Parabolic resonances in 3 degree of freedom near-integrable Hamiltonian systems
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)].
scientific article; zbMATH DE number 1738968
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