A Numerical Study of Arnold Diffusion in a Priori Unstable Systems

This paper concerns the problem of the numerical detection of Arnold diffusion in a priori unstable systems. Specifically, we introduce a new definition of Arnold diffusion which is adapted to the numerical investigation of the problem, and is based on the numerical computation of the stable and unstable manifolds of the system. Examples of this Arnold diffusion are provided in a model system. In this model, we also find that Arnold diffusion behaves as an approximate Markovian process, thus it becomes possible to compute diffusion coefficients. The values of the diffusion coefficients satisfy the scaling \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D(\epsilon)\simeq \epsilon^2}$$\end{document} . We also find that this law is correlated to the validity of the Melnikov approximation: in fact, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D(\epsilon)\simeq \epsilon^2}$$\end{document} law is valid up to the same critical value of ε for which the error terms of Melnikov approximations have a sharp increment.