Synchronization of branching chain of coupled maps: application to the logistic map

TL;DR

This paper explores synchronization behaviors in a chain of coupled logistic maps, identifying distinct parent-child and sibling synchronization mechanisms and analyzing their dependence on the parent node's state distribution.

Contribution

It provides a theoretical and numerical analysis of two types of synchronization in coupled chaotic maps organized in a chain, highlighting differences and computational challenges.

Findings

Parent-child and sibling synchronization differ significantly.

Synchronization depends on the parent node's state distribution.

Numerical instabilities can be mitigated with adjusted computational schemes.

Abstract

We investigate the synchronization dynamics in a chain of coupled chaotic maps organized in a single-parent family tree, whose properties can be captured considering each parent node connected to two children, one of which also serves as the parent for the subsequent node. Our analysis focuses on two distinct synchronization behaviors: parent-child synchronization, defined by the vanishing distance between successive nodes along the chain, and sibling synchronization, corresponding to the convergence of the states of two child nodes. Our findings reveal significant differences in these two type of synchronization mechanisms, which are closely associated with the probability distribution of the state of parent node. Theoretical analysis and simulations with the logistic map support our findings. We further investigate numerical aspects of the implementation corresponding to cases for which the simulated regimes differ from the theoretically predicted one due to computational finite accuracy. We perform a detailed study on how instabilities are numerically suppressed or amplified along the chain. In some cases, a properly adjusted computational scheme can solve this problem.