#### For citation:

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

# Nonlinear phenomena in Kuramoto networks with dynamical couplings

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.

- Winfree AT. Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 1967;16(1):15–42. DOI: 10.1016/0022-5193(67)90051-3.
- Kuramoto Y. Self-entrainment of a population of coupled non-linear oscillators. In: Araki H. International Symposium on Mathematical Problems in Theoretical Physics. Vol. 39 of Lecture Notes in Physics. Springer, Berlin, Heidelberg; 1975. P. 420–422. DOI: 10.1007/BFb0013365.
- Kuramoto Y. Chemical Oscillations, Waves, and Turbulence. Springer, Berlin, Heidelberg; 1984. 158 p. DOI: 10.1007/978-3-642-69689-3.
- Sakaguchi H, Kuramoto Y. A soluble active rotater model showing phase transitions via mutual entertainment. Prog. Theor. Phys. 1986;76(3):576–581. DOI: 10.1143/PTP.76.576.
- Strogatz SH. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D. 2000;143(1–4):1–20. DOI: 10.1016/S0167-2789(00)00094-4.
- Acebron JA, Bonilla LL, Perez Vicente CJ, Ritort F, Spigler R. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 2005;77(1):137–185. DOI: 10.1103/RevModPhys.77.137.
- Pikovsky A, Rosenblum M. Dynamics of globally coupled oscillators: Progress and perspectives. Chaos. 2015;25(9):097616. DOI: 10.1063/1.4922971.
- Rodrigues FA, Peron TKDM, Ji P, Kurths J. The Kuramoto model in complex networks. Phys. Rep. 2016;610:1–98. DOI: 10.1016/j.physrep.2015.10.008.
- Maslennikov OV, Nekorkin VI. Adaptive dynamical networks. Phys. Usp. 2017;60(7):694–704. DOI: 10.3367/UFNe.2016.10.037902.
- Gross T, Blasius B. Adaptive coevolutionary networks: a review. J. R. Soc. Interface. 2008;5(20): 259–271. DOI: 10.1098/rsif.2007.1229.
- Maistrenko YL, Lysyansky B, Hauptmann C, Burylko O, Tass PA. Multistability in the Kuramoto model with synaptic plasticity. Phys. Rev. E. 2007;75(6):066207. DOI: 10.1103/PhysRevE.75.066207.
- Takahashi YK, Kori H, Masuda N. Self-organization of feed-forward structure and entrainment in excitatory neural networks with spike-timing-dependent plasticity. Phys. Rev. E. 2009;79(5):051904. DOI: 10.1103/PhysRevE.79.051904.
- Picallo CB, Riecke H. Adaptive oscillator networks with conserved overall coupling: Sequential firing and near-synchronized states. Phys. Rev. E. 2011;83(3):036206. DOI: 10.1103/PhysRevE.83.036206.
- Seliger P, Young SC, Tsimring LS. Plasticity and learning in a network of coupled phase oscillators. Phys. Rev. E. 2002;65(4):041906. DOI: 10.1103/PhysRevE.65.041906.
- Niyogi RK, English LQ. Learning-rate-dependent clustering and self-development in a network of coupled phase oscillators. Phys. Rev. E. 2009;80(6):066213. DOI: 10.1103/PhysRevE.80.066213.
- Timms L, English LQ. Synchronization in phase-coupled Kuramoto oscillator networks with axonal delay and synaptic plasticity. Phys. Rev. E. 2014;89(3):032906. DOI: 10.1103/PhysRevE.89.032906.
- Ren Q, Zhao J. Adaptive coupling and enhanced synchronization in coupled phase oscillators. Phys. Rev. E. 2007;76(1):016207. DOI: 10.1103/PhysRevE.76.016207.
- Hou JL, Zhao J. The order-oscillation induced by negative feedback in the adaptive scheme. Phys. Lett. A. 2010;374(7):929–932. DOI: 10.1016/j.physleta.2009.12.016.
- Ren Q, He M, Yu X, Long Q, Zhao J. The adaptive coupling scheme and the heterogeneity in intrinsic frequency and degree distributions of the complex networks. Phys. Lett. A. 2014;378(3): 139–146. DOI: 10.1016/j.physleta.2013.10.031.
- Aoki T, Aoyagi T. Co-evolution of phases and connection strengths in a network of phase oscillators. Phys. Rev. Lett. 2009;102(3):034101. DOI: 10.1103/PhysRevLett.102.034101.
- Tanaka T, Aoki T, Aoyagi T. Dynamics in co-evolving networks of active elements. Forma. 2009;24:17–22.
- Aoki T, Aoyagi T. Self-organized network of phase oscillators coupled by activity-dependent interactions. Phys. Rev. E. 2011;84(6):066109. DOI: 10.1103/PhysRevE.84.066109.
- Kasatkin DV, Nekorkin VI. Dynamics of the phase oscillators with plastic couplings. Radiophys. Quantum El. 2016;58(11):877–891. DOI: 10.1007/s11141-016-9662-1.
- Kasatkin DV, Nekorkin VI. Dynamics of a network of interacting phase oscillators with dynamic couplings. Izvestiya VUZ. Applied Nonlinear Dynamics. 2015;23(4):58–70 (in Russian). DOI: 10.18500/0869-6632-2015-23-4-58-70.
- Emelianova AA, Nekorkin VI. On the intersection of a chaotic attractor and a chaotic repeller in the system of two adaptively coupled phase oscillators. Chaos. 2019;29(11):111102. DOI: 10.1063/1.5130994.
- Emelianova AA, Nekorkin VI. The third type of chaos in a system of two adaptively coupled phase oscillators. Chaos. 2020;30(5):051105. DOI: 10.1063/5.0009525.
- Emelianova AA, Nekorkin VI. Emergence and synchronization of a reversible core in a system of forced adaptively coupled Kuramoto oscillators. Chaos. 2021;31(3):033102. DOI: 10.1063/5.0038833.
- Gonchenko SV, Turaev DV. On three types of dynamics and the notion of attractor. Proc. Steklov Inst. Math. 2017;297(1):116–137. DOI: 10.1134/S0081543817040071.
- Gonchenko SV, Turaev DV, Shilnikov LP. On Newhouse domains of two-dimensional diffeomorphisms that are close to a diffeomorphism with a structurally unstable heteroclinic contour. Proc. Steklov Inst. Math. 1997;216:70–118.
- Nekorkin VI, Kasatkin DV. Dynamics of a network of phase oscillators with plastic couplings. AIP Conf. Proc. 2016;1738(1):210010. DOI: 10.1063/1.4951993.
- Kasatkin DV, Yanchuk S, Scholl E, Nekorkin VI. Self-organized emergence of multilayer structure and chimera states in dynamical networks with adaptive couplings. Phys. Rev. E. 2017;96(6):062211. DOI: 10.1103/PhysRevE.96.062211.
- Berner R, Scholl E, Yanchuk S. Multiclusters in networks of adaptively coupled phase oscillators. SIAM J. Appl. Dyn. Syst. 2019;18(4):2227–2266. DOI: 10.1137/18M1210150.
- Berner R, Fialkowski J, Kasatkin D, Nekorkin V, Yanchuk S, Scholl E. Hierarchical frequency clusters in adaptive networks of phase oscillators. Chaos. 2019;29(10):103134. DOI: 10.1063/1.5097835.
- Kasatkin DV, Nekorkin VI. The effect of topology on organization of synchronous behavior in dynamical networks with adaptive couplings. Eur. Phys. J. Spec. Top. 2018;227(10–11):1051–1061. DOI: 10.1140/epjst/e2018-800077-7.
- Berner R, Polanska A, Scholl E, Yanchuk S. Solitary states in adaptive nonlocal oscillator networks. Eur. Phys. J. Spec. Top. 2020;229(12–13):2183–2203. DOI: 10.1140/epjst/e2020-900253-0.
- Panaggio MJ, Abrams DM. Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity. 2015;28(3):R67. DOI: 10.1088/0951-7715/28/3/R67.
- Gutierrez R, Amann A, Assenza S, Gomez-Gardenes J, Latora V, Boccaletti S. Emerging meso- and macroscales from synchronization of adaptive networks. Phys. Rev. Lett. 2011;107(23):234103. DOI: 10.1103/PhysRevLett.107.234103.
- Assenza S, Gutierrez R, Gomez-Gardenes J, Latora V, Boccaletti S. Emergence of structural patterns out of synchronization in networks with competitive interactions. Sci. Rep. 2011;1(1):99. DOI: 10.1038/srep00099.
- Avalos-Gaytan V, Almendral JA, Papo D, Schaeffer SE, Boccaletti S. Assortative and modular networks are shaped by adaptive synchronization processes. Phys. Rev. E. 2012;86(1):015101(R). DOI: 10.1103/PhysRevE.86.015101.
- Avalos-Gaytan V, Almendral JA, Leyva I, Battiston F, Nicosia V, Latora V, Boccaletti S. Emergent explosive synchronization in adaptive complex networks. Phys. Rev. E. 2018;97(4):042301. DOI: 10.1103/PhysRevE.97.042301.
- Kasatkin DV, Klinshov VV, Nekorkin VI. Itinerant chimeras in an adaptive network of pulse coupled oscillators. Phys. Rev. E. 2019;99(2):022203. DOI: 10.1103/PhysRevE.99.022203.
- Berner R, Sawicki J, Scholl E. Birth and stabilization of phase clusters by multiplexing of adaptive networks. Phys. Rev. Lett. 2020;124(8):088301. DOI: 10.1103/PhysRevLett.124.088301.
- Kasatkin DV, Nekorkin VI. Synchronization of chimera states in a multiplex system of phase oscillators with adaptive couplings. Chaos. 2018;28(9):093115. DOI: 10.1063/1.5031681.
- Andrzejak RG, Ruzzene G, Malvestio I. Generalized synchronization between chimera states. Chaos. 2017;27(5):053114. DOI: 10.1063/1.4983841.
- Maksimenko VA, Makarov VV, Bera BK, Ghosh D, Dana SK, Goremyko MV, Frolov NS, Koronovskii AA, Hramov AE. Excitation and suppression of chimera states by multiplexing. Phys. Rev. E. 2016;94(5):052205. DOI: 10.1103/PhysRevE.94.052205.
- Makarov VV, Koronovskii AA, Maksimenko VA, Hramov AE, Moskalenko OI, Buldu JM, Boccaletti S. Emergence of a multilayer structure in adaptive networks of phase oscillators. Chaos, Solitons & Fractals. 2016;84:23–30. DOI: 10.1016/j.chaos.2015.12.022.
- Kachhvah AD, Dai X, Boccaletti S, Jalan S. Interlayer Hebbian plasticity induces first-order transition in multiplex networks. New J. Phys. 2020;22:122001. DOI: 10.1088/1367-2630/abcf6b.

- 1717 reads