The stability of boundary equilibria of three-dimensional Filippov systems

TL;DR

This paper investigates the stability of boundary equilibria in three-dimensional Filippov systems, revealing that stability depends on a global reinjection mechanism, with detailed analysis of a four-parameter family of piecewise-linear hybrid systems.

Contribution

It characterizes the stability of boundary equilibria in 3D Filippov systems and links it to a global reinjection mechanism, supported by numerical analysis.

Findings

Stability is governed by a global reinjection mechanism.

Detailed numerical study of a four-parameter family.

Characterization of boundary equilibrium stability in 3D systems.

Abstract

For three-dimensional piecewise-smooth systems of ordinary differential equations, this paper characterises the stability of points that belong to a switching surface and are equilibria of exactly one of the two neighbouring pieces of the system. Stability is challenging to characterise when nearby orbits repeatedly switch between regular motion on one side of the switching surface, and sliding motion on the switching surface, as defined via Filippov's convention. We prove that in this case stability is governed by the behaviour of a global reinjection mechanism of a four-parameter family of piecewise-linear hybrid systems, and perform a detailed numerical study of this family.