Synchronization and partial synchronization of linear maps

TL;DR

This paper investigates synchronization phenomena in low-dimensional chaotic linear maps, revealing discontinuous and continuous transitions, singular behaviors, and riddled basins, with implications for understanding complex dynamics in such systems.

Contribution

It provides a detailed analysis of synchronization transitions in low-dimensional chaotic maps, highlighting differences between Bernoulli and tent maps and introducing the concept of riddled basins in these systems.

Findings

Lyapunov exponents identify synchronization transition points.

Discontinuous transition in Bernoulli maps, continuous in tent maps.

Presence of riddled basin attractors and coexistence of behaviors.

Abstract

We study synchronization of low-dimensional ($d=2,3,4$) chaotic piecewise linear maps. For Bernoulli maps we find Lyapunov exponents and locate the synchronization transition, that numerically is found to be discontinuous (despite continuously vanishing Lyapunov exponent(s)). For tent maps, a limit of stability of the synchronized state is used to locate the synchronization transition that numerically is found to be continuous. For nonidentical tent maps at the partial synchronization transition, the probability distribution of the synchronization error is shown to develop highly singular behavior. We suggest that for nonidentical Bernoulli maps (and perhaps some other discontinuous maps) partial synchronization is merely a smooth crossover rather than a well defined transition. More subtle analysis in the $d=4$ case locates the point where the synchronized state becomes stable. In some cases, however, a riddled basin attractor appears, and synchronized and chaotic behaviors coexist. We also suggest that similar riddling of a basin of attractor might take place in some extended systems where it is known as stable chaos.