Bifurcations from families of periodic solutions in piecewise differential systems

TL;DR

This paper investigates bifurcations from families of periodic solutions in piecewise differential systems using Lyapunov-Schmidt reduction and averaging theory to identify conditions for the emergence of isolated periodic solutions.

Contribution

It introduces a method combining Lyapunov-Schmidt reduction and averaging theory to analyze bifurcations in piecewise differential systems with families of periodic solutions.

Findings

Identifies conditions for bifurcation of isolated periodic solutions.

Extends bifurcation analysis to piecewise systems with families of solutions.

Provides a framework for studying stability of bifurcating solutions.

Abstract

Consider a differential system of the form $$ x'=F_0(t,x)+\sum_{i=1}^k \varepsilon^i F_i(t,x)+\varepsilon^{k+1} R(t,x,\varepsilon), $$ where $F_i:\mathbb{S}^1 \times D \to \mathbb{R}^m$ and $R:\mathbb{S}^1 \times D \times (-\varepsilon_0,\varepsilon_0) \to \mathbb{R}^m$ are piecewise $C^{k+1}$ functions and $T$-periodic in the variable $t$. Assuming that the unperturbed system $x'=F_0(t,x)$ has a $d$-dimensional submanifold of periodic solutions with $d<m$, we use the Lyapunov-Schmidt reduction and the averaging theory to study the existence of isolated $T$-periodic solutions of the above differential system.