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  -