Cite this article as:
Bukh A. V., Strelkova G. I., Anishchenko V. S. Spiral Wave Patterns in Two-Layer 2D Lattices of Nonlocally Coupled Discrete Oscillators. Synchronization of Spiral Wave Chimeras. Izvestiya of Saratov University. New series. Series Physics, 2019, vol. 19, iss. 3, pp. 166-177. DOI: https://doi.org/10.18500/1817-3020-2019-19-3-166-177
Spiral Wave Patterns in Two-Layer 2D Lattices of Nonlocally Coupled Discrete Oscillators. Synchronization of Spiral Wave Chimeras
The paper describes the spatio-temporal dynamics of a lattice that is given by a 2D N × N network of nonlocally coupled Nekorkin maps which model neuronal activity. The network behavior is studied for periodic and no-flux boundary conditions. It is shown that for certain values of the coupling parameters, rotating spiral waves and spiral wave chimeras can be observed in the considered lattice. We analyze and compare statistical and dynamical characteristics of the local oscillators from coherence and incoherence clusters of a spiral wave chimera. Furthermore, effects of mutual and external synchronization of spiral wave structures in two coupled such lattices are studied. We show numerically that spiral wave structures, including spiral wave chimeras, can be synchronized and establish the mechanism of their synchronization. Our numerical studies indicate that when the coupling strength between the lattices is sufficiently weak, only a certain part of oscillators of the interacting networks is imperfectly synchronized, while the other part demonstrates a partially synchronous behavior. If the spatiotemporal patterns in the lattices do not include incoherent cores, imperfect synchronization is realized for most oscillators above a certain value of the coupling strength. In the regime of spiral wave chimeras, the imperfect synchronization of all oscillators cannot be achieved even for sufficiently large values of the coupling strength.
1. Kaneko K. Pattern dynamics in spatiotemporal chaos. Physica D, 1967, vol. 34, pp. 1–41.
2. Afraimovich V., Nekorkin V., Osipov G., Shalfeev V. Stability, Structures And Chaos In Nonlinear Synchronization Networks. Singapore, World Scientifi c, 1995. 260 p.
3. Epstein I. R., Pojman J. A. An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos. Oxford, Oxford University Press, 1998. 480 p.
4. Strogatz S. H. Exploring complex networks. Nature, 2001, vol. 410, pp. 268.
5. Nekorkin V. I., Velarde M. G. Synergetic Phenomena in Active Lattices. Berlin, Springer, 2002. 359 p.
6. Dorogovtsev S. N., Mendes J. F. Evolution of networks. Advances in physics, 2002, vol. 51, pp. 1079–1187.
7. Newman M. E. The structure and function of complex networks. SIAM review, 2003, vol. 45, pp. 167–256.
8. Ben-Naim E., Frauenfelder H., Toroczkai Z. Complex networks. Berlin, Springer, 2004. 520 p.
9. Boccaletti S., Latora V., Moreno Y., Chavez M., Hwang D.-U. Complex networks: Structure and dynamics. Physics reports, 2006, vol. 424, pp. 175–308.
10. Martens E. A., Laing C. R., Strogatz S. H. Solvable model of spiral wave chimeras. Phys. Rev. Lett., 2010, vol. 104, pp. 044101.
11. Barabá si A.-L., Pósfai M. Network science. Cambridge, Cambridge University Press, 2016. 475 p.
12. Pecora L. M., Sorrentino F., Hagerstrom A. M., Murphy T. E., Roy R. Cluster synchronization and isolated desynchronization in complex networks with symmetries. Nat. Commun., 2014, vol. 5, pp. 4079.
13. Nekorkin V. I., Kazantsev V. B., Velarde M. G. Mutual synchronization of two lattices of bistable elements. Phys. Lett. A, 1997, vol. 236, pp. 505–512.
14. Nekorkin V. I., Voronin M. L., Velarde M. G. Clusters in an assembly of globally coupled bistable oscillators. Eur. Phys. J. B, 1999, vol. 9, pp. 533–543.
15. Belykh V. N., Belykh I. V., Hasler M. Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems. Phys. Rev. E, 2000, vol. 62, pp. 6332–6345.
16. Belykh V. N., Belykh I. V., Mosekilde E. Cluster synchronization modes in an ensemble of coupled chaotic oscillators. Phys. Rev. E, 2001, vol. 63, pp. 036216.
17. Nekorkin V., Kazantsev V., Velarde M. Synchronization in two-layer bistable coupled map lattices. Physica D, 2001, vol. 151, pp. 1–26.
18. Akopov A., Astakhov V., Vadivasova T., Shabunin A., Kapitaniak T. Frequency synchronization of clusters in coupled extended systems. Phys. Lett. A, 2005, vol. 334, pp. 169–172.
19. Hogan J., Krauskopf A. R., Bernado M. di, Wilson R. E., Osigna H. M., Homer M. E., Champneys A. R. Nonlinear Dynamics and Chaos. Where do we go from here? Florida, US, CRC Press, 2002. 376 p.
20. Kuramoto Y., Battogtokh D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom. Complex Syst., 2002, vol. 5, pp. 380–385.
21. Abrams D. M., Strogatz S. H. Chimera states for coupled oscillators. Phys. Rev. Lett., 2004, vol. 93, pp. 174102.
22. Panaggio M. J., Abrams D. M. Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity, 2015, vol. 28, pp. R67–R87.
23. Domenico M. De, Solé -Ribalta A., Cozzo E., Kivelä M., Moreno Y., Porter M. A., Gó mez S., Arenas A., Mathematical formulation of multilayer networks. Phys. Rev. X, 2013, vol. 3, pp. 041022.
24. Kivel M., Arenas A., Barthelemy M., Gleeson J. P., Moreno Y., Porter M. A. Multilayer networks. J. Complex Netw., 2014, vol. 2, pp. 203–271.
25. Boccaletti S., Bianconi G., Criado R., Genio C. del, Gmez-Gardees J., Romance M., Sendia-Nadal I., Wang Z., Zanin M. The structure and dynamics of multilayer networks. Phys. Rep., 2014, vol. 544, pp. 1 – 122.
26. Gambuzza L. V., Frasca M., Gó mez-Gardeñ es J. Intralayer synchronization in multiplex networks. Europhys. Lett., 2015, vol. 110, pp. 20010.
27. Ghosh S., Kumar A., Zakharova A., Jalan S. Birth and death of chimera: Interplay of delay and multiplexing. Europhys. Lett., 2016, vol. 115, pp. 60005.
28. Sevilla-Escoboza R., Sendia-Nadal I., Leyva I., Gutirrez R., Buld J. M., Boccaletti S. Inter-layer synchronization in multiplex networks of identical layers. Chaos, 2016, vol. 26, pp. 065304.
29. Ghosh S., Jalan S. Emergence of chimera in multiplex network. Int J Bifurc Chaos, 2016, vol. 26, pp. 1650120.
30. Maksimenko V. A., Makarov V. V., Bera B. K., Ghosh D., Dana S. K., 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, pp. 052205.
31. Mikhaylenko M., Ramlow L., Jalan S., Zakharova A. Weak multiplexing in neural networks: Switching between chimera and solitary states. Chaos, 2019, vol. 29, pp. 023122.
32. Andrzejak R. G., Ruzzene G., Malvestio I. Generalized synchronization between chimera states. Chaos, 2017, vol. 27, pp. 053114.
33. Singh A., Ghosh S., Jalan S., Kurths J. Synchronization in delayed multiplex networks. Europhys. Lett., 2015, vol. 111, pp. 30010.
34. Boccaletti S., Almendral J., Guan S., Leyva I., Liu Z., Sendia-Nadal I., Wang Z., Zou Y. Explosive transitions in complex networks structure and dynamics: Percolation and synchronization. Phys. Rep., 2016, vol. 660, pp. 1 – 94.
35. 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, pp. 111102.
36. Bukh A. V., Strelkova G. I., Anishchenko V. S. Synchronization of chimera states in coupled networks of nonlinear chaotic oscillators. Russ. J. Nonlin. Dyn., 2018, vol. 14, pp. 419–433.
37. Strelkova G., Vadivasova T., Anishchenko V. Synchronization of chimera states in a network of many unidirectionally coupled layers of discrete maps. Regul. Chaot. Dyn., 2018, vol. 23, pp. 948 – 960.
38. Rybalova E., Vadivasova T., Strelkova G., Anishchenko V., Zakharova A. Forced synchronization of a multilayer heterogeneous network of chaotic maps in the chimera state mode. Chaos, 2019, vol. 29, pp. 033134.
39. Leyva I., Sevilla-Escoboza R., Sendia-Nadal I., Gutirrez R., Buld J., Boccaletti S. Inter-layer synchronization in non-identical multi-layer networks. Sci. Rep., 2017, vol. 7, pp. 45475.
40. Sawicki J., Omelchenko I., Zakharova A., Schö ll E. Delay controls chimera relay synchronization in multiplex networks. Phys. Rev. E, 2018, vol. 98, pp. 062224.
41. Zhang X., Boccaletti S., Guan S., Liu Z. Explosive synchronization in adaptive and multilayer networks. Phys. Rev. Lett., 2015, vol. 114, pp. 038701.
42. Leyva I., Sendia-Nadal I., Sevilla-Escoboza R., Vera- Avila V. P., Chholak P., Boccaletti S. Relay synchronization in multiplex networks. Sci. Rep., 2018, vol. 8, pp. 8629.
43. Leyva I., Sendia-Nadal I., Boccaletti S. Explosive synchronization in mono and multilayer networks. Disc. and Cont. Dyn. Syst. Ser. B, 2018, vol. 25, pp. 1931.
44. Kachhvah A. D., Jalan S. Delay regulated explosive synchronization in multiplex networks. New J. Phys., 2019, vol. 21, pp. 015006.
45. Omelchenko I., Maistrenko Y., Hö vel P., Schö ll E. Loss of coherence in dynamical networks: Spatial chaos and chimera states. Phys. Rev. Lett., 2011, vol. 106, pp. 234102.
46. Omelchenko I., Riemenschneider B., Hö vel P., Maistrenko Y., Schö ll E. Transition from spatial coherence to incoherence in coupled chaotic systems. Phys. Rev. E, 2012, vol. 85, pp. 026212.
47. Hagerstrom A. M., Murphy T. E., Roy R., Hö vel P., Omelchenko I., Schö ll E. Experimental observation of chimeras in coupled-map lattices. Nat. Phys., 2012, vol. 8, pp. 658–661.
48. Tinsley M. R., Nkomo S., Showalter K. Chimera and phase-cluster states in populations of coupled chemical oscillators. Nat. Phys., 2012, vol. 8, pp. 662–665.
49. Larger L., Penkovsky B., Maistrenko Y. Virtual chimera states for delayed-feedback systems. Phys. Rev. Lett., 2013, vol. 111, pp. 054103.
50. Martens E. A., Thutupalli S., Fourriè re A., Hallatschek O. Chimera states in mechanical oscillator networks. Proc. Natl. Acad. Sci. USA, 2013, vol. 110, pp. 10563–10567.
51. Panaggio M. J., Abrams D. M. Chimera states on a fl at torus. Phys. Rev. E, 2013, vol. 110, pp. 094102.
52. Dudkowski D., Maistrenko Y., Kapitaniak T. Different types of chimera states: An interplay between spatial and dynamical chaos. Phys. Rev. E, 2014, vol. 90, pp. 032920.
53. Maistrenko Y. L., Vasylenko A., Sudakov O., Levchenko R., Maistrenko V. L. Cascades of multiheaded chimera states for coupled phase oscillators. Int J Bifurc Chaos. 2014, vol. 24, pp. 1440014.
54. Yeldesbay A., Pikovsky A., Rosenblum M. Chimeralike states in an ensemble of globally coupled oscillators. Phys. Rev. Lett., 2014, vol. 112, pp. 144103.
55. Kapitaniak T., Kuzma P., Wojewoda J., Czolczynski K., Maistrenko Y. Imperfect chimera states for coupled pendula. Sci. Rep., 2014, vol. 4, pp. 6379.
56. Hizanidis J., Panagakou E., Omelchenko I., Schö ll E., Hö vel P., Provata A. Chimera states in population dynamics: Networks with fragmented and hierarchical connectivities. Phys. Rev. E, 2015, vol. 92, pp. 012915.
57. Semenova N., Zakharova A., Schö ll E., Anishchenko V. Does hyperbolicity impede emergence of chimera states in networks of nonlocally coupled chaotic oscillators? Europhys. Lett., 2015, vol. 112, pp. 40002.
58. Olmi S., Martens E. A., Thutupalli S., Torcini A. Intermittent chaotic chimeras for coupled rotators. Phys. Rev. E, 2015, vol. 92, pp. 030901.
59. Panaggio M. J., Abrams D. M. Chimera states on the surface of a sphere. Phys. Rev. E, 2015, vol. 91, pp. 022909.
60. Kemeth F. P., Haugland S. W., Schmidt L., Kevrekidis I. G., Krischer K. A classifi cation scheme for chimera states. Chaos, 2016, vol. 26, pp. 094815.
61. Schö ll E. Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics. Eur. Phys. J. Spec. Top., 2016, vol. 225, pp. 891–919.
62. Ulonska S., Omelchenko I., Zakharova A., Schö ll E. Chimera states in networks of van der pol oscillators with hierarchical connectivities. Chaos, 2016, vol. 26, pp. 094825.
63. Semenova N., Zakharova A., Anishchenko V., Schö ll E. Coherence-resonance chimeras in a network of excitable elements. Phys. Rev. Lett., 2016, vol. 117, pp. 014102.
64. Semenov V., Zakharova A., Maistrenko Y., Schö ll E. Delayed-feedback chimera states: Forced multiclusters and stochastic resonance. Europhys. Lett., 2016, vol. 115, pp. 10005.
65. Hizanidis J., Kouvaris N. E., Zamora-Lò pez G., Dı̀az- Guilera A., Antonopoulos C. G. Chimera-like states in modular neural networks. Sci. Rep., 2016, vol. 6, pp. 19845.
66. Majhi S., Perc M., Ghosh D. Chimera states in uncoupled neurons induced by a multilayer structure. Sci. Rep., 2016, vol. 6, pp. 39033.
67. Sawicki J., Omelchenko I., Zakharova A., Schö ll E. Chimera states in complex networks: interplay of fractal topology and delay. Eur. Phys. J. Spec. Top., 2017, vol. 226, pp. 1883–1892.
68. 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, pp. 1857–1866.
69. Bogomolov S. A., Slepnev A. V., Strelkova G. I., Schö ll 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., 2016, vol. 43, pp. 25–36.
70. Tian C.-H., Zhang X.-Y., Wang Z.-H., Liu Z.-H. Diversity of chimera-like patterns from a model of 2d arrays of neurons with nonlocal coupling. Front. Phys., 2017, vol. 12, pp. 128904.
71. Schmidt A., Kasimatis T., Hizanidis J., Provata A., Hö vel P. Chimera patterns in two-dimensional networks of coupled neurons. Phys. Rev. E, 2017, vol. 95, pp. 032224.
72. Shepelev I., Bukh A., Vadivasova T., Anishchenko V., Zakharova A. Double-well chimeras in 2d lattice of chaotic bistable elements. Commun. Nonlinear Sci. Numer. Simul., 2018, vol. 54, pp. 50–61.
73. Hildebrand M., Cui J., Mihaliuk E., Wang J., Showalter K. Synchronization of spatiotemporal patterns in locally coupled excitable media. Phys. Rev. E, 2003, vol. 68, pp. 026205.
74. Kuramoto Y., Shima S.-i. Rotating spirals without phase singularity in reaction-diffusion systems. Prog. Theor. Phys. Suppl., 2003, vol. 150, pp. 115–125.
75. Shima S.-i., Kuramoto Y. Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. Phys. Rev. E, 2004, vol. 69, pp. 036213.
76. Tang X., Yang T., Epstein I., Liu Y., Zhao Y., Gao Q. Novel type of chimera spiral waves arising from decoupling of a diffusible component. J. Chem. Phys., 2014, vol. 141, pp. 024110.
77. Xie J., Knobloch E., Kao H.-C. Twisted chimera states and multicore spiral chimera states on a two-dimensional torus. Phys. Rev. E, 2015, vol. 92, pp. 042921.
78. Totz J. F., Rode J., Tinsley M. R., Showalter K., Engel H. Spiral wave chimera states in large populations of coupled chemical oscillators. Nat. Phys., 2018, vol. 14, pp. 282–285.
79. Laing C. R. The dynamics of chimera states in heterogeneous kuramoto networks. Physica D, 2009, vol. 238, pp. 1569–1588.
80. Nkomo S., Tinsley M. R., Showalter K. Chimera states in populations of nonlocally coupled chemical oscillators. Phys. Rev. Lett., 2013, vol. 110, pp. 244102.
81. Gu C., St-Yves G., Davidsen J. Spiral wave chimeras in complex oscillatory and chaotic systems. Phys. Rev. Lett., 2013, vol. 111, pp. 134101.
82. Kuramoto Y., Shima S.-i., Battogtokh D., Shiogai Y. Mean-fi eld theory revives in self-oscillatory fi elds with non-local coupling. Prog. Theor. Phys. Suppl., 2006, vol. 161, pp. 127–143.
83. Li B.-W., Dierckx H. Spiral wave chimeras in locally coupled oscillator systems. Phys. Rev. E, 2016, vol. 93, pp. 020202.
84. Weiss S., Deegan R. D. Weakly and strongly coupled belousov-zhabotinsky patterns. Phys. Rev. E, 2017, vol. 95, pp. 022215.
85. Kundu S., Majhi S., Muruganandam P., Ghosh D. Diffusion induced spiral wave chimeras in ecological system. Eur. Phys. J. Spec. Top., 2018, vol. 227, pp. 983–993.
86. Guo S., Dai Q., Cheng H., Li H., Xie F., Yang J. Spiral wave chimera in two-dimensional nonlocally coupled fi tzhughnagumo systems. Chaos, Solitons & Fractals, 2018, vol. 114, pp. 394–399.
87. Nekorkin V., Vdovin L. Map-based model of the neural activity. Prikladnaya nelineynaya dinamika [Applied nonlinear dynamics], 2007, vol. 15, pp. 36–60.
88. Izhikevich E. M. Which model to use for cortical spiking neurons? IEEE Trans. Neural Netw., 2004, vol. 15, pp. 1063–1070.
89. Pikovsky A., Politi A. Lyapunov Exponents. Cambridge, Cambridge University Press, 2016. 295 p.
90. Haugland S. W., Schmidt L., Krischer K. Self-organized alternating chimera states in oscillatory media. Sci. Rep., 2015, vol. 5, pp. 9883.
91. 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, pp. 093108.
92. Pikovsky A., Rosenblum M., Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge, Cambridge University Press, 2003. 433 p.
