Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians

TL;DR

This paper develops a new method for constructing resonant normal forms with exponentially small remainders for Gevrey Hamiltonians, extending previous analytic results and providing bounds on stability times and manifold splitting.

Contribution

It introduces a novel approach to normal form construction for Gevrey Hamiltonians, applicable to non-analytic cases, with implications for stability and invariant manifold analysis.

Findings

Normal forms with exponentially small remainders for Gevrey Hamiltonians

Exponential bounds on stability times of action variables

Exponential bounds on splitting of invariant manifolds

Abstract

In this paper, we give a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency. The construction is based on the special case of a periodic frequency, a Diophantine result concerning the approximation of a vector by independent periodic vectors and a technique of composition of periodic averaging. It enables us to deal with non-analytic Hamiltonians, and in this first part we will focus on Gevrey Hamiltonians and derive normal forms with an exponentially small remainder. This extends a result which was known for analytic Hamiltonians, and only in the periodic case for Gevrey Hamiltonians. As applications, we obtain an exponentially large upper bound on the stability time for the evolution of the action variables and an exponentially small upper bound on the splitting of invariant manifolds for hyperbolic tori, generalizing corresponding results for analytic Hamiltonians.