Biasymptotically Quasi-Periodic Solutions for Time-Dependent Hamiltonians

We consider time-dependent perturbations of integrable and near-integrable Hamiltonians. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of biasymptotically quasi-periodic solutions. That is, orbits converging to some quasi-periodic solutions in the future (as $t \rightarrow +\infty$) and the past (as $t \rightarrow -\infty$).
Concerning the proof, thanks to the implicit function theorem, we prove the existence of a family of orbits converging to some quasi-periodic solutions in the future and another family of motions converging to some quasi-periodic solutions in the past. Then, we look at the intersection between these two families when $t = 0$. Under suitable hypotheses on the Hamiltonian’s regularity and the perturbation’s smallness, it is a large set, and each point gives rise to biasymptotically quasi-periodic solutions.