Biasymptotically quasiperiodic solutions for time-dependent Hamiltonians

In a previous work [Asymptotically quasiperiodic solutions for time-dependent Hamiltonians, arXiv preprint arXiv:2211.06623 (2022)], we consider time-dependent perturbations of a Hamiltonian vector field having an invariant torus supporting quasiperiodic solutions. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of an asymptotic KAM torus. An asymptotic KAM torus is a time-dependent family of embedded tori converging as time tends to infinity to the invariant torus associated with the unperturbed system. Now, it is quite natural to wonder when we have the existence of a biasymptotic KAM torus. That is a continuous time-dependent family of embedded tori converging in the future and the past to suitable quasiperiodic invariant tori. In this work, we go one step further. We analyze time-dependent perturbations of integrable and near-integrable Hamiltonians. Assuming the perturbation decays polynomially fast in time, we prove the existence of orbit converging to some quasiperiodic solutions in the future and the past.