Chaotic dynamics from a pseudo-linear system

Abstract

Abstract Investigating the possibility of applying techniques from linear systems theory to the setting of non-linear systems has been the focus of many papers. The pseudo-linear (PL) form representation of non-linear dynamical systems has led to the concept of non-linear eigenvalues (NEValues) and non-linear eigenvectors (NEVectors). When the NEVectors do not depend on the state vector of the system, then the NEValues determine the global qualitative behaviour of a non-linear system throughout the state space. The aim of this paper is to use this fact to construct a non-linear 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 Hénon-like type, which means that they are the closure of the unstable manifold of a saddle periodic orbit. We also show how PL forms can be used to control the chaotic system and to synchronize two identical chaotic systems.