  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.

Dynamical Systems