Synchronization of dynamical systems: an approach using a Computer Algebra System

TL;DR

This paper explores the synchronization phenomenon in dynamical systems using the free Computer Algebra System Maxima, providing theoretical insights, graphical animations, and applicable procedures for various systems like Lorentz attractor and coupled pendula.

Contribution

It offers a detailed theoretical approach to synchronization in dynamical systems with practical implementation using Maxima, adaptable to any first-order differential system.

Findings

Synchronization demonstrated in Lorentz attractor and pendula models

Procedures applicable to any system of first-order differential equations

Graphical animations illustrate synchronization phenomena

Abstract

The synchronization between two dynamical systems is one of the most appealing phenomena occurring in Nature. Already observed by Huygens in the case of two pendula, it is a current area of research in the case of chaotic systems, with numerous applications in Physics, Biology or Engineering. We present an elementary but detailed exploration of the theory behind this phenomenon, including some graphical animations, with the aid of the free CAS Maxima, but the code can be easily ported to other CASs. The examples used are the Lorentz attractor and a pair of coupled pendula because these are well-known models of dynamical systems, but the procedures are applicable to any system described by a system of first-order differential equations.