Persistence of invariant tori for almost periodically forced reversible systems

Shengqing Hu

doi:10.3934/dcds.2020188

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2020, Volume 40, Issue 7: 4497-4518. Doi: 10.3934/dcds.2020188

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$ \frac12 $

-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles

Persistence of invariant tori for almost periodically forced reversible systems

Shengqing Hu^,

Department of Mathematics, Nanjing University, Nanjing 210093, China

^* Corresponding author: Shengqing Hu
^* Corresponding author: Shengqing Hu

Received: August 31, 2019

Revised: December 31, 2019

Published: April 2020

This work was partially supported by the China Postdoctoral Science Foundation (Grant No. 003056)

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Abstract

In this paper, nearly integrable system under almost periodic perturbations is studied

$ \left\{\begin{array}{l} \dot{x} = \omega_0+y+f(t, x, y), \\ \dot{y} = g(t, x, y), \end{array}\right. $

where $ x\in\mathbb{T}^n, \, y\in\mathbb{R}^n $, $ \omega_0\in\mathbb{R}^n $ is the frequency vector, and the perturbations $ f, g $ are real analytic almost periodic functions in $ t $ with the infinite frequency $ \omega = (\cdots, \omega_\lambda, \cdots)_{\lambda\in\mathbb{Z}} $. We also assume that the above system is reversible with respect to the involution $ \mathcal{M}_0:(x, y)\rightarrow (-x, y) $. By KAM iterative method, we prove the existence of invariant tori for the above reversible system. As an application, we discuss the existence of almost periodic solutions and the boundedness of all solutions for a second-order nonlinear differential equation.

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Mathematics Subject Classification: Primary: 37J40; Secondary: 70K43, 70H12, 34C11.

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