Stability of nearly-integrable systems with dissipation

TL;DR

This paper analyzes the stability of nearly-integrable Hamiltonian systems with added dissipation, identifying conditions for linear and exponential stability times and applying results to specific examples.

Contribution

It introduces a framework for assessing stability times in dissipative nearly-integrable systems under resonance conditions, with explicit bounds and applications.

Findings

Established linear and exponential stability times under small dissipation and perturbation.

Derived stability bounds depending on resonance assumptions and parameter smallness.

Applied theoretical results to concrete examples demonstrating stability behaviors.

Abstract

We study the stability of a vector field associated to a nearly-integrable Hamiltonian dynamical system to which a dissipation is added. Such a system is governed by two parameters, named the perturbing and dissipative parameters, and it depends on a drift function. Assuming that the frequency of motion satisfies some resonance assumption, we investigate the stability of the dynamics, and precisely the variation of the action variables associated to the conservative model. According to the structure of the vector field, one can find linear and exponential stability times, which are established under smallness con- ditions on the parameters. We also provide some applications to concrete examples, which exhibit a linear or exponential stability behavior.