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Persistence of invariant tori for almost periodically forced reversible systems

  • * Corresponding author: Shengqing Hu

    * Corresponding author: Shengqing Hu

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

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  • 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.

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

    Citation:

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