Role of multistability in the transition to chaotic phase synchronization

Abstract

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 Rössler systems and model maps are given. © 1999 American Institute of Physics. ͓S1054-1500͑99͒00201-3͔The interaction between two or more chaotic oscillators can produce a number of different synchronization phenomena, depending on the degree to which the oscillators adjust their motion in accordance with one another. A variety of technical applications of such synchronization phenomena are presently being considered, e.g., for secure communication and for the surveillance and control of systems that operate in a chaotic regime. Chaotic synchronization may also play an essential role for the regulation of many biological systems where an individual functional unit displays complex dynamics. In the present paper we investigate the bifurcations that can occur when chaotic synchronization is established between two period-doubling systems with different basic periodicities. We show how this synchronization gives rise to a nested structure of phase synchronized regions.

Scaling features of multimode motions in coupled chaotic oscillators

“…We consider here only two families having the largest basins of attraction, namely, ''in-phase'' and ''out-of-phase'' attractors [8]. In the first case the phase difference of x 1 ðtÞ and x 2 ðtÞ vanishes for x 1 ¼ x 2 (the corresponding periodic regimes are labeled as 2 i C 0 , where i ¼ 1; 2; 3; .…”

“…1), a band-merging bifurcation of the chaotic attractors CA 0 and CA 1 leads to the appearance of a new regime CA R . The attractor CA R contains the trajectories of CA 0 and CA 1 and has two positive Lyapunov exponents [8]. Therefore, hyperchaotic oscillations takes place.…”

“…Here, the parameters a, b, and l govern the dynamics of each subsystem, and c is the coupling parameter. [8]. Let us briefly recall those aspects of their bifurcation diagram on the ðd; lÞ parameter plane (Fig.…”

“…The phenomenon of multistability is clearly expressed in the synchronization of coupled oscillators that individually follows the period-doubling route to chaos. This is an example of so-called phase multistability: The synchronous solutions have different phase relationships between the oscillations [8]. In particular, if two periodic oscillators with n subharmonics x 0 =2 n (n ¼ 1; 2; .…”

The transition to chaotic phase synchronization

“…Along with changes in the variation of the average frequency, the transition between phase-locked and un-locked chaos is also reÀected in a speci¿c variation of the Lyapunov exponents, in characteristic changes of the spectrum of the forced chaotic oscillator, and through changes in the size and form of its Poincaré section [14,15]. It is generally known that the edge of the synchronization domain is made up by a dense set of saddle-node bifurcations [6,13,16,17].…”