Synchronization in a class of chaotic systems

TL;DR

This paper investigates synchronization conditions for a class of chaotic systems with nonlinearity, proposing a linear coupling method and deriving parameter conditions to ensure synchronization, applicable to various practical chaotic systems.

Contribution

It introduces a formal approach to determine coupling parameters for synchronizing non-identical chaotic systems with piecewise linear nonlinearity.

Findings

Derived conditions for coupling parameters guaranteeing synchronization.

Proposed a formal solution approach for the difference dynamics.

Applicable to a wide class of chaotic systems.

Abstract

In this work, the synchronization problem of a master-slave system of autonomous ordinary differential equations (ODEs) is considered. Here, the systems are, chaotic with a nonlinearity represented by a piecewise linear function, non-identical and linearly coupled. The idea behind our methodology is quite simple: we couple the systems with a linear function of the difference between the states of the systems and we propose a formal solution for the ODE that governs the evolution of that difference and then we determine what the parameters should be of the coupling function, so that the solution of that ODE is a fixed point close to zero. As the main result, we obtain conditions for the coupling function that guarantize the synchronization, based on a suitable descomposition of the system joined to a fixed point theorem. The scheme seems to be valid for a wide class of chaotic systems of great practical utility.