T. E. Vadivasova

Nonlinear dynamics of chaotic and stochastic systems : tutorial and modern developments

From the contents: Tutorial * Dynamical Systems * Fluctuations in Dynamic Systems * Synchronization of Periodic Systems * Dynamical Chaos * Routes to Chaos * Synchronization of Chaos * Controlling Chaos * Reconstruction of Dynamical Systems * Stochastic Dynamics * Stochastic Resonance * Synchronization of Stochastic Systems * The Beneficial Role of Noise in Excitable Systems * Noise Induced Transport.

SYNCHRONIZATION OF CHAOS

This paper is devoted to the problem of synchronization of dynamical systems in chaotic oscillations regimes. The authors attempt to use the ideas of synchronization and its mechanisms on a certain class of chaotic oscillations. These are chaotic oscillations for which one can pick out basic frequencies in their power spectra. The physical and computer experiments were carried out for the system of two coupled auto-oscillators. The experimental installation permitted one to realize both unidirectional coupling (external synchronization) and symmetrical coupling (mutual synchronization). An auto-oscillator with an inertial nonlinearity was chosen as a partial subsystem. It possesses a chaotic attractor of spiral type in its phase space. It is known that such chaotic oscillations have a distinguished peak in the power spectrum at the frequency f0 (basic frequency). In the experiments, one could make the basic frequencies of partial oscillators equal or different. The bifurcation diagrams on the plane of con...

Phase dynamics of two coupled oscillators under external periodic force

The effect of synchronization has been studied in a system of two coupled Van der Pol oscillators under external harmonic force. The analysis has been carried out using the phase approach. The mechanisms of complete and partial synchronization have been established. The main type of bifurcation described in this paper is the saddle-node bifurcation of invariant curves that corresponds to the saddle-node bifurcation of two-dimensional tori in the complete system of differential equations for the dynamical system under study.

Role of multistability in the transition to chaotic phase synchronization

In this paper we describe the transition to phase synchronization for systems of coupled nonlinear oscillators that individually follow the Feigenbaum route to chaos. A nested structure of phase synchronized regions of different attractor families is observed. With this structure, the transition to nonsynchronous behavior is determined by the loss of stability for the most stable synchronous mode. It is shown that the appearance of hyperchaos and the transition from lag synchronization to phase synchronization are related to the merging of chaotic attractors from different families. Numerical examples using Rossler systems and model maps are given. (c) 1999 American Institute of Physics.

Correlation analysis of dynamical chaos

We study correlation and spectral properties of chaotic self-sustained oscillations of different types. It is shown that some classical models of stochastic processes can be used to describe behavior of autocorrelation functions of chaos. The influence of noise on chaotic systems is also considered.

DYNAMICS OF TWO COUPLED CHUA'S CIRCUITS

Two resistively coupled Chuas circuits are studied in this paper. We consider the case when the two circuits are identical, and the case when there is a detuning between the basic frequencies of the two circuits. In the first case, the multistability (i.e. the coexistence of attractors) evolution scenario is studied. In the second case, we study the influence of detuning on the multistability, and on the mutual synchronization in the chaotic regimes. We present bifurcation diagrams of the main synchronization regions. We also compare the dynamics of coupled Chuas circuits with the dynamics of other coupled systems.

SYNCHRONIZATION IN DRIVEN CHAOTIC SYSTEMS : DIAGNOSTICS AND BIFURCATIONS

Abstract We investigate generic aspects of chaos synchronization in an externally forced Rossler system. By comparing different diagnostic methods, we show the existence of a well-defined cut-off of synchronization associated with the transition from weak to fully developed chaos. Chaotic synchronization is found to be lost at this cut-off only after the last band-merging bifurcation has occurred. Everywhere at the boundary of phase synchronization, one of the Lyapunov exponents becomes equal to zero. Two types of chaotic behavior, differing by the number of vanishing Lyapunov exponents, are observed outside the synchronization region.

Time-delayed feedback control of coherence resonance near subcritical Hopf bifurcation: theory versus experiment.

Using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation, we investigate the influence of time delay on noise-induced oscillations. It is shown that for appropriate choices of time delay, either suppression or enhancement of coherence resonance can be achieved. Analytical calculations are combined with numerical simulations and experiments on an electronic circuit.

Studying hyperbolicity in chaotic systems

Abstract On the basis of method [1] proposed for diagnosing 2-dimensional chaotic saddles we present a numerical procedure to distinguish hyperbolic and nonhyperbolic chaotic attractors in three-dimensional flow systems. This technique is based on calculating the angles between stable and unstable manifolds along a chaotic trajectory in R 3 . We show for three-dimensional flow systems that this serves as an efficient characteristic for exploring chaotic differential systems. We also analyze the effect of noise on the structure of angle distribution for both 2-dimensional invertible maps and a 3-dimensional continuous system.

Phase multistability of synchronous chaotic oscillations

The paper describes the sequence of bifurcations leading to multistability of periodic and chaotic synchronous attractors for the coupled Rossler systems which individually demonstrate the Feigenbaum route to chaos. We investigate how a frequency mismatch affects this phenomenon. The role of a set of coexisting synchronous regimes in the transitions to and between different forms of synchronization is studied.