# American Institute of Mathematical Sciences

January  2020, 19(1): 541-585. doi: 10.3934/cpaa.2020027

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

 School of Mathematics, Shandong University, Jinan, Shandong 250100, China

* Corresponding author

Received  July 2018 Revised  January 2019 Published  July 2019

Fund Project: The first author was supported by a CSC scholarship (CSC student ID: 201606240030). The third author was partially supported by the National Natural Science Foundation of China (Grant Nos. 11171185, 11571201).

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 $\epsilon,$ there exists a Cantor set $\mathcal{E}\subset (0, \epsilon_0)$ of positive Lebesgue measure with sufficiently small $\epsilon_0$ such that for each $\epsilon\in\mathcal{E}$ 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 $n$-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 Pöschel-Rüssmann KAM method, our results show that Pöschel-Rüssmann 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.

Citation: Wen Si, Fenfen Wang, Jianguo Si. Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method. Communications on Pure & Applied Analysis, 2020, 19 (1) : 541-585. doi: 10.3934/cpaa.2020027
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