Parabolic resonances in 3 degree of freedom near-integrable Hamiltonian systems
type: Item
instance of: scholarly article
title: Parabolic resonances in 3 degree of freedom near-integrable Hamiltonian systems
zbMATH Open document ID: 1036.70010
DOI: 10.1016/S0167-2789(02)00374-3
author: Anna Litvak-Hinenzon
author: V. Rom-Kedar
published in: Physica D
publication date: 2002-05-12T00:00:00Z
review text: In 1964 \textit{V.I. Arnold} [Sov. Math., Dokl. 5, 581-585 (1964; Zbl 0135.42602)] introduced an interesting phenomenon leading to instability of Hamiltonian systems with more than 2 degrees of freedom (called also Arnold diffusion). In the paper under review an another kind of complexity occurring in high-dimensional systems is described. The authors consider near-integrable Hamiltonian systems with 3 degrees of freedom (d.o.f.). They assume that the corresponding nonperturbed integrable system possesses a normally parabolic \(m\)-resonant (\(m=1,2\)) \(2\)-torus. The numerical experiments suggest that the trajectories of the perturbed system starting near the parabolic resonant torus exhibit a complicated behavior. This is related to the appearance of a low-dimensional resonant torus with degenerate normal stability, called a parabolic \(m\)-resonance (PR). The authors construct a simple phenomenological model which is a near-integrable \(3\) d.o.f. Hamiltonian system containing free parameters \(\alpha_1\), \(\alpha_2\), \(\alpha_3\) which govern in the nonperturbed case the topology of integrable surfaces and their stability. The thorough analysis of energy-momenta bifurcation diagram shows the presence of a whole family of parabolic resonant tori of mixed stability types. This in turn leads the authors to the classification (illustrated with help of numericall simulations) of the behavior and instability of different types of PR orbits which appear as a result of a small perturbation. Remarkably, PRs are quite persistent for smooth near-integrable 3 d.o.f. Hamiltonian systems as shown in the another paper by the same authors [Nonlinearity 15, 1149-1177 (2002; Zbl 1055.37062)].
reviewed by: Alexei Tsygvintsev
Mathematics Subject Classification ID: 70H08
Mathematics Subject Classification ID: 70K55
zbMATH DE Number: 1738968
zbMATH Keywords: parabolic resonant torus
zbMATH Keywords: degenerate normal stability
zbMATH Keywords: parabolic \(m\)-resonance
zbMATH Keywords: energy-momenta bifurcation diagram
zbMATH Keywords: chaos
MaRDI profile type: Publication
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full work available at URL: s0167-2789(02)00374-3
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description: scientific article; zbMATH DE number 1738968