Chirikov and Nekhoroshev diffusion estimates: bridging the two sides of the river

Abstract

We present theoretical and numerical results pointing towards a strong connection between the estimates for the diffusion rate along simple resonances in multidimensional nonlinear Hamiltonian systems that can be obtained using the heuristic theory of Chirikov and a more formal one due to Nekhoroshev. We show that, despite a wide-spread impression, the two theories are complementary rather than antagonist. Indeed, although Chirikov's 1979 review has thousands of citations, almost all of them refer to topics such as the resonance overlap criterion, fast diffusion, the Standard or Whisker Map, and not to the constructive theory providing a formula to measure diffusion along a single resonance. However, as will be demonstrated explicitly below, Chirikov's formula provides values of the diffusion coefficient which are quite well comparable to the numerically computed ones, provided that it is implemented on the so-called optimal normal form derived as in the analytic part of Nekhoroshev's theorem. On the other hand, Chirikov's formula yields unrealistic values of the diffusion coefficient, in particular for very small values of the perturbation, when used in the original Hamiltonian instead of the optimal normal form. In the present paper, we take advantage of this complementarity in order to obtain accurate theoretical predictions for the local value of the diffusion coefficient along a resonance in a specific 3DoF nearly integrable Hamiltonian system. Besides, we compute numerically the diffusion coefficient and a full comparison of all estimates is made for ten values of the perturbation parameter, showing a very satisfactory agreement.

25 pages, 9 figures. NOTICE: this is the author's version of a work that was accepted for publication in Physica D. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication

References in corpus (3)

• Chirikov Diffusion in the Asteroidal Three-Body Resonance (5,-2,-2)
• On the connection between the Nekhoroshev theorem and Arnold Diffusion
• Testing the accuracy of the overlap criterion

Cited by in corpus (14)

• The path to instability in compact multi-planetary systems
• Intramolecular vibrational energy redistribution and the quantum ergodicity transition: a phase space perspective
• Chaotic Diffusion in the Gliese-876 Planetary System
• On the relevance of chaos for halo stars in the Solar Neighbourhood
• On the relevance of chaos for halo stars in the solar neighbourhood II
• Effects of chaos on the detectability of stellar streams
• Diffusion and Drift in Volume-Preserving Maps
• Relevance of the resonance junctions on the Arnold web to dynamical tunneling and eigenstate delocalization
• Semi-analytic computations of the speed of Arnold diffusion along single resonances in a priori stable Hamiltonian systems
• Shannon Entropy Applied to the Planar Restricted Three-Body Problem
• Partial barriers to chaotic transport in 4D symplectic maps
• Stochastic approach to diffusion inside the chaotic layer of a resonance
• Advanced Techniques in Stability Analysis of Trans-Neptunian Objects
• Arnold web and dynamical tunneling in a four-site Bose-Hubbard model