Asymptotically quasiperiodic solutions for time-dependent Hamiltonians

TL;DR

This paper extends previous results on asymptotically quasiperiodic solutions to time-dependent Hamiltonian systems, removing decay and smallness constraints by leveraging Hamiltonian geometry.

Contribution

It generalizes earlier work by relaxing decay conditions and eliminating smallness assumptions for time-dependent Hamiltonian perturbations.

Findings

Existence of asymptotically quasiperiodic solutions in Hamiltonian systems.

Removal of decay and smallness constraints in the perturbation.

Application of Hamiltonian geometric properties to prove results.

Abstract

In 2015, M. Canadell and R. de la Llave consider a time-dependent perturbation of a vector field having an invariant torus supporting quasiperiodic solutions. Under a smallness assumption on the perturbation and assuming the perturbation decays (when t goes to infinity) exponentially fast in time, they proved the existence of motions converging in time (when t goes to infinity) to quasiperiodic solutions associated with the unperturbed system (asymptotically quasiperiodic solutions). In this paper, we generalize this result in the particular case of time-dependent Hamiltonian systems. The exponential decay in time is relaxed (due to the geometrical properties of Hamiltonian systems) and the smallness assumption on the perturbation is removed.