On the connection between the Nekhoroshev theorem and Arnold Diffusion

TL;DR

This paper uses Nekhoroshev theorem techniques to estimate Arnold diffusion rates in a Hamiltonian system, revealing how the diffusion coefficient depends on the optimal remainder size and small parameter epsilon.

Contribution

It establishes a quantitative relationship between the diffusion coefficient and the optimal remainder, connecting analytical estimates with numerical results in Hamiltonian dynamics.

Findings

Diffusion coefficient scales as D ∝ ||R_{opt}||^3

Optimal remainder scales as ||R_{opt}|| ∝ exp(1/ε^{0.21})

Comparison with numerical results confirms theoretical estimates

Abstract

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschl\'{e} et al. (2000). A resonant normal form is constructed by a computer program and the size of its remainder $||R_{opt}||$ at the optimal order of normalization is calculated as a function of the small parameter $\epsilon$. We find that the diffusion coefficient scales as $D\propto||R_{opt}||^3$, while the size of the optimal remainder scales as $||R_{opt}|| \propto\exp(1/\epsilon^{0.21})$ in the range $10^{-4}\leq\epsilon \leq 10^{-2}$. A comparison is made with the numerical results of Lega et al. (2003) in the same model.