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

## TL;DR

This paper proves the persistence of invariant tori in nearly integrable Hamiltonian systems with almost periodic time-dependent perturbations, extending KAM theory to more general time dependencies.

## Contribution

It establishes the existence of invariant tori and almost periodic solutions in Hamiltonian systems under almost periodic perturbations, broadening the scope of classical results.

## Key findings

- Existence of invariant tori in almost periodic Hamiltonian systems.
- Existence of almost periodic solutions for second order differential equations.
- Boundedness of all solutions in the considered systems.

## 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.