Arkady Pikovsky

Phase synchronization of chaotic oscillators by external driving

Abstract We extend the notion of phase locking to the case of chaotic oscillators. Different definitions of the phase are discussed. and the phase dynamics of a single self-sustanined chaotic oscillator subjected to external force is investigated. We describe regimes where the amplitude of the oscillator remains chaotic and the phase is synchronized by the external force. This effect is demonstrated for periodic and noisy driving. This phase synchronization is characterized via direct calculation of the phase, as well as by implicit indications, such as the resonant growth of the discrete component in the power spectrum and the appearance of a macroscopic average field in an ensemble of driven oscillators. The Rossler and the Lorenz systems are shown to provide examples of different phase coherence properties, with different response to the external force. A relation between the phase synchronization and the properties of the Lyapunov spectrum is discussed.

Noise-Induced Dynamics in Bistable Systems with Delay

Noise-induced dynamics of a prototypical bistable system with delayed feedback is studied theoretically and numerically. For small noise and magnitude of the feedback, the problem is reduced to the analysis of the two-state model with transition rates depending on the earlier state of the system. Analytical solutions for the autocorrelation function and the power spectrum have been found. The power spectrum has a peak at the frequency corresponding to the inverse delay time, whose amplitude has a maximum at a certain noise level, thus demonstrating coherence resonance. The linear response to the external periodic force also has maxima at the frequencies corresponding to the inverse delay time and its harmonics.

On the interaction of strange attractors

This paper deals with the dynamics of diffusively coupled strange attractors. Such interaction tends to equalize their instantaneous states and, for large coupling constant, results in a homogeneous state that is chaotic in time. The stability of this state depends on the relation between the Lyapunov exponent and the coupling constant. Statistical properties are determined for weakly inhomogeneous disturbances near a stable homogeneous regime. The inhomogeneous state beyond the stability threshold is treated by using the mean-field approximation. We show that both cases of soft (supercritical) and hard (subcritical) excitation of the inhomogeneous state may occur.

Destruction of Anderson Localization by a Weak Nonlinearity

We study numerically the spreading of an initially localized wave packet in a one-dimensional discrete nonlinear Schrödinger lattice with disorder. We demonstrate that above a certain critical strength of nonlinearity the Anderson localization is destroyed and an unlimited subdiffusive spreading of the field along the lattice occurs. The second moment grows with time proportional, variant t alpha, with the exponent alpha being in the range 0.3-0.4. For small nonlinearities the distribution remains localized in a way similar to the linear case.

PHASE SYNCHRONIZATION IN REGULAR AND CHAOTIC SYSTEMS

In this contribution we present a brief introduction to the theory of synchronization of selfsustained oscillators. Classical results for synchronization of periodic motions and eects of noise on this process are reviewed and compared with recently found phase synchronization phenomena in chaotic oscillators. The basic notions of phase and frequency locking are reconsidered within a common framework. The application of phase synchronization to data analysis is discussed.

Symmetry breaking bifurcation for coupled chaotic attractors

The authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one-dimensional mappings. They show that the probability density of the asymmetric component satisfies a scaling law. The exponent in this scaling law varies continuously with the distance from the bifurcation point, and is determined by the spectrum of local Lyapunov exponents of the uncoupled map. Finally they show that the topology of the invariant set is rather unusual: though the attractor for supercritical coupling is a line, it is surrounded by a strange invariant set which is dense in a two-dimensional neighbourhood of the attractor.

Characterizing strange nonchaotic attractors

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. Second, we propose to 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 nonzero probability for positive local Lyapunov exponents to occur. (c) 1995 American Institute of Physics.

Chapter 9 Phase synchronization: From theory to data analysis

Publisher Summary The chapter describes particular experiments and searching for phase synchronization. The phase synchronization of chaotic system is the appearance of a certain relation between the phases of interacting systems (or between the phase of a system and that of an external force), while the amplitudes can remain chaotic and are noncorrelated. The properties of phase synchronization in chaotic systems are similar to those of synchronization in periodic noisy oscillators. Synchronization plays an important role in several neurological diseases such as epilepsies and pathological tremors. The chapter reviews the ideas and results of theoretical studies of the synchronization phenomena that are used to time series analysis. The chapter presents techniques of the bivariate data analysis and illustrates them by examples of physiological data. These examples are given in the ascending order of the signal analysis complexity.

Synchronization: From pendulum clocks to chaotic lasers and chemical oscillators

Many natural and human-made nonlinear oscillators exhibit the ability to adjust their rhythms due to weak interaction: two lasers, being coupled, start to generate with a common frequency; cardiac pacemaker cells fire simultaneously; violinists in an orchestra play in unison. Such coordination of rhythms is a manifestation of a fundamental nonlinear phenomenon--synchronization. Discovered in the 17th century by Christiaan Huygens, it was observed in physics, chemistry, biology and even social behaviour, and found practical applications in engineering and medicine. The notion of synchronization has been recently extended to cover the adjustment of rhythms in chaotic systems, large ensembles of oscillating units, rotating objects, continuous media, etc. In spite of essential progress in theoretical and experimental studies, synchronization remains a challenging problem of nonlinear sciences.

System Size Resonance in Coupled Noisy Systems and in the Ising Model

We consider an ensemble of coupled nonlinear noisy oscillators demonstrating in the thermodynamic limit an Ising-type transition. In the ordered phase and for finite ensembles stochastic flips of the mean field are observed with the rate depending on the ensemble size. When a small periodic force acts on the ensemble, the linear response of the system has a maximum at a certain system size, similar to the stochastic resonance phenomenon. We demonstrate this effect of system size resonance for different types of noisy oscillators and for different ensembles---lattices with nearest neighbors coupling and globally coupled populations. The Ising model is also shown to demonstrate the system size resonance.