Multistable switching dynamical system with the location of symmetric equilibria

TL;DR

This paper introduces a piecewise linear switching dynamical system in three dimensions that exhibits multistability and complex scroll attractors due to hyperbolic focus-saddle equilibria, with applications in chaos generation.

Contribution

It presents a novel class of multistable switching systems with symmetric equilibria, demonstrating how discrete control modes induce multistability and complex attractors in R^3.

Findings

System exhibits multiple scroll attractors

Unstable hyperbolic focus-saddle equilibria drive multistability

Chaotic behavior arises from stretching and folding mechanisms

Abstract

A switching dynamical system by means of piecewise linear systems in R^3 that presents multistability is presented. The flow of the system displays multiple scroll attractors due to the unstable hyperbolic focus-saddle equilibria with stability index of type I, i.e., a negative real eigenvalue and a pair of complex conjugated eigenvalues with positive real part. This class of systems is constructed by a discrete control mode changing the equilibrium point regarding the location of their states. The scrolls are generated when the stable and unstable eigenspaces of each adjacent equilibrium point generate the stretching and folding mechanisms to generate chaos, i.e., the unstable manifold in the first subsystem carry the trajectory towards the stable manifold of the immediate adjacent subsystem.