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

TL;DR

This paper establishes conditions for the persistence of zeros in perturbed functions and applies these results to analyze the bifurcation of periodic solutions in higher order differential systems using Lyapunov-Schmidt reduction.

Contribution

It introduces new sufficient conditions for zero persistence in perturbed functions and computes bifurcation functions for periodic solutions up to fifth order.

Findings

Derived explicit bifurcation functions up to order 5.

Provided criteria for the persistence of periodic solutions under perturbations.

Analyzed cases with a continuum of zeros in bifurcation functions.

Abstract

In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form $$g(z,\varepsilon)=g_0(z)+\sum_{i=1}^k \varepsilon^i g_i(z)+\mathcal{O}(\varepsilon^{k+1}),$$ for $|\varepsilon|\neq0$ sufficiently small. Here $g_i:\mathcal{D}\rightarrow\mathbb{R}^n$, for $i=0,1,\ldots,k$, 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 $$ x'=F_0(t,x)+\sum_{i=1}^k \varepsilon^i F_i(t,x)+\mathcal{O}(\varepsilon^{k+1}), \quad (t,z)\in\mathbb{S}^1\times\mathcal{D}. $$ It is assumed that the unperturbed differential system has a sub-manifold of periodic solutions $\mathcal{Z}$, $\textrm{dim}(\mathcal{Z})\leq n$. 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.