Optimal stability and instability for near-linear Hamiltonians

TL;DR

This paper establishes optimal conditions for stability and instability in near-linear Hamiltonian systems, introduces a new concept of effective stability, and confirms a related conjecture in the linear case.

Contribution

It provides a general stability result for perturbed linear integrable Hamiltonians, constructs an optimal instability example, and introduces and partially characterizes effectively stable Hamiltonians.

Findings

Proved a broad stability theorem for linear Hamiltonian perturbations.

Constructed an example demonstrating optimal instability.

Confirmed a conjecture on effective stability in the linear case.

Abstract

In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover, in the same spirit as the notion of KAM stable integrable Hamiltonians, we will introduce a notion of effectively stable integrable Hamiltonians, conjecture a characterization of these Hamiltonians and show that our result prove this conjecture in the linear case.