V. V. Astakhov

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 chaotic oscillators by periodic parametric perturbations

Abstract We show that the synchronization of coupled chaotic oscillators can be achieved by means of periodic parametric perturbations of the coupling element. The possibility of synchronization is demonstrated for the simple case of two identical nonautonomous oscillators with piecewise linear characteristics both analytically and numerically. Using linear analysis we have determined the stability conditions for symmetric oscillations.

Controlling spatiotemporal chaos in a chain of the coupled logistic maps

A method of controlling spatiotemporal chaos in a chain of the logistic maps, based on the Ott, Grebogi, Yorke (OGY) approach, is proposed in the paper. Forms of spatiotemporal disturbances of a system parameter are determined analytically to stabilize desired spatiotemporal states. The results are illustrated by numerical experiments. Transitions of the system under control from the regime of spatiotemporal chaos to the regular homogeneous regime with temporal period one, two, four, and also to more complicated regular regimes with different spatial and temporal periods have been demonstrated. >

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.

ANTIPHASE SYNCHRONIZATION IN SYMMETRICALLY COUPLED SELF-OSCILLATORS

We consider the antiphase synchronization in symmetrically coupled self-oscillators. As model, two Chua’s circuits coupled via a capacity are used. Linear analysis in the vicinity of the symmetric subspace gives the stability conditions for antiphase oscillations. Numerical oscillations demonstrate controlled antiphase synchronization at dierent values of the parameters of the 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.

Modeling chemical reactions by forced limit-cycle oscillator: Synchronization phenomena and transition to chaos

Abstract The lattice limit-cycle (LLC) model is introduced as a minimal mean-field scheme which can model reactive dynamics on lattices (low dimensional supports) producing non-linear limit cycle oscillations. Under the influence of an external periodic force the dynamics of the LLC may be drastically modified. Synchronization phenomena, bifurcations and transitions to chaos are observed as a function of the strength of the force. Taking advantage of the drastic change on the dynamics due to the periodic forcing, it is possible to modify the output/product or the production rate of a chemical reaction at will, simply by applying a periodic force to it, without the need to change the support properties or the experimental conditions.

DEVELOPING CHAOS ON BASE OF TRAVELING WAVES IN A CHAIN OF COUPLED OSCILLATORS WITH PERIOD-DOUBLING. SYNCHRONIZATION AND HIERARCHY OF MULTISTABILITY FORMATION

The work is devoted to the analysis of dynamics of traveling waves in a chain of self-oscillators with period-doubling route to chaos. As a model we use a ring of Chuas circuits symmetrically coupled via a resistor. We consider how complicated are temporal regimes with parameters changing influences on spatial structures in the chain. We demonstrate that spatial periodicity exists until transition to chaos through period-doubling and tori birth bifurcations of regular regimes. Temporal quasi-periodicity does not induce spatial quasi-periodicity in the ring. After transition to chaos exact spatial periodicity is changed by the spatial periodicity in the average. The periodic spatial structures in the chain are connected with synchronization of oscillations. For quantity researching of the synchronization we propose a measure of chaotic synchronization based on the coherence function and investigate the dependence of the level of synchronization on the strength of coupling and on the chaos developing in the system. We demonstrate that the spatial periodic structure is completely destroyed as a consequence of loss of coherence of oscillations on base frequencies.

In-phase and Antiphase Complete Chaotic Synchronization in Symmetrically Coupled Discrete Maps

We consider in-phase and antiphase synchronization of chaos in a system of coupled cubic maps. Regions of stability and robustness of the regime of in-phase complete synchronization was found. It was demonstrated that the loss of the synchronization is accompanied by bubbling and riddling phenomena. The mechanisms of these phenomena are connected with bifurcations of the main family of periodic orbits and orbits appeared from them. We found that in spite of the in-phase synchronization, the antiphase self-synchronization of chaos is impossible for discrete maps with symmetric diffusive coupling. For achieving antiphase synchronization we used method of controlled synchronization by addition feedback. The region ofthe controlled antiphase synchronization and phenomena which accompany the loss of the synchronization are presented.

Non-bifurcational mechanism of loss of chaos synchronization in coupled non-identical systems

Abstract We show that in the weakly non-identical coupled systems, the loss of synchronization (the destruction of a chaotic attractor located in the vicinity of the invariant subspace of identical systems) can be initiated by the smooth shift of one of these orbits out of the chaotic attractor.