Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction

In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form g(z,ε)=g0(z)+∑i=1kεigi(z)+O(εk+1), for |ε|≠0 sufficiently small. Here gi:D→Rn, for i=0,1,...,k, are smooth functions being D⊂Rn an open bounded set. Then we use this result to compute the bifurcation functions which allow us to study the periodic solutions of the following T-periodic smooth differential system x′=F0(t,x)+∑i=1kεiFi(t,x)+O(εk+1),(t,z)∈S1×D. It is assumed that the unperturbed differential system has a sub-manifold of periodic solutions Z, dim⁡(Z)⩽n. We also study the case when the bifurcation functions have a continuum of zeros. Finally we provide the explicit expressions of the bifurcation functions up to order 5.