The speed of Arnold diffusion

Abstract

A detailed numerical study is presented of the slow diffusion (Arnold diffusion) taking place around resonance crossings in nearly integrable Hamiltonian systems of three degrees of freedom in the so-called ‘Nekhoroshev regime’. The aim is to construct estimates regarding the speed of diffusion based on the numerical values of a truncated form of the so-called remainder of a normalized Hamiltonian function, and to compare them with the outcomes of direct numerical experiments using ensembles of orbits. In this comparison we examine, one by one, the main steps of the so-called analytic and geometric parts of the Nekhoroshev theorem. Thus: (i) we review and implement an algorithm Efthymiopoulos (2008) [45] for Hamiltonian normalization in multiply resonant domains which is implemented as a computer program making calculations up to a high normalization order. (ii) We compute the dependence of the optimal normalization order on the small parameter ϵ in a specific model and compare the result with theoretical estimates on this dependence. (iii) We examine in detail the consequences of assuming simple convexity conditions for the unperturbed Hamiltonian on the geometry of the resonances and on the phase space structure around resonance crossings. (iv) We discuss the dynamical mechanisms by which the remainder of the optimal Hamiltonian normal form drives the diffusion process. Through these steps, we are led to two main results: (i) We construct in our concrete example a convenient set of variables, proposed first by Benettin and Gallavotti (1986) [12], in which the phenomenon of Arnold diffusion in doubly resonant domains can be clearly visualized. (ii) We determine, by numerical fitting of our data, the dependence of the local diffusion coefficient D on the size ‖Ropt‖ of the optimal remainder function, and we compare this with a heuristic argument based on the assumption of normal diffusion. We find a power law D∝‖Ropt‖2(1+b), where the constant b has a small positive value depending also on the multiplicity of the resonance considered.