Normal forms \`a la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium

TL;DR

This paper extends Moser's classical normal form theorem to aperiodically time-dependent Hamiltonians near hyperbolic equilibria, especially when the perturbation decays over time, broadening the theorem's applicability.

Contribution

It generalizes Moser's normal form theorem to include aperiodic and decaying time-dependent Hamiltonians, providing new insights into their local dynamics.

Findings

Normal form exists for aperiodically time-dependent Hamiltonians near hyperbolic equilibria.

Decay in perturbation enhances the normal form result.

Extension of classical theory to more general time-dependent settings.

Abstract

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is obtained in the case in which the perturbing function exhibits a time decay.