Arnold diffusion and Nekhoroshev theory

TL;DR

This paper explores Arnold diffusion in Hamiltonian systems with three or more degrees of freedom, focusing on the mechanisms, invariant objects, and quantitative methods to analyze the diffusion speed, supported by numerical examples.

Contribution

It provides a detailed introduction to Arnold diffusion, linking it to Nekhoroshev theory, and discusses methods for quantifying diffusion speed using normal form techniques and numerical algorithms.

Findings

Identification of phase space invariant objects driving diffusion

Quantitative estimates of Arnold diffusion speed

Validation of methods through numerical examples

Abstract

Starting with Arnold's pioneering work, the term "Arnold diffusion" has been used to describe the slow diffusion taking place in the space of the actions in Hamiltonian nonlinear dynamical systems with three or more degrees of freedom. The present text is an elaborated transcript of the introductory course given in the Milano I-CELMECH school on the topic of Arnold diffusion and its relation to Nekhoroshev theory. The course introduces basic concepts related to our current understanding of the mechanisms leading to Arnold diffusion. Emphasis is placed upon the identification of those invariant objects in phase space which drive chaotic diffusion, such as the stable and unstable manifolds emanating from (partially) hyperbolic invariant objects. Besides a qualitative understanding of the diffusion mechanisms, a precise quantification of the speed of Arnold diffusion can be achieved by methods based on canonical perturbation theory, i.e. by the construction of a suitable normal form at optimal order. As an example of such methods, we discuss the (quasi-)stationary-phase approximation for the selection of remainder terms acting as driving terms for the diffusion. Finally, we discuss the efficiency of such methods through numerical examples in which the optimal normal form is determined by a computer-algebraic implementation of a normalization algorithm.