ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Pikovsky A. S. Synchronization of oscillators with hyperbolic chaotic phases. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 1, pp. 78-87. DOI: 10.18500/0869-6632-2021-29-1-78-87

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Synchronization of oscillators with hyperbolic chaotic phases

Pikovsky Arkady Samuilovich, Potsdam University

Topic and aim. Synchronization in populations of coupled oscillators can be characterized with order parameters that describe collective order in ensembles. A dependence of the order parameter on the coupling constants is well-known for coupled periodic oscillators. The goal of the study is to extend this analysis to ensembles of oscillators with chaotic phases, moreover with phases possessing hyperbolic chaos. Models and methods. Two models are studied in the paper. One is an abstract discrete-time map, composed with a hyperbolic Bernoulli transformation and with Kuramoto dynamics. Another model is a system of coupled continuous-time chaotic oscillators, where each individual oscillator has a hyperbolic attractor of Smale–Williams type. Results. The discrete-time model is studied with the Ott–Antonsen ansatz, which is shown to be invariant under the application of the Bernoulli map. The analysis of the resulting map for the order parameter shows, that the asynchronouis state is always stable, but the synchronous one becomes stable above a certain coupling strength. Numerical analysis of the continuous-time model reveals a complex sequence of transitions from an asynchronous state to a completely synchronous hyperbolic chaos, with intermediate stages that include regimes with periodic in time mean field, as well as with weakly and strongly irregular mean field variations. Discussion. Results demonstrate that synchronization of systems with hyperbolic chaos of phases is possible, although a rather strong coupling is required. The approach can be applied to other systems of interacting units with hyperbolic chaotic dynamics.

А.П. выражает благодарность Российскому научному фонду (исследования в разделе 2, грант № 17-12-01534) и DFG (грант PI 220/21-1). Численные эксперименты в разделе 1 были поддержаны лабораторией динамических систем и приложений НИУ ВШЭ министерства науки и высшего образования Российской Федерации (грант № 075-15-2019-1931).
  1. Pikovsky A., Rosenblum M., and Kurths J. Synchronization. A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge, 2001. DOI: 10.1017/CBO9780511755743.
  2. Kuznetsov S.P. Example of a physical system with a hyperbolic attractor of the Smale–Williams Type. Phys. Rev. Lett., 2005, vol. 95, no. 14, p. 144101. DOI: 10.1103/PhysRevLett.95.144101.
  3. Kuznetsov S.P. Hyperbolic Chaos: A Physicist’s View. Higher Education Press, Beijing and Springer-Verlag: Berlin, Heidelberg, 2012, 336 p. DOI: 10.1007/978-3-642-23666-2.
  4. Kuznetsov S. and Pikovsky A. Autonomous coupled oscillators with hyperbolic strange attractors. Physica D, 2007, vol. 232, no. 2, pp. 87–102. DOI: 10.1016/j.physd.2007.05.008.
  5. Acebr´on J.A., Bonilla L.L., Vicente C.J.P., Ritort F., and Spigler R. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys., 2005, vol. 77, no. 1, p. 137. DOI: 10.1103/RevModPhys.77.137.
  6. Ott E. and Antonsen T.M. Low dimensional behavior of large systems of globally coupled oscillators. Chaos, 2008, vol. 18, no. 3, p. 037113. DOI: 10.1063/1.2930766.
  7. Pikovsky A., Rosenblum M., and Kurths J. Synchronization in a population of globally coupled chaotic oscillators. Europhys. Lett., 1996, vol. 34, no. 3, pp. 165–170. DOI: 10.1209/epl/i1996-00433-3.