Improving the averaging theory for computing periodic solutions of the differential equations

Abstract

For ${m=1,2,3,}$ we consider differential systems of the form

$$x'\,=\,F_{0}(t,x)\,+\,\sum_{i=1}^{m}\,\varepsilon^{i} F_{i}(t,x)+\varepsilon^{m+1} R(t,x,\varepsilon),$$

where ${F_i:\mathbb{R} \times \mathcal{D}\,\rightarrow\,\mathbb{R}^{n}}$ and ${R:\mathbb{R}\,\times\,\mathcal{D}\,\times\,(-\varepsilon_{0},\varepsilon_{0})\,\rightarrow\,\mathbb{R}^{n}}$ are ${\mathcal{C}^{m+1}}$ functions, and ${T}$ -periodic in the first variable, being ${\mathcal{D}}$ an open subset of ${\mathbb{R}^{n}}$, and ${\varepsilon}$ a small parameter. For such system, we assume that the unperturbed system x′  =  F ₀(t, x) has a k-dimensional manifold of periodic solutions with k ≤ n. We weaken the sufficient assumptions for studying the periodic solutions of the perturbed system when ${|\varepsilon|\, > \,0}$ is sufficiently small.