Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method

Abstract

This paper focuses on almost-periodic time-dependent perturbations of a class of almost-periodically forced systems near non-hyperbolic equilibrium points in two cases: (a) elliptic case, (b) degenerate case (including completely degenerate). In elliptic case, it is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to perturbation parameter \begin{document}$ \epsilon, $\end{document} there exists a Cantor set \begin{document}$ \mathcal{E}\subset (0, \epsilon_0) $\end{document} of positive Lebesgue measure with sufficiently small \begin{document}$ \epsilon_0 $\end{document} such that for each \begin{document}$ \epsilon\in\mathcal{E} $\end{document} the system has an almost-periodic response solution. In degenerate case, we prove that, firstly, the almost-periodically perturbed degenerate system in one-dimensional case admits an almost-periodic response solution under nonzero average condition on perturbation and some weak non-resonant condition; Secondly, imposing further restriction on smallness of the perturbation besides nonzero average, we prove the almost-periodically forced degenerate system in \begin{document}$ n $\end{document} -dimensional case has an almost-periodic response solution under small perturbation without any non-resonant condition; Finally, almost-periodic response solution can still be obtained with weakened nonzero average condition by used Herman method but non-resonant condition should be strengthened. Some proofs of main results are based on a modified Poschel-Russmann KAM method, our results show that Poschel-Russmann KAM method can be applied to study the existence of almost-periodic solutions for almost-periodically forced non-conservative systems. Our results generalize the works in [ 14 , 13 , 23 , 20 ] from quasi-periodic case to almost-periodic case and also give rise to the reducibility of almost-periodic perturbed linear differential systems.