ISSN 0869-6632 (Print)
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Kasatkin D. V., Emelianova A. A., Nekorkin V. I. Nonlinear phenomena in Kuramoto networks with dynamical couplings. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 4, pp. 635-675. DOI: 10.18500/0869-6632-2021-29-4-635-675

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Nonlinear phenomena in Kuramoto networks with dynamical couplings

Kasatkin Dmitry Vladimirovich, Institute of Applied Physics of the Russian Academy of Sciences
Emelianova Anastasiia Aleksandrovna, Institute of Applied Physics of the Russian Academy of Sciences
Nekorkin Vladimir Isaakovich, Institute of Applied Physics of the Russian Academy of Sciences

The purpose of this study is to acquaint the reader with one of the effective approaches to describing processes in adaptive networks, built in the framework of the well-known Kuramoto model. Methods. The solution to this problem is based on the analysis of the results of works devoted to the study of the dynamics of oscillatory networks with adaptive couplings. Main classes of models of dynamical couplings used in the description of adaptive networks are considered, and the dynamical and structural effects caused by the presence of the corresponding law of coupling adaptation are analyzed. Results. Principles of constructing models of adaptive networks based on the phase description developed by Kuramoto are presented. Materials presented in the review show that the Kuramoto system with dynamic couplings demonstrates a wide range of fundamentally new phenomena and modes. Considered networks include well-known models of dynamical couplings that implement various laws of adaptation of inter-element interactions depending on the states of the elements, in particular, on their relative phase difference. For each network model, a class of possible solutions is established, and general properties of collective dynamics are identified, due to the presence of adaptability of couplings. One of the features of such networks is the multistability of behavior, determined by the possibility of the formation in the network of many different cluster states, including chimera ones. It was found that the implemented coupling adaptation mechanism affects not only the configuration of clusters formed in the network, but also the nature of phase distributions within them. The processes of cluster formation are accompanied by a restructuring of the interaction topology, leading to the formation of hierarchical and modular structures. Conclusion. In conclusion, we briefly summarize the results presented in the review.

The work was performed as a part of the State Assignment of the Institute of Applied Physics RAS, project No. 0030-2021-0011, and was supported by the Russian Foundation for Basic Research (grant No. 20-52-12021)
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