Dynamical analysis of a chaos generator

TL;DR

This paper explores the use of nonlinear eigenvalues to analyze and control a chaos-generating dynamical system, demonstrating bifurcations, chaos emergence, and synchronization techniques.

Contribution

It introduces a method to analyze chaos via nonlinear eigenvalues and demonstrates control and synchronization of chaotic systems using pseudo linear forms.

Findings

System exhibits continual stretching and folding behavior.

Chaos arises through period doubling bifurcations.

Chaotic attractors are Hénon-like and controllable.

Abstract

Investigating the possibility of applying techniques from linear systems theory to the setting of nonlinear systems has been the focus of many papers. The pseudo linear form representation of nonlinear dynamical systems has led to the concept of nonlinear eigenvalues and nonlinear eigenvectors. When the nonlinear eigenvectors do not depend on the state vector of the system, then the nonlinear eigenvalues determine the global qualitative behaviour of a nonlinear system throughout the state space. The aim of this paper is to use this fact to construct a nonlinear dynamical system of which the trajectories of the system show continual stretching and folding. We first prove that the system is globally bounded. Next, we analyse the system numerically by studying bifurcations of equilibria and periodic orbits. Chaos arises due to a period doubling cascade of periodic attractors. Chaotic attractors are presumably of H\'enon-like type, which means that they are the closure of the unstable manifold of a saddle periodic orbit. We also show how pseudo linear forms can be used to control the chaotic system and to synchronize two identical chaotic systems.