A four dimensional map for escape from resonance: negative energy modes and nonlinear instability

TL;DR

This paper investigates nonlinear instability caused by negative energy modes in Hamiltonian systems, using a four-dimensional map to model how such modes lead to phase space escape and Arnold diffusion.

Contribution

It introduces a four-dimensional map that models Hamiltonian systems with negative energy modes and analyzes their nonlinear instability and escape mechanisms.

Findings

Negative energy modes can cause nonlinear instability despite spectral stability.

The four-dimensional map demonstrates how negative energy modes lead to phase space escape.

The model captures Arnold diffusion phenomena in Hamiltonian systems.

Abstract

Positive definiteness of a Hamiltonian expanded about an equilibrium point provides only a necessary condition for stability, a criterion known as Dirichlet's theorem. The reason that this criterion is not necessary for stability is because of the possible existence of negative energy modes, which are linearly stable modes of oscillation that have negative energy. When such modes are present, the Hamiltonian is, in general, indefinite. Although such systems with negative energy modes are linearly stable (spectral stable), they are unstable to infinitesimal perturbations under the nonlinear dynamics. In the present work we study this kind of nonlinear instability with the simplest nontrivial four dimensional area-preserving map, which has a cubic degree of freedom, that was designed to mimic the behavior of a Hamiltonian system with one positive and one negative energy mode, and a quadratic degree of freedom, that allows eventual escapes in phase space, usually called as Arnold diffusion.