Let's take a time-periodic Hamiltonian $H(t,x,y)$ on $\mathbb{R}^2$ and apply an arbitrarily small time-independent perturbation to $H$ via $$ \tilde H (t,x,y) = H(t,x,y) + \epsilon V(x,y), $$ where $V$ is a smooth function, $\epsilon >0$ small. Are there any properties of the solutions of the perturbed system $\tilde H$ that carry over or influence the behaviour of the solutions of the unperturbed system $H$?

Any hints or references to the literature are very much appreciated.