Dynamics and Stability of Non-Smooth Dynamical Systems with Two Switches

TL;DR

This paper extends Filippov dynamics to non-smooth systems with singular switching manifolds involving two switches, providing new theorems on their qualitative behavior and stability.

Contribution

It introduces a methodology using blow-ups and singular perturbation to analyze systems with double discontinuities, advancing the understanding of their dynamics.

Findings

Extended Filippov dynamics to singular cases with two switches.

Derived theorems describing fundamental dynamics in affine cases.

Established Peixoto-like theorems for structural stability.

Abstract

One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. At this work, we leverage a methodology that, through blow-ups and singular perturbation, allows the extension of Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, non-linear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto like theorems characterizing semi-local structural stability.