# Almost periodic solutions for an asymmetric oscillation

## TL;DR

This paper investigates the existence and boundedness of almost periodic solutions for a nonlinear asymmetric oscillation differential equation driven by an almost periodic forcing function, extending invariant curve theorems to this context.

## Contribution

The authors establish variants of the invariant curve theorem for planar almost periodic mappings and apply these to prove the existence of almost periodic solutions in an asymmetric oscillation model.

## Key findings

- Existence of almost periodic solutions under certain conditions
- Boundedness of all solutions for the differential equation
- Extension of invariant curve theorem to almost periodic mappings

## Abstract

In this paper we study the dynamical behaviour of the differential equation \begin{equation*} x''+ax^+ -bx^-=f(t), \end{equation*} where $x^+=\max\{x,0\}$,\ $x^-=\max\{-x,0\}$, $a$ and $b$ are two different positive constants, $f(t)$ is a real analytic almost periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar almost periodic mappings, which was proved recently by the authors (see \cite{Huang}).\ Then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric oscillation.