TY - JOUR A1 - Omel'chenko, Oleh T1 - Mathematical framework for breathing chimera states JF - Journal of nonlinear science N2 - About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence-incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott-Antonsen reduction technique. KW - Coupled oscillators KW - Breathing chimera states KW - Coherence-incoherence KW - patterns KW - Ott-Antonsen equation KW - Periodic solutions KW - Stability Y1 - 2022 U6 - https://doi.org/10.1007/s00332-021-09779-1 SN - 0938-8974 SN - 1432-1467 VL - 32 IS - 2 PB - Springer CY - New York ER - TY - JOUR A1 - Turukina, L. V. A1 - Pikovskij, Arkadij T1 - Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - We show that a hyperbolic chaos can be observed in resonantly coupled oscillators near a Hopf bifurcation, described by normal-form-type equations for complex amplitudes. The simplest example consists of four oscillators, comprising two alternatively activated, due to an external periodic modulation, pairs. In terms of the stroboscopic Poincare map, the phase differences change according to an expanding Bernoulli map that depends on the coupling type. Several examples of hyperbolic chaos for different types of coupling are illustrated numerically. KW - Coupled oscillators KW - Hyperbolic chaos Y1 - 2011 U6 - https://doi.org/10.1016/j.physleta.2011.02.017 SN - 0375-9601 VL - 375 IS - 11 SP - 1407 EP - 1411 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 - 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 -