Chaotic Synchronization and Riddled Basins of Attraction
O. Popovych, Yu.L. Maistrenko, V.L. Maistrenko, and E. Mosekilde
During the last few years it has become clear that interacting chaotic systems may display different forms of synchronization, and a variety of applications of chaotic synchronization for monitoring dynamical systems and for new types of communication are presently being investigated. Some of the main questions that arise in this connection relate to the stability of the synchronized state and to the form of its basin of attraction. Recent investigations of these problems have led to the discovery of several new phenomena, including on-off intermittency and riddled basins of attraction.
On-off intermittency denotes an extreme form of behavior where the trajectory from time to time exhibits bursts in which it moves far away from the synchronized state, to sooner or later be reinjected into the proximity of this state so that the process can repeat itself in an apparently random manner. Riddled basins of attraction arise under conditions where the synchronized state is attracting on average, but where low-periodic orbits embedded in this state are transversely unstable. As a result of this instability a dense set of tongues emanate from the synchronized state in which the trajectory diverges or approaches other stable states.
We have investigated these phenoemna for different types of coupled one-dimensional maps. In particular, we have shown how the presence of an absorbing area inside the domain of attraction can account for the global dynamics of the system. In this way we have been able to explain under which conditions on-off intermittency can arise. At the same time the presence of an absorbing area also explains the distinction between local and global riddling of the basin. We have also shown how the concept of intermingled absorbing areas comes into play.
Chaotic synchronzation may have a variety of technical applications. However, the phenomenon of chaotic phase synchronization is also of interest for interacting biological oscillators, e.g. beta-cells in the pancreas or nephrons in the kidney. In view of such applications it is important to extend the analysis from coupled one-dimensional maps to systems of coupled differential equations.
The purpose of the present project is to perform analyses of chaotic synchronization and riddled basins of attraction for ensembles of nonlinear maps and for interacting time continuous systems exhibiting chaotic dynamics.
For further details see also our book on Chaotic Synchronization.