Bifurcations of finite-time stable limit cycles from focus boundary equilibria in impacting systems, Filippov systems and sweeping processes

TL;DR

This paper presents a universal theorem for bifurcations of finite-time stable limit cycles from focus boundary equilibria in various piecewise-smooth systems, including impacting, Filippov, and sweeping processes.

Contribution

It introduces a general theorem that unifies bifurcation analysis of limit cycles across different classes of piecewise-smooth systems.

Findings

Derived linearized systems for bifurcation analysis

Provided closed-form formulas for system coefficients

Validated results with illustrative examples

Abstract

We establish a theorem on bifurcation of limit cycles from a focus boundary equilibrium of an impacting system, which is universally applicable to prove bifurcation of limit cycles from focus boundary equilibria in other types of piecewise-smooth systems, such as Filippov systems and sweeping processes. Specifically, we assume that one of the subsystems of the piecewise-smooth system under consideration admits a focus equilibrium that lie on the switching manifold at the bifurcation value of the parameter. In each of the three cases, we derive a linearized system which is capable to conclude about the occurrence of a finite-time stable limit cycle from the above-mentioned focus equilibrium when the parameter crosses the bifurcation value. Examples illustrate how conditions of our theorems lead to closed-form formulas for the coefficients of the linearized system.