Consider the equation $\frac{dx}{dt}=\sin 2\pi x+c \sin 2\pi t$ with $|c|<1$, I want to show that $\lim\limits_{t\to\infty} \frac{x(t)}{t}=0$.
For a given $x\in \mathbb R$, there is a unique trajectory passing through $(0,x)$. Let $A(x)$ be the value of this trajectory at $t=1$. Since $\sin 2\pi x+c \sin 2\pi t$ is a double periodical function. $A:\mathbb R\to\mathbb R$ is a lifting of a map $S^1\to S^1$. Therefore the limit we want to calculate is just the rotation number of $A$. But I have no idea how to calculate this number.
On the lines $x=\frac{n}2+\frac14$ the right side has a constant sign $(-1)^n$. This makes the corresponding intervals around $x=k+\frac12$ into trapping regions. Thus the solution is constrained to one of these intervals, with the associated consequences for the limit.