Michael A. Zaks

Phase synchronization of chaotic oscillations in terms of periodic orbits.

We consider phase synchronization of chaotic continuous-time oscillator by periodic external force. Phase-locking regions are defined for unstable periodic cycles embedded in chaos, and synchronization is described in terms of these regions. A special flow construction is used to derive a simple discrete-time model of the phenomenon. It allows to describe quantitatively the intermittency at the transition to phase synchronization. (c) 1997 American Institute of Physics.

Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems.

We study the stochastic dynamics of an ensemble of N globally coupled excitable elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is disturbed by independent Gaussian noise. In simulations of the Langevin dynamics we characterize the collective behavior of the ensemble in terms of its mean field and show that with the increase of noise the mean field displays a transition from a steady equilibrium to global oscillations and then, for sufficiently large noise, back to another equilibrium. In the course of this transition diverse regimes of collective dynamics ranging from periodic subthreshold oscillations to large-amplitude oscillations and chaos are observed. In order to understand the details and mechanisms of these noise-induced dynamics we consider the thermodynamic limit N-->infinity of the ensemble, and derive the cumulant expansion describing temporal evolution of the mean field fluctuations. In Gaussian approximation this allows us to perform the bifurcation analysis; its results are in good qualitative agreement with dynamical scenarios observed in the stochastic simulations of large ensembles.

Role of unstable periodic orbits in phase and lag synchronization between coupled chaotic oscillators

An increase of the coupling strength in the system of two coupled Rössler oscillators leads from a nonsynchronized state through phase synchronization to the regime of lag synchronization. The role of unstable periodic orbits in these transitions is investigated. Changes in the structure of attracting sets are discussed. We demonstrate that the onset of phase synchronization is related to phase-lockings on the surfaces of unstable tori, whereas transition from phase to lag synchronization is preceded by a decrease in the number of unstable periodic orbits.

Coarsening in a bistable system with long-delayed feedback

The correspondence between long-delayed systems and one-dimensional spatially extended media enables a direct interpretation of purely temporal phenomena in terms of spatio- temporal patterns. On the basis of this result, we provide the evidence of a characteristic spatio- temporal dynamics —coarsening— in a long-delayed bistable system. Nucleation, propagation and annihilation of fronts, leading eventually to a single phase, are observed in an experiment based on a laser with opto-electronic feedback. A numerical and analytical study of a general phenomenological model is also performed and compared with the experimental findings. Copyright c EPLA, 2012

Periodic Stationary Patterns Governed by a Convective Cahn--Hilliard Equation

We investigate bifurcations of stationary periodic solutions of a convective Cahn--Hilliard equation,

TWO MECHANISMS OF THE TRANSITION TO CHAOS IN FINITE-DIMENSIONAL MODELS OF CONVECTION

u_t + Duu_x + (u - u^3 + u_{xx})_{xx} = 0

On the Correlation Dimension of the Spectral Measure for the Thue-Morse Sequence

, describing phase separation in driven systems, and study the stability of the main family of these solutions. For the driving parameter

Steady stokes flow with long-range correlations, fractal fourier spectrum, and anomalous transport.

D D_0

Comment on "Time-averaged properties of unstable periodic orbits and chaotic orbits in ordinary differential equation systems".

, the periodic stationary solutions are stable if their wavelength belongs to a certain stability interval. It is therefore shown that in a driven phase-separating system that undergoes spinodal decomposition the coarsening can be stopped by the driving force, and formation of stable periodic structures is possible. The modes that destroy the stability at the boundaries of the stability interval are also found.

On the generalized dimensions for the Fourier spectrum of the Thue-Morse sequence

Abstract The onset of chaos in systems of ordinary differential equations possessing a stationary solution with a one-dimensional unstable manifold is studied both numerically and qualitatively with the help of an auxiliary piecewise monotomic discontinuous recursion relation. A connection is established between the route to chaos and the ratio of two leading eigenvalues of the vector field linearized near the fixed point. This connection is confirmed by numerical data obtained from the investigation of differential equations originating from a hydrodynamical problem. Two routes are considered — a well-known mechanism suggested by Lorenz and another one which is due to the accumulation of bifurcations corresponding to the emergence of homoclinic orbits of a saddle-point. The asymptotical properties of the latter route prove to be entirely determined by the above ratio.