Bifurcations from families of periodic solutions in piecewise differential systems

Abstract

Consider a differential system of the form x′=F0(t,x)+∑i=1kεiFi(t,x)+εk+1R(t,x,ε),where Fi:S1×D→Rm and R:S1×D×(−ε0,ε0)→Rm are piecewise Ck+1 functions and T-periodic in the variable t. Assuming that the unperturbed system x′=F0(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.