"Many systems of physical interest are Hamiltonian systems subject to non-autonomous perturbations decaying in time, such as a molecule interacting with a laser pulse or certain models in celestial mechanics where a perturbing body drifts away. In analogy with classical KAM theory, where one asks whether the invariant tori of an integrable system persist under a small autonomous perturbation, it is natural to ask whether, in this context, the invariant tori of the unperturbed system persist, in a suitable asymptotic sense, in the non-autonomous dynamics. Such persistence results have recently been obtained in the case of Lagrangian tori, and it is the aim of this talk to discuss the lower-dimensional case.

We consider a Hamiltonian system carrying a lower-dimensional invariant torus supporting quasiperiodic solutions, perturbed by a term decaying in time. Under suitable decay rates we obtain an asymptotic KAM torus: an invariant manifold in the extended phase space whose dynamics converge, as time tends to infinity, to the quasiperiodic solutions of the unperturbed system. Moreover, we describe the associated transverse dynamics and show that the linearized flow along the orbits contained in this manifold is asymptotically conjugate to an explicit model cocycle, so the asymptotic KAM torus retains the elliptic, hyperbolic or even parabolic transverse behaviour of the unperturbed torus.

This is joint work with Donato Scarcella."