Improved stability estimates at elliptic equilibria of Hamiltonian systems

TL;DR

This paper enhances the theoretical bounds on the stability duration of elliptic equilibria in Hamiltonian systems by refining the estimates derived from Birkhoff normal form, considering resonance properties and providing comparative examples.

Contribution

It introduces improved a-priori stability bounds that incorporate resonance effects and quadratic terms, advancing the understanding of stability times in Hamiltonian systems.

Findings

New stability estimates depend on resonance properties.

Improved bounds show longer stability times than traditional estimates.

Comparative example demonstrates the effectiveness of the new bounds.

Abstract

This paper deals with an improvement of the "a-priori stability bounds" on the variation of the action variables and on the stability time obtained from a given Birkhoff normal form around the elliptic equilibrium point of an Hamiltonian system satisfying a non-resonance condition of finite order N. In particular, we improve the standard a-priori lower bound on the stability time from a purely linear dependence on the inverse of the polynomial norm of the remainder of the normal form to the sum of a linear term (which is still present but with a different constant coefficient) and a quadratic one. The prevalence between the linear and the quadratic term depends on the resonance properties of all the monomials in the remainder of the normal form with degree from N to a finite order M. We also provide a comparative example of the new estimates and the traditional a priori ones in the framework of computer-assisted proofs.