Grigory V. Osipov

The synchronization of chaotic systems

Abstract Synchronization of chaos refers to a process wherein two (or many) chaotic systems (either equivalent or nonequivalent) adjust a given property of their motion to a common behavior due to a coupling or to a forcing (periodical or noisy). We review major ideas involved in the field of synchronization of chaotic systems, and present in detail several types of synchronization features: complete synchronization, lag synchronization, generalized synchronization, phase and imperfect phase synchronization. We also discuss problems connected with characterizing synchronized states in extended pattern forming systems. Finally, we point out the relevance of chaos synchronization, especially in physiology, nonlinear optics and fluid dynamics, and give a review of relevant experimental applications of these ideas and techniques.

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.

Synchronization in Oscillatory Networks

Basics on Synchronization and Paradigmatic Models.- Basic Models.- Synchronization Due to External Periodic Forcing.- Synchronization of Two Coupled Systems.- Synchronization in Geometrically Regular Ensembles.- Ensembles of Phase Oscillators.- Chains of Coupled Limit-Cycle Oscillators.- Ensembles of Chaotic Oscillators with a Periodic-Doubling Route to Chaos, R#x00F6 ssler Oscillators.- Intermittent-Like Oscillations in Chains of Coupled Maps.- Regular and Chaotic Phase Synchronization of Coupled Circle Maps.- Controlling Phase Synchronization in Oscillatory Networks.- Chains of Limit-Cycle Oscillators.- Chains and Lattices of Excitable Luo-Rudy Systems.- Synchronization in Complex Networks and Influence of Noise.- Noise-Induced Synchronization in Ensembles of Oscillatory and Excitable Systems.- Networks with Complex Topology.

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.

Cluster synchronization in oscillatory networks

Synchronous behavior in networks of coupled oscillators is a commonly observed phenomenon attracting a growing interest in physics, biology, communication, and other fields of science and technology. Besides global synchronization, one can also observe splitting of the full network into several clusters of mutually synchronized oscillators. In this paper, we study the conditions for such cluster partitioning into ensembles for the case of identical chaotic systems. We focus mainly on the existence and the stability of unique unconditional clusters whose rise does not depend on the origin of the other clusters. Also, conditional clusters in arrays of globally nonsymmetrically coupled identical chaotic oscillators are investigated. The design problem of organizing clusters into a given configuration is discussed.

Introduction to Focus Issue: synchronization in complex networks.

Synchronization in large ensembles of coupled interacting units is a fundamental phenomenon relevant for the understanding of working mechanisms in neuronal networks, genetic networks, coupled electrical and laser networks, coupled mechanical systems, networks in social sciences, and others. It relates to mathematical and computational analysis of the existence of different states and its stability, clustering, bifurcations and chaos, robustness and sensitivity analysis, etc., at the intersection between synchronization and pattern formation in complex networks. This interdisciplinary oriented Focus Issue presents recent progress in this area with contributions on generic methods, specific model studies, and applications.

Synchronization phenomena in mixed media of passive, excitable, and oscillatory cells

We study collective phenomena in highly heterogeneous cardiac cell culture and its models. A cardiac culture is a mixture of passive (fibroblasts), oscillatory (pacemakers), and excitable (myocytes) cells. There is also heterogeneity within each type of cell as well. Results of in vitro experiments are modelled by Luo-Rudy and FitzHugh-Nagumo systems. For oscillatory and excitable media, we focus on the transitions from fully incoherent behavior to partially coherent behavior and then to global synchronization as the coupling strength is increased. These regimes are characterized qualitatively by spatiotemporal diagrams and quantitatively by profiles of dependence of individual frequencies on coupling. We find that synchronization clusters are determined by concentric and spiral waves. These waves arising due to the heterogeneity of medium push covered cells to oscillate in synchrony. We are also interested in the influence of passive and excitable elements on the oscillatory characteristics of low- and high-dimensional ensembles of cardiac cells. The mixture of initially silent excitable and passive cells shows the transitions to oscillatory behavior. In the media of oscillatory and passive or excitable cells, the effect of oscillation death is observed.

Cluster synchronization and spatio-temporal dynamics in networks of oscillatory and excitable Luo-Rudy cells

We study collective phenomena in nonhomogeneous cardiac cell culture models, including one- and two-dimensional lattices of oscillatory cells and mixtures of oscillatory and excitable cells. Individual cell dynamics is described by a modified Luo-Rudy model with depolarizing current. We focus on the transition from incoherent behavior to global synchronization via cluster synchronization regimes as coupling strength is increased. These regimes are characterized qualitatively by space-time plots and quantitatively by profiles of local frequencies and distributions of cluster sizes in dependence upon coupling strength. We describe spatio-temporal patterns arising during this transition, including pacemakers, spiral waves, and complicated irregular activity.

Spatiotemporal control of heart rate in a rabbit heart

Sinoatrial node is responsible for the origin of the wave of excitation, which spreads throughout the heart and orchestrates cardiac contraction via calcium-mediated excitation-contraction coupling. P wave represents the spread of excitation in the atria. It is well known that the autonomic nervous system controls the heart rate by dynamically altering both cellular ionic fluxes and the anatomical location of the leading pacemaker. In this study, we used isolated rabbit right atria and mathematical model of the pacemaker region of the rabbit heart. Application of isoproterenol resulted in dose-dependent acceleration of the heart rate and superior shift of the leading pacemaker. In the mathematical model, such behavior could be reproduced by a gradient of expression in β1-adrenergic receptors along the superior-inferior axis. Application of acetylcholine resulted in preferentially inferior shift of pacemaker and slowing of the heart rate. The mathematical model reproduced this behavior with imposing a gradient of expression of acetylcholine-sensitive potassium channel. We conclude that anatomical shift of the leading pacemaker in the rabbit heart could be achieved through gradient of expression of β1-adrenergic receptors and I(K,ACh).

Fibroblasts alter spiral wave stability.

We consider a three-domain model of cardiac tissue consisting of fibroblasts, myocytes, and extracellular space. We show in the one dimensional case that the fibroblasts with different resting potentials may alter restitution properties of tissue. On this basis we demonstrated that in two dimensional slice of cardiac tissue, a spiral wave break up can be caused purely by the influence of fibroblasts and, vice-versa, initially unstable spiral can be stabilized by fibroblasts depending on the value of their resting potential.