# 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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##### References:
 [1] Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9 [2] Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569 [3] Paolo Perfetti. Hamiltonian equations on $\mathbb{T}^\infty$ and almost-periodic solutions. Conference Publications, 2001, 2001 (Special) : 303-309. doi: 10.3934/proc.2001.2001.303 [4] Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701 [5] Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635 [6] Claire Chavaudret, Stefano Marmi. Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition. Journal of Modern Dynamics, 2012, 6 (1) : 59-78. doi: 10.3934/jmd.2012.6.59 [7] J. Colliander, M. Keel, Gigliola Staffilani, H. Takaoka, T. Tao. Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 665-686. doi: 10.3934/dcds.2008.21.665 [8] Claire Chavaudret, Stefano Marmi. Erratum: Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition. Journal of Modern Dynamics, 2015, 9: 285-287. doi: 10.3934/jmd.2015.9.285 [9] Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 [10] Danijela Damjanovic and Anatole Katok. Local rigidity of actions of higher rank abelian groups and KAM method. Electronic Research Announcements, 2004, 10: 142-154. [11] Pablo Amster, Mónica Clapp. Periodic solutions of resonant systems with rapidly rotating nonlinearities. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 373-383. doi: 10.3934/dcds.2011.31.373 [12] Michela Procesi. Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 541-552. doi: 10.3934/dcds.2005.13.541 [13] Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703 [14] Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6425-6462. doi: 10.3934/dcdsb.2021026 [15] Xiang Li, Zhixiang Li. Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay. Communications on Pure & Applied Analysis, 2011, 10 (2) : 687-700. doi: 10.3934/cpaa.2011.10.687 [16] Nobu Kishimoto. Resonant decomposition and the $I$-method for the two-dimensional Zakharov system. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4095-4122. doi: 10.3934/dcds.2013.33.4095 [17] Shouchuan Hu, Nikolas S. Papageorgiou. Positive solutions for resonant (p, q)-equations with concave terms. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2639-2656. doi: 10.3934/cpaa.2018125 [18] Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857 [19] Zhenguo Liang, Jiansheng Geng. Quasi-periodic solutions for 1D resonant beam equation. Communications on Pure & Applied Analysis, 2006, 5 (4) : 839-853. doi: 10.3934/cpaa.2006.5.839 [20] Tiantian Ma, Zaihong Wang. Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1563-1581. doi: 10.3934/dcds.2013.33.1563

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