Synchronization: Theory and Application

Dynamical systems with symmetries show a number of atypical behaviours for generic dynamical systems. As coupled cell systems often possess symmetries, these behaviours are important for understanding dynamical effects in such systems. In particular the presence of symmetries gives invariant subspaces that interact with attractors to give new types of instability and intermittent attractor. In this paper we review and extend some recent work (Ashwin, Rucklidge and Sturman 2002) on robust non-ergodic attractors consists of cycles between invariant subspaces, called ‘cycling chaos’ by (1995).

By considering a simple model of coupled oscillators that show such cycles, we investigate the difference in behaviour between what we call free-running and phase-resetting (discontinuous) models. The difference is shown most clearly when observing the types of attractors created when an attracting cycle loses stability at a resonance. We describe both scenarios — giving intermittent stuck-on chaos for the free-running model, and an infinite family of periodic orbits for the phase-resetting case. These require careful numerical simulation to resolve quantities that routinely get as small as 10⁻¹⁰⁰⁰.

We characterise the difference between these models by considering the rates at which the cycles approach the invariant subspaces. Finally, we demonstrate similar behaviour in a continuous version of the phase-resetting model that is less amenable to analysis and raise some open questions.

A new chaos-based technique for modelling the diversity of approximately periodic signals is introduced and exploited, combined with generalized chaotic synchronization phenomena, for the solution of temporal pattern recognition problems.

Basics of the theory of direct chaotic communications is presented. We introduce the notion of chaotic radio pulse and consider signal structures and modulation methods applicable in direct chaotic schemes. Signal processing in noncoherent and coherent receivers is discussed. The efficiency of direct chaotic communications is investigated by means of numerical simulation. Potential application areas are analyzed, including multiple access systems.

Dominance of Milnor attractors in high-dimensional dynamical systems is reviewed, with the use of globally coupled maps. From numerical simulations, the threshold number of degrees of freedom for such prevalence of Milnor attractors is suggested to be 5 ~ 10, which is also estimated from an argument of combinatorial explosion of basin boundaries. Chaotic itinerancy is revisited from the viewpoint of Milnor attractors. Relevance to neural networks is discussed.

The paper presents a two-dimensional version of the Feigenbaum-Kadanoff-Shenker renormalization group equation. Several universality classes of critical behavior are discussed, which may occur at the onset of chaotic or strange nonchaotic attractors via quasiperiodicity at the golden-mean frequency ratio. Parameter space arrangement and respective scaling properties are discussed and illustrated.

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained, providing a special kind of dynamical patterns called clusters. The simplest, coherent clusters arise when all oscillators display the same temporal behavior. Others, more complicated clusters are developed when population of the oscillators splits into subgroups such that all oscillators within a given group move in synchrony. Considering a system of mean-field coupled logistic maps, we study in details the transition from coherence to clustering and demonstrate that there are four different mechanisms of the desynchronization: riddling and blowout bifurcations, appearance of symmetric and asymmetric clusters. We also investigate the cluster-splitting bifurcation when the underlying dynamics is periodic. For the system of three and four coupled Rössler oscillators, we prove the existence of clusters and describe related bifurcations and in-cluster dynamics.

By controling the excretion of water and salts, the kidneys play an important role in regulating the blood pressure and maintaining a proper environment for the cells of the body. This control depends to a large extent on mechanisms that are associated with the individual functional unit, the nephron. However, a variety of cooperative phenomena arising through interactions among the nephrons may also be important. The purpose of this chapter is to present experimental evidence for a coupling between nephrons that are connected via a common piece of afferent arteriole, to develop a mathematical model that can account for the observed synchronization phenomena, and to discuss the possible physiological significance of these phenomena. We are particularly interested in synchronization effects that can occur among neighboring nephrons that individually display irregular (or chaotic) dynamics in their pressure and flow regulation.

The onset of collective synchronous behavior in globally coupled ensembles of oscillators is discussed. We present a formalism that is applicable to general ensembles of heterogeneous, continuous time dynamical units that, when uncoupled, are chaotic, periodic, or a mixture of both. A discussion of convergence issues, important for the proper implementation of our method, is included. Our work leads to a quantitative prediction for the critical coupling value at the onset of collective synchrony and for the growth rate of the resulting coherent state.

Time delayed-feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems. The method is based on applying feedback proportional to the deviation of the current state of the system from its state one period in the past so that the control signal vanishes when the stabilization of the desired orbit is attained. A brief review of experimental implementations, applications for theoretical models, and most important modifications of the method is presented. Some recent results concerning the theory of the delayed feedback control as well as an idea of using unstable degrees of freedom in a feedback loop to avoid a well known topological limitation of the method are described in details.