Bifurcations from families of periodic solutions in piecewise differential systems

Abstract

Consider a differential system of the form $$x^{\prime }=F_0(t,x)+\sum _{i=1}^k{\epsilon }^i{F}_i(t,x)+{\epsilon }^{k+1}R(t,x,\epsilon ),$$ where $F_i:{\mathbb{S}}^1\times D\to {\mathbb{R}}^m$ and $R:{\mathbb{S}}^1\times D\times (-{\epsilon }_0,{\epsilon }_0)\to {\mathbb{R}}^m$ are piecewise $C^{k+1}$ functions and $T$-periodic in the variable $t$. Assuming that the unperturbed system $x^{\prime }=F_0(t,x)$ 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.