  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 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. That is a time-dependent family of embedded tori
converging as time tends to infinity to the quasiperiodic invariant torus of
the unperturbed system.
  In this paper, the dynamic on the invariant torus associated with the
unperturbed Hamiltonian is arbitrary. Therefore, we need to assume exponential
decay in time in order to prove the existence of a time-dependent family of
embedded tori converging in time to the invariant torus associated with the
unperturbed system. The proof relies on the implicit function theorem, and the
most complicated and original part rests on the solution of the associated
linearized problem.
