The original question is to show that for small $\varepsilon>0$, all solution of ODE $(\dot q(t),\dot p(t))=(p(t),-q(t)^3+\varepsilon\sin t)$ is bounded. Using KAM theory we can show that for a fixed compact set $K\subset\mathbb R^2$ we can find a $\varepsilon_0(K)>0$, when $0<\varepsilon<\varepsilon_0$ all solution with initial condition in $K$ is bounded.
I want to show that such $\varepsilon_0>0$ can be chosen to be independent of the compact set. And using change of variable all $\varepsilon$ is true iff $\varepsilon=1$ is true.
Is there any simply way to show that without directly solve the equation?