For Hamiltonian systems with two degrees of freedom, close to nonlinear integrable systems, we discuss rearrangements in the degenerate resonance zones in terms of the averaged system normalized near the resonance. For degeneracy order n = 3 we describe typical rearrangements of phase portraits connected with passage from Poincaré–Birkhoff chains to vortex pairs and Kármán vortex streets. Bibliography: 16 titles.
Similar content being viewed by others
References
V. I. Arnlold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974); English transl.: Springer, New York etc. (1978).
H. Poincaré, “ Sur un théorème de géometrie,” Circ. Mat. Palermo 33, 357-407 (1912).
G. D. Birkhoff, “Proof of Poincaré’s geometric theorem,” Trans. Am. Math. Soc. 14, 14–22 (1913).
A. A. Karabanov and A. D. Morozov, “On degenerate resonances in Hamiltonian systems with two degrees of freedom,” Chaos Solitons Fractals 69, 201-208 (2014).
J. Guckenheimer and Ph. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, Berlin etc. (1983).
A. D. Morozov, Quasi-Conservative Systems: Cycles, Resonances and Chaos, World Sci., Singapore (1998).
A. D. Morozov Resonances, Cycles, and Chaos in Quasiconservative Systems [in Russian], Izhevsk etc. (2005).
A. D. Morozov and L. P. Shil’nikov, “On nonconservative periodic systems close to twodimensional Hamiltonian” [in Russian], Prikl. Mat. Mekh. 47, No. 3, 385-394 (1983); English transl.: J. Appl. Math. Mech. 47, 327-334 (1984).
A. D. Morozov, “On bifurcations in degenerate resonance zones,” Regul. Chaotic Dyn. 19, No. 4, 451-459 (2014).
A. D. Morozov, “On degenerate resonances and “vortex pairs,” Regul. Chaotic Dyn. 13, No. 1, 27-36 (2008).
J. E. Howard and J. Humpherys, “Nonmonotonic twist maps,” Physica D 80, 256-276 (1995).
C. Simó, “Invariant curves of analytic perturbed nontwist area preserving maps,” Regul. Chaotic Dyn. 3, No. 3, 180-195 (1998).
A. A. Karabanov and A. D. Morozov, “On averaging near degenerate resonance in fourdimensional two-frequency problem” [in Russian], Tr. Sredn. Mat. Obshch. 6, No. 1, 292–301 (2004).
A. D. Morozov and S. A. Boykova, “On investigation of the degenerate resonances,” Regul. Chaotic Dyn. 4, No. 1, 70-82 (1999).
A. D. Morozov, “Degenerate resonances in Hamiltonian systems with 3/2 degrees of freedom,” Chaos 12, No. 3, 539–548 (2002).
A. D. Morozov and T. N. Dragunov, Visualization and Analysis of Invariant Sets of Dynamical Systems [in Russian], Izhevsk etc. (2003).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 85, June 2016, pp. 147-151.
Rights and permissions
About this article
Cite this article
Karabanov, A.A., Morozov, A.D. Degenerate Resonances in Hamiltonian Systems: From Poincaré–Birkhoff Chains to Vortex Pairs and Kármán Vortex Streets. J Math Sci 219, 155–159 (2016). https://doi.org/10.1007/s10958-016-3092-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-016-3092-7