On the stability of boundary equilibria in Filippov systems

TL;DR

This paper investigates the stability of boundary equilibria in Filippov systems, establishing conditions under which stability properties of the original system can be inferred from a simplified affine approximation, and exploring complex dynamics in specific examples.

Contribution

The paper proves the equivalence of exponential stability between a Filippov system and its affine approximation near boundary equilibria, and shows stability is preserved under small perturbations, extending known results to non-homogeneous systems.

Findings

Exponential stability of boundary equilibria in Filippov systems can be characterized by their affine approximations.

Exponential and asymptotic stability are equivalent for the affine approximation system.

A four-dimensional example exhibits chaotic-like convergence to the boundary equilibrium.

Abstract

The leading-order approximation to a Filippov system $f$ about a generic boundary equilibrium $x^*$ is a system $F$ that is affine one side of the boundary and constant on the other side. We prove $x^*$ is exponentially stable for $f$ if and only if it is exponentially stable for $F$ when the constant component of $F$ is not tangent to the boundary. We then show exponential stability and asymptotic stability are in fact equivalent for $F$. We also show exponential stability is preserved under small perturbations to the pieces of $F$. Such results are well known for homogeneous systems. To prove the results here additional techniques are required because the two components of $F$ have different degrees of homogeneity. The primary function of the results is to reduce the problem of the stability of $x^*$ from the general Filippov system $f$ to the simpler system $F$. Yet in general this problem remains difficult. We provide a four-dimensional example of $F$ for which orbits appear to converge to $x^*$ in a chaotic fashion. By utilising the presence of both homogeneity and sliding motion the dynamics of $F$ can in this case be reduced to the combination of a one-dimensional return map and a scalar function.