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Strange nonchaotic attractors typically appear in quasiperiodically driven nonlinear systems. Two methods of their characterization are proposed. The first one is based on the bifurcation analysis of the systems, resulting from periodic approximations of the quasiperiodic forcing. Secondly, we propose th characterize their strangeness by calculating a phase sensitivity exponent, that measures the sensitivity with respect to changes of the phase of the external force. It is shown, that phase sensitivity appears if there is a non-zero probability for positive local Lyapunov exponents to occur.

We develop an effective description of noise-induced oscillations based on deterministic phase dynamics. The phase equation is constructed to exhibit correct frequency and distribution density of noise-induced oscillations. In the simplest one-dimensional case the effective phase equation is obtained analytically, whereas for more complex situations a simple method of data processing is suggested. As an application an effective coupling function is constructed that quantitatively describes periodically forced noise-induced oscillations.

An effective dynamical description of a general class of stochastic phase oscillators is presented. For this, the effective phase velocity is defined either by the stochastic phase oscillators invariant probability density or its first passage times. Using the first approach the effective phase exhibits the correct frequency and invariant distribution density, whereas the second approach models the proper phase resetting curve. The discrepancy of the effective models is most pronounced for noise-induced oscillations and is related to non-monotonicity of the stochastic phase variable due to fluctuations.

Phase compactons
(2006)

We study the phase dynamics of a chain of autonomous, self-sustained, dispersively coupled oscillators. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons - solitary waves with a compact support - and kovatons - compact formations of glued together kink-antikink pairs that propagate with a unique speed, but may assume an arbitrary width. We demonstrate that lattice solitary waves, though not exactly compact, have tails which decay at a superexponential rate. They are robust and collide nearly elastically and together with wave sources are the building blocks of the dynamics that emerges from typical initial conditions. In finite lattices, after a long time, the dynamics becomes chaotic. Numerical studies of the complex Ginzburg-Landau lattice show that the non-dispersive coupling causes a damping and deceleration, or growth and acceleration, of compactons. A simple perturbation method is applied to study these effects. (c) 2006 Elsevier B.V. All rights reserved

We consider the effect of external noise on the dynamics of limit cycle oscillators. The Lyapunov exponent becomes negative under influence of small white noise, what means synchronization of two or more identical systems subject to common noise. We analytically study the effect of small nonidentities in the oscillators and in the noise, and derive statistical characteristics of deviations from the perfect synchrony. Large white noise can lead to desynchronization of oscillators, provided they are nonisochronous. This is demonstrated for the Van der Pol-Duffing system

We study the stability of self-sustained oscillations under the influence of external noise. For small-noise amplitude a phase approximation for the Langevin dynamics is valid. A stationary distribution of the phase is used for an analytic calculation of the maximal Lyapunov exponent. We demonstrate that for small noise the exponent is negative, which corresponds to synchronization of oscillators. (c) 2004 Elsevier B.V. All rights reserved

Common noise acting on a population of identical oscillators can synchronize them. We develop a description of this process which is not limited to the states close to synchrony, but provides a global picture of the evolution of the ensembles. The theory is based on the Watanabe-Strogatz transformation, allowing us to obtain closed stochastic equations for the global variables. We show that at the initial stage, the order parameter grows linearly in time, while at the later stages the convergence to synchrony is exponentially fast. Furthermore, we extend the theory to nonidentical ensembles with the Lorentzian distribution of natural frequencies and determine the stationary values of the order parameter in dependence on driving noise and mismatch.

We consider general properties of groups of interacting oscillators, for which the natural frequencies are not in resonance. Such groups interact via nonoscillating collective variables like the amplitudes of the order parameters defined for each group. We treat the phase dynamics of the groups using the Ott-Antonsen ansatz and reduce it to a system of coupled equations for the order parameters. We describe different regimes of cosynchrony in the groups. For a large number of groups, heteroclinic cycles, corresponding to a sequential synchronous activity of groups and chaotic states where the order parameters oscillate irregularly, are possible.

We propose a novel method of reconstructing the topology and interaction functions for a general oscillator network. An ensemble of initial phases and the corresponding instantaneous frequencies is constructed by repeating random phase resets of the system dynamics. The desired details of network structure are then revealed by appropriately averaging over the ensemble. The method is applicable for a wide class of networks with arbitrary emergent dynamics, including full synchrony.

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.