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 $$g(z,\epsilon )=g_0(z)+\sum _{i=1}^k{\epsilon }^i{g}_i(z)+\mathcal{O}({\epsilon }^{k+1}),$$ for $\mathrm{\mid }\epsilon \mathrm{\mid }\mathrm{\ne }0$ sufficiently small. Here $g_i:\mathcal{D}\to {\mathbb{R}}^n$, for $i=0,1,\dots ,k$, are smooth functions being $\mathcal{D}\subset {\mathbb{R}}^n$ an open bounded set. Then we use this result to compute the bifurcation functions which controls the periodic solutions of the following $T$-periodic smooth differential system $$x^{\mathrm{\prime }}=F_0(t,x)+\sum _{i=1}^k{\epsilon }^i{F}_i(t,x)+\mathcal{O}({\epsilon }^{k+1}),(t,z)\in {\mathbb{S}}^1\times \mathcal{D}\mathrm{.}$$ It is assumed that the unperturbed differential system has a sub-manifold of periodic solutions $\mathcal{Z}$, $\text{dim}(\mathcal{Z})\le n$. 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.