Persistence of invariant tori in integrable Hamiltonian systems under almost periodic perturbations

Abstract

In this paper we are concerned with the existence of invariant tori in nearly integrable Hamiltonian systems \begin{equation*} H=h(y)+f(x,y,t), \end{equation*} where $y\in D\subseteq\mathbb{R}^n$ with $D$ being a closed bounded domain, $x\in \mathbb{T}^n$, $f(x,y,t)$ is a real analytic almost periodic function in $t$ with the frequency ${{\omega}}=(\cdots,{{\omega}}_\lambda,\cdots)_{\lambda\in \mathbb{Z}}\in \mathbb{R}^{\mathbb{Z}}$. As an application, we will prove the existence of almost periodic solutions and the boundedness of all solutions for the second order differential equations with superquadratic potentials depending almost periodically on time.