TY - JOUR A1 - Lueck, S. A1 - Pikovskij, Arkadij T1 - Dynamics of multi-frequency oscillator ensembles with resonant coupling JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - We study dynamics of populations of resonantly coupled oscillators having different frequencies. Starting from the coupled van der Pol equations we derive the Kuramoto-type phase model for the situation, where the natural frequencies of two interacting subpopulations are in relation 2 : 1. Depending on the parameter of coupling, ensembles can demonstrate fully synchronous clusters, partial synchrony (only one subpopulation synchronizes), or asynchrony in both subpopulations. Theoretical description of the dynamics based on the Watanabe-Strogatz approach is developed. KW - Oscillator populations KW - Kuramoto model KW - Resonant interaction Y1 - 2011 U6 - https://doi.org/10.1016/j.physleta.2011.06.016 SN - 0375-9601 VL - 375 IS - 28-29 SP - 2714 EP - 2719 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Burylko, Oleksandr A1 - Pikovskij, Arkadij T1 - Desynchronization transitions in nonlinearly coupled phase oscillators JF - Physica :D, Nonlinear phenomena N2 - We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs. KW - Coupled oscillators KW - Oscillator ensembles KW - Kuramoto model KW - Nonlinear coupling KW - Bifurcations Y1 - 2011 U6 - https://doi.org/10.1016/j.physd.2011.05.016 SN - 0167-2789 VL - 240 IS - 17 SP - 1352 EP - 1361 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Dynamics of heterogeneous oscillator ensembles in terms of collective variables JF - Physica :D, Nonlinear phenomena N2 - We consider general heterogeneous ensembles of phase oscillators, sine coupled to arbitrary external fields. Starting with the infinitely large ensembles, we extend the Watanabe-Strogatz theory, valid for identical oscillators, to cover the case of an arbitrary parameter distribution. The obtained equations yield the description of the ensemble dynamics in terms of collective variables and constants of motion. As a particular case of the general setup we consider hierarchically organized ensembles, consisting of a finite number of subpopulations, whereas the number of elements in a subpopulation can be both finite or infinite. Next, we link the Watanabe-Strogatz and Ott-Antonsen theories and demonstrate that the latter one corresponds to a particular choice of constants of motion. The approach is applied to the standard Kuramoto-Sakaguchi model, to its extension for the case of nonlinear coupling, and to the description of two interacting subpopulations, exhibiting a chimera state. With these examples we illustrate that, although the asymptotic dynamics can be found within the framework of the Ott-Antonsen theory, the transients depend on the constants of motion. The most dramatic effect is the dependence of the basins of attraction of different synchronous regimes on the initial configuration of phases. KW - Coupled oscillators KW - Oscillator ensembles KW - Kuramoto model KW - Nonlinear coupling KW - Watanabe-Strogatz theory KW - Ott-Antonsen theory Y1 - 2011 U6 - https://doi.org/10.1016/j.physd.2011.01.002 SN - 0167-2789 VL - 240 IS - 9-10 SP - 872 EP - 881 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Komarov, Maxim A1 - Pikovskij, Arkadij T1 - The Kuramoto model of coupled oscillators with a bi-harmonic coupling function JF - Physica : D, Nonlinear phenomena N2 - We study synchronization in a Kuramoto model of globally coupled phase oscillators with a bi-harmonic coupling function, in the thermodynamic limit of large populations. We develop a method for an analytic solution of self-consistent equations describing uniformly rotating complex order parameters, both for single-branch (one possible state of locked oscillators) and multi-branch (two possible values of locked phases) entrainment. We show that synchronous states coexist with the neutrally linearly stable asynchronous regime. The latter has a finite life time for finite ensembles, this time grows with the ensemble size as a power law. (C) 2014 Elsevier B.V. All rights reserved. KW - Kuramoto model KW - Bi-harmonic coupling function KW - Multi-branch entrainment KW - Synchronization Y1 - 2014 U6 - https://doi.org/10.1016/j.physd.2014.09.002 SN - 0167-2789 SN - 1872-8022 VL - 289 SP - 18 EP - 31 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Vlasov, Vladimir A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij T1 - Dynamics of weakly inhomogeneous oscillator populations: perturbation theory on top of Watanabe-Strogatz integrability JF - Journal of physics : A, Mathematical and theoretical N2 - As has been shown by Watanabe and Strogatz (WS) (1993 Phys. Rev. Lett. 70 2391), a population of identical phase oscillators, sine-coupled to a common field, is a partially integrable system: for any ensemble size its dynamics reduce to equations for three collective variables. Here we develop a perturbation approach for weakly nonidentical ensembles. We calculate corrections to the WS dynamics for two types of perturbations: those due to a distribution of natural frequencies and of forcing terms, and those due to small white noise. We demonstrate that in both cases, the complex mean field for which the dynamical equations are written is close to the Kuramoto order parameter, up to the leading order in the perturbation. This supports the validity of the dynamical reduction suggested by Ott and Antonsen (2008 Chaos 18 037113) for weakly inhomogeneous populations. KW - Kuramoto model KW - oscillator populations KW - integrability KW - perturbation theory Y1 - 2016 U6 - https://doi.org/10.1088/1751-8113/49/31/31LT02 SN - 1751-8113 SN - 1751-8121 VL - 49 PB - IOP Publ. Ltd. CY - Bristol ER - TY - GEN A1 - Munyaev, Vyacheslav A1 - Smirnov, Lev A. A1 - Kostin, Vasily A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Analytical approach to synchronous states of globally coupled noisy rotators T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We study populations of globally coupled noisy rotators (oscillators with inertia) allowing a nonequilibrium transition from a desynchronized state to a synchronous one (with the nonvanishing order parameter). The newly developed analytical approaches resulted in solutions describing the synchronous state with constant order parameter for weakly inertial rotators, including the case of zero inertia, when the model is reduced to the Kuramoto model of coupled noise oscillators. These approaches provide also analytical criteria distinguishing supercritical and subcritical transitions to the desynchronized state and indicate the universality of such transitions in rotator ensembles. All the obtained analytical results are confirmed by the numerical ones, both by direct simulations of the large ensembles and by solution of the associated Fokker-Planck equation. We also propose generalizations of the developed approaches for setups where different rotators parameters (natural frequencies, masses, noise intensities, strengths and phase shifts in coupling) are dispersed. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1188 KW - coupled rotators KW - synchronization transition KW - hysteresis KW - Kuramoto model KW - noisy systems Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-524261 SN - 1866-8372 IS - 2 ER - TY - JOUR A1 - Munyaev, Vyacheslav A1 - Smirnov, Lev A. A1 - Kostin, Vasily A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Analytical approach to synchronous states of globally coupled noisy rotators JF - New Journal of Physics N2 - We study populations of globally coupled noisy rotators (oscillators with inertia) allowing a nonequilibrium transition from a desynchronized state to a synchronous one (with the nonvanishing order parameter). The newly developed analytical approaches resulted in solutions describing the synchronous state with constant order parameter for weakly inertial rotators, including the case of zero inertia, when the model is reduced to the Kuramoto model of coupled noise oscillators. These approaches provide also analytical criteria distinguishing supercritical and subcritical transitions to the desynchronized state and indicate the universality of such transitions in rotator ensembles. All the obtained analytical results are confirmed by the numerical ones, both by direct simulations of the large ensembles and by solution of the associated Fokker-Planck equation. We also propose generalizations of the developed approaches for setups where different rotators parameters (natural frequencies, masses, noise intensities, strengths and phase shifts in coupling) are dispersed. KW - coupled rotators KW - synchronization transition KW - hysteresis KW - Kuramoto model KW - noisy systems Y1 - 2019 VL - 22 IS - 2 PB - Springer Science CY - New York ER -