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

© 2017 IOP Publishing Ltd & London Mathematical Society Printed in the UK. In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form g(z, ϵ) = go(Z) + σki=l ϵi gi(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) + σki = l ϵi Fi (t, x) + O(ϵk+1), (t, x) ϵ 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.