A general framework for boundary equilibrium bifurcations of Filippov systems

TL;DR

This paper introduces a comprehensive framework for analyzing boundary equilibrium bifurcations in Filippov systems, providing new normal forms, stability criteria, and insights into complex dynamics like chaos and multiple attractors.

Contribution

It develops a general normal form for BEBs in any dimension, linking stability to the number of unstable directions, and explores the rich dynamics near BEBs including chaos and multistability.

Findings

Normal form for BEBs in arbitrary dimensions

Connection between stability and unstable directions

BEBs can generate chaos and multiple attractors

Abstract

As parameters are varied a boundary equilibrium bifurcation (BEB) occurs when an equilibrium collides with a discontinuity surface in a piecewise-smooth system of ODEs. Under certain genericity conditions, at a BEB the equilibrium either transitions to a pseudo-equilibrium (on the discontinuity surface) or collides and annihilates with a coexisting pseudo-equilibrium. These two scenarios are distinguished by the sign of a certain inner product. Here it is shown that this sign can be determined from the number of unstable directions associated with the two equilibria by using techniques developed by Feigin. A new normal form is proposed for BEBs in systems of any number of dimensions. The normal form involves a companion matrix, as does the leading order sliding dynamics, and so the connection to the stability of the equilibria is explicit. In two dimensions the parameters of the normal form distinguish, in a simple way, the eight topologically distinct cases for the generic local dynamics at a BEB. A numerical exploration in three dimensions reveals that BEBs can create multiple attractors and chaotic attractors, and that the equilibrium at the BEB can be unstable even if both equilibria are stable. The developments presented here stem from seemingly unutilised similarities between BEBs in discontinuous systems (specifically Filippov systems as studied here) and BEBs in continuous systems for which analogous results are, to date, more advanced.