A theory of modulational diffusion
type: Item
instance of: scholarly article
title: A theory of modulational diffusion
zbMATH Open document ID: 0584.58033
DOI: 10.1016/0167-2789(85)90091-0
author: M. A. Lieberman
author: Boris V. Chirikov
author: D. L. Shepelyansky
author: Franco Vivaldi
published in: Physica D
publication date: 1985-01-01T00:00:00Z
review text: An analytical theory of the phenomenon of ''modulational diffusion'' has been developed in a book by \textit{A. J. Lichtenberg} and the second author [Regular and stochastic motion (1983; Zbl 0506.70016)]. In the present paper this approach is continued and compared to numerical simulation. For this reason the authors consider the two degrees of freedom nonautonomous Hamiltonian \[ H(I_ 1,I_ 2,\theta_ 1,\theta_ 2)=\frac{I^ 2_ 1}{2}+\frac{I^ 2_ 2}{2}-\epsilon \cos (\theta_ 1+\lambda \sin \omega t)-\mu \cos (\theta_ 1-\theta_ 2). \] Here \(I_ i\) and \(\theta_ i\) denote action and angle variables; the first two terms describe unperturbed (integrable) motion of two oscillators, the third term introduces phase modulation of the first oscillator, whereas the last term couples the motion of both oscillators. If \(\epsilon \omega^{-2}\lambda^{-1/2}\) is sufficiently large (i.e., if the modulation frequency is sufficiently small) a modulational layer of overlapping resonances is produced, and the coupling to the second oscillator produces a long time diffusion along this layer. The diffusion rate for this process is evaluated analytically (within some approximations) and numerically. The diffusion coefficient is found to vary in a series of descending steps as the frequency difference between the two oscillators is increased.
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 70K30
zbMATH DE Number: 3936040
zbMATH Keywords: chaos
zbMATH Keywords: modulational diffusion
zbMATH Keywords: thick layer diffusion
zbMATH Keywords: long time predection
zbMATH Keywords: diffusion coefficient
zbMATH Keywords: numerical simulation
Wikidata QID: Q62593783
reviewed by: Volker Perlick
MaRDI profile type: Publication
full work available at URL: 0167-2789(85)90091-0
OpenAlex ID: W2126415302
cites work: Regular and stochastic motion
cites work: Q3317878
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description: scientific article; zbMATH DE number 3936040