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.
Since the reference in the comment by Piyush Grover is quite old, I think it makes sense to add this 2012 work by Lin Wang: Converse KAM theory revisited, which moreover considers flows and not maps. In short: very little can be said. Wang gives negative examples, showing that Lagrangian tori can be destroyed by arbitrarily small and fairly well behaved perturbations.
Here are some results (based on previous works by Wang) contained in the work. Assuming that $H_0$ is an integrable Hamiltonian with $d ≥ 2$ degrees of freedom and a rotation vector $\omega$, then one has the following theorems.
Theorem 1.1: There exists a sequence of $C^\infty$ Hamiltonians $\{H_n\}_{n∈N}$ such that $H_n \to H_0$ in the $C^{2d−δ}$ topology and the Hamiltonian flow generated by $H_n$ does not admit the Lagrangian torus with the rotation vector $\omega$.
Theorem 1.2: There exists a sequence of $C^\omega$ Hamiltonians $\{H_n\}_{n∈N}$ such that $H_n \to H_0$ in the $C^{d+1−δ}$ topology and the Hamiltonian flow generated by ${H_n}$ does not admit the Lagrangian torus with the rotation vector $\omega$.
Theorem 1.3: All Lagrangian tori can be destructed by analytic perturbations which are arbitrarily small in the $C^{d−δ}$ topology.
