doi.org/10.1038/srep03289
  8. Aguirre J., Sevilla-Escoboza R., Gutiérrez R., Papo D., Buldú J. M. Synchronization of interconnected networks: The role of connector nodes. Phys. Rev. Lett., 2014, vol. 112, 248701. DOI:  https://doi.org/10.1103/PhysRevLett.112.248701
  9. Sevilla-Escoboza R., Sendina-Nadal I., Leyva I., Gutierrez R., Buldu J. M., Boccaletti S. On the Interlayer synchronization in multiplex networks of identical layers. Chaos, 2016, vol. 26, iss. 6, 065304. DOI:  https://doi.org/10.1063/1.4952967
  10. Leyva I., Sevilla-Escoboza R., Sendiña-Nadal I., Gutiérrez R., Buldú J. M., Boccaletti S. Inter-layer synchronization in non-identical multi-layer networks. Sci. Rep., 2017, vol. 7, iss. 1, 45475. DOI:  https://doi.org/10.1038/srep45475
  11. Kuramoto Y., Battogtokh D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonl. Phenom. Complex Syst., 2002, vol. 4, pp. 380‒385.
  12. Abrams D. M., Strogatz S. H. Chimera states for coupled oscillators. Phys. Rev. Lett., 2004, vol. 93, iss. 17, 174102. DOI:  https://doi.org/10.1103/PhysRevLett.93.174102
  13. 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, 234102. DOI:  https://doi.org/10.1103/PhysRevLett.106.234102
  14. Tinsley M. R., Nkomo S., Showalter K. Chimera and phase cluster states in populations of coupled chemical oscillators. Nature Physics, 2012, vol. 8, iss. 9, pp. 662‒666. DOI:  https://doi.org/10.1038/nphys2371
  15. Martens E. A., Thutupalli S., Fourrire A., Hallatschek O. Chimera states in mechanical oscillator networks. Proc. Nat. Acad. Sci. USA, 2013, vol. 110, iss. 26, pp. 10563‒10567. DOI:  https://doi.org/10.1073/pnas.1302880110
  16. Zakharova A., Kapeller M., Schöll E. Chimera death: Symmetry breaking in dynamical networks. Phys. Rev. Lett., 2014, vol. 112, 154101. DOI:  https://doi.org/10.1103/PhysRevLett.112.154101
  17. Panaggio M. J., Abrams D. M. Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity, 2015, vol. 28, iss. 3, R67‒R87. DOI:  https://doi.org/10.1088/0951-7715/28/3/r67
  18. Strelkova G. I., Anishchenko V. S. Spatio-temporal structures in ensembles of coupled chaotic systems. Phys. Usp., 2020, vol. 63, no. 2, pp. 145‒161. DOI:  https://doi.org/10.3367/UFNe.2019.01.038518
  19. Yeldesbay A., Pikovsky A., Rosenblum M. Chimeralike States in an Ensemble of Globally Coupled Oscillators. Phys. Rev. Lett., 2014, vol. 112, iss. 14, 144103. DOI:  https://doi.org/10.1103/PhysRevLett.112.144103
  20. Sethia G. C., Sen A. Chimera States: The Existence Criteria Revisited. Phys. Rev. Lett., 2014, vol. 112, 144101. DOI:  https://doi.org/10.1103/PhysRevLett.112.144101
  21. Laing C. R. Chimera in networks with purely local coupling. Phys. Rev. E, 2015, vol. 92, 050904. DOI:  https://doi.org/10.1103/PhysRevE.92.050904
  22. Clerc M. G., Coulibaly S., Ferrë M. A., Garcïa-Nustes M. A., Rojas R. G. Chimera-type states induced by local coupling. Phys. Rev. E, 2016, vol. 93, iss. 5, 052204. DOI:  https://doi.org/10.1103/PhysRevE.93.052204
  23. 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, 012915. DOI:  https://doi.org/10.1103/PhysRevE.92.012915
  24. Bera B. K., Majhi S., Ghosh D., Perc M. Chimera states: Effects of different coupling topologies. EPL, 2017, vol. 118, iss. 1, 10001. DOI:  https://doi.org/10.1209/0295-5075/118/10001
  25. Maksimenko V. A., Makarov V. V., Bera B. K., Dibakar Ghosh, Syamal Kumar Dana, Goremyko M. V., Frolov N. S., Koronovskii A. A., Hramov A. E. Excitation and suppression of chimera states by multiplexing. Phys. Rev. E, 2016, vol. 94, 052205. DOI:  https://doi.org/10.1103/PhysRevE.94.052205
  26. Majhi S., Perc M., Ghosh D. Chimera states in a multilayer network of coupled and uncoupled neurons. Chaos, 2017, vol. 27, 073109. DOI:  https://doi.org/10.1063/1.4993836
  27. Andrzejak R. G., Ruzzene G., Malvestio I. Generalized synchronization between chimera states. Chaos, 2017, vol. 27, iss. 5, 053114. DOI:  https://doi.org/10.1063/1.4983841
  28. Bukh A., Rybalova E., Semenova N., Strelkova G., Anishchenko V. New type of chimera and mutual synchronization of spatiotemporal structures in two coupled ensembles of nonlocally interacting chaotic maps. Chaos, 2017, vol. 27, 5009375. DOI:  https://doi.org/10.1063/1.5009375
  29. Ghosh D., Zakharova A., Jalan S. Non-identical multiplexing promotes chimera states. Chaos, Solitons and Fractals, 2018, vol. 106, pp. 56‒60. DOI:  https://doi.org/10.1016/j.chaos.2017.11.010
  30. Strelkova G. I., Vadivasova T. E., Anishchenko V. S. Synchronization of chimera states in a network of many unidirectionally coupled layers of discrete maps. Regular and Chaotic Dynamics, 2018, vol. 23, pp. 948‒960. DOI:  https://doi.org/10.1134/S1560354718070092
  31. Rybalova E. V., Vadivasova T. E., Strelkova G. I., Anishchenko V. S., Zakharova A. S. Forced synchronization of a multilayer heterogeneous network of chaotic maps in the chimera state mode. Chaos, 2019, vol. 29, iss. 3, 033134. DOI:  https://doi.org/10.1063/1.5090184
  32. Abrams D. M., Strogatz S. H. Chimera states in a ring of nonlocally coupled oscillators. Int. J. of Bif. Chaos, 2006, vol. 16, iss. 1, pp. 21‒37. DOI:  https://doi.org/10.1142/S0218127406014551
  33. Wolfrum M., Omel’chenko O. E., Yanchuk S., Maistrenko Y. L. Spectral properties of chimera states. Chaos, 2011, vol. 21, iss. 1, 013112. DOI:  https://doi.org/10.1063/1.3563579
  34. Omel’chenko O. E., Wolfrum M., Yanchuk S., Maistrenko Y. L., Sudakov O. Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locallycoupled phase oscillators. Phys. Rev. E, 2012, vol. 85, 036210. DOI:  https://doi.org/10.1103/PhysRevE.85.036210
  35. Ashwin P., Building H., Burylko O. Weak chimeras in minimal networks of coupled phase oscillators. Chaos, 2015, vol. 25, iss. 1, 4905197. DOI:  https://doi.org/10.1063/1.4905197
  36. Omelchenko I., Omel’chenko O. E., Hövel P., Schöll E. When nonlocal coupling between oscillators becomes stronger: patched synchrony or multichimera states. Phys. Rev. Lett., 2013, vol. 110, iss. 22, 224101. DOI:  https://doi.org/10.1103/PhysRevLett.110.224101
  37. Omelchenko I., Zakharova A., Hövel P., Siebert J., Schöll E. Nonlinearity of local dynamics promotes multi-chimeras. Chaos, 2015, vol. 25, iss. 8, pp. 1‒8. DOI:  https://doi.org/10.1063/1.4927829
  38. Vüllings A., Hizanidis J., Omelchenko I., Hövel P. Clustered chimera states in systems of type-I excitability. New Journal of Physics, 2016, vol. 16, pp. 1‒14. DOI:  https://doi.org/10.1088/1367-2630/16/12/123039
  39. Tanaka H., Lichtenberg A., Oishi S. First order phase transition resulting from finite inertia in coupled oscillator systems. Phys. Rev. Lett., 1997, vol. 78, iss. 11, pp. 2104–2107. DOI:  https://doi.org/10.1103/PhysRevLett.78.2104
  40. Acebrón J. A., Bonilla L. L., Spigler R. Synchronization in populations of globally coupled oscillators with inertial effects. Phys. Rev. E, 2000, vol. 62, pp. 3437–3454. DOI:  https://doi.org/10.1103/physreve.62.3437
  41. Ha S.-Y., Kim Y., Li Z. Large-time dynamics of Kuramoto oscillators under the effects of inertia and frustration. SIAM Journal on Appl. Dynamical Systems, 2014, vol. 13, iss. 1, pp. 466–492. DOI:  https://doi.org/10.1137/130926559
  42. Belykh I. V., Brister B. N., Belykh V. N. Bistability of patterns of synchrony in Kuramoto oscillators with inertia. Chaos, 2016, vol. 26, iss. 9, 094822. DOI:  https://doi.org/10.1063/1.4961435
  43. Wisenfeld K., Colet P., Strogatz S. Synchronization transitions in a disordered Josephson series array. Phys. Rev. Lett., 1996, vol. 76, pp. 404‒407. DOI:  https://doi.org/10.1103/PhysRevLett.76.404
  44. Trees B., Saranathon V., Stroud D. Synchronization in disordered Josephson junction arrays: Small-world connections and the Kuramoto model. Phys. Rev. E, 2005, vol. 71, 016215. DOI:  https://doi.org/10.1103/PhysRevE.71.016215
  45. Hizanidis J., Lazarides N., Neofotistos G., Tsironis G. Chimera states and synchronization in magnetically driven SQUID metamaterials. Eur. Phys. J. Special Topics, 2016, vol. 225, pp. 1231–1243. DOI:  https://doi.org/10.1140/epjst/e2016-02668-9
  46. Mishra A., Saha S., Hens C., Roy P. K., Bose M., Louodop P., Cerdeira H. A., Dana S. K. Coherent libration to coherent rotational dynamics via chimeralike states and clustering in a Josephson junction array. Phys. Rev. E, 2017, vol. 95, iss. 1, 010201. DOI:  https://doi.org/10.1103/PhysRevE.95.010201
  47. Mishra A., Saha S., Roy P. K., Kapitaniak T., Dana S. K. Multicluster oscillation death and chimeralike states in globally coupled Josephson Junctions. Chaos, 2017, vol. 27, iss. 2, 023110. DOI:  https://doi.org/10.1063/1.4976147
  48. Kapitaniak T., Kuzma P., Wojewoda J., Czolczynski K., Maistrenko Y. Imperfect chimera states for coupled pendula. Sci. Rep., 2014, vol. 4, 6379. DOI:  https://doi.org/10.1038/srep06379
  49. Dudkowski D., Jaros P., Czokzynski K., Kapitaniak T. Small amplitude chimeras for coupled clocks. Nonlinear Dynamics, 2020, vol. 102, iss. 3, pp. 1‒12. DOI:  https://doi.org/10.1007/s11071-020-05990-z
  50. Filatrella G., Nielsen A. H., Pedersen N. F. Analysis of a power grid using a Kuramoto-like model. Eur. Phys. J., 2008, vol. 61, pp. 485–491. DOI:  https://doi.org/10.1140/epjb/e2008-00098-8
  51. Nishikawa T., Motter A. E. Comparative analysis of existing models for power grid synchronization. New Journal of Physics, 2015, vol. 17, iss. 1, 015012. DOI:  https://doi.org/10.1088/1367-2630/7/1/015012
  52. Grzybowski J. M. V., Macau E. E. N., Yoneyama T. On synchronization in power-grids modelled as networks of second-order Kuramoto oscillators. Chaos, 2016, vol. 26, iss. 11, 113113. DOI:  https://doi.org/10.1063/1.4967850
  53. Goldschmidt R. J., Pikovsky A., Politi A. Blinking chimeras in globally coupled rotators. Chaos, 2019, vol. 29, iss. 7, 5105367. DOI:  https://doi.org/10.1063/1.5105367
  54. Olmi S., Martens E. A., Thutupalli S., Torcini A. Intermittent chaotic chimeras for coupled rotators. Phys. Rev. E, 2015, vol. 92, 030901. DOI:  https://doi.org/10.1103/PhysRevE.92.030901
  55. Olmi S. Chimera states in coupled Kuramoto oscillators with inertia. Chaos, 2015, vol. 25, iss. 12, 123125. DOI:  https://doi.org/10.1063/1.4938734
  56. Jaros P., Maistrenko Y., Kapitaniak T. Chimera states on the route from coherence to rotating waves. Phys. Rev. E, 2015, vol. 91, iss. 2, 022907. DOI:  https://doi.org/10.1103/PhysRevE.91.022907
  57. 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, 011103. DOI:  https://doi.org/10.1063/1.5019792
  58. Maistrenko Y., Penkovsky B., Rosenblum M. Solitary state at the edge of synchrony in ensembles with attractive and repulsive interaction. Phys. Rev. E, 2014, vol. 89, iss. 6, 060901. DOI:  https://doi.org/10.1103/PhysRevE.89.060901
  59. Teichmann E., Rosenblum M. Solitary states and partial synchrony in oscillatory ensembles with attractive and repulsive interactions featured. Chaos, 2019, vol. 29, iss. 9, 093124. DOI:  https://doi.org/10.1063/1.5118843
  60. Shepelev I. A., Vadivasova T. E. Solitary states in a 2D lattice of bistable elements with global and close to global interaction. Russ. J. Nonlin. Dyn., 2017, vol. 13, iss. 3, pp. 317‒329. DOI: https://doi.org/10.20537/nd1703002
  61. 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. Special Topics, 2017, vol. 226, pp. 1857–1866. DOI:  https://doi.org/10.1140/epjst/e2017-70023-1
  62. Semenova N., Vadivasova T., Anishchenko V. Mechanism of solitary state appearance in an ensemble of nonlocally coupled Lozi maps. Eur. Phys. J. Special Topics, 2018, vol. 227, pp. 1173‒1183. DOI:  https://doi.org/10.1140/epjst/e2018-800035-y
  63. Rybalova E. V., Strelkova G. I., Anishchenko V. S. Mechanism of realizing a solitary state chimera in a ring of nonlocally coupled chaotic maps. Chaos, Solitons and Fractals, 2018, vol. 115, pp. 300‒305. DOI:  https://doi.org/10.1016/j.chaos.2018.09.003