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Analytic structure of the Henon-Heiles Hamiltonian in integrable and nonintegrable regimes. (English) Zbl 0492.70019


MSC:

70H05 Hamilton’s equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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References:

[1] M. Tabor, The Onset of Chaos in Dynamical Systems, Advances in Chemical Physics, Vol. 46 (Wiley, New York, 1981).
[2] M. J. Ablowitz, A. Ramani, and H. Segur, J. Math. Phys. 21, 715 (1980).JMAPAQ0022-2488
[3] S. Kowalevskaya, Acta Math. 14, 81 (1890); ACMAA80001-5962
[4] V. V. Golubov, Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point (State Publishing House, Moscow, 1953).
[5] H. Segur, ”Solitons and the Inverse Scattering Transform,” Lectures given at the International School of Physics, ’Enrico Fermi,’ Varenna, Italy, July 7–19, 1980.
[6] M. Tabor and J. Weiss, ”Analytical Structure of the Lorenz System,” Phys. Rev. A 24, 2157 (1981).PLRAAN1050-2947
[7] I. C. Percival and J. M. Green, ”Hamiltonian Maps in the Complex Plane,” Princeton Plasma Physics Laboratory Preprint, PPPL-1744 Jan. 1981 (unpublished).
[8] B. Mandelbrot, Fractals: Form, Chance and Dimension (Freeman, San Francisco, 1977). · Zbl 0376.28020
[9] M. Henon and C. Heiles, Astron. J. 69, 73 (1964).ANJOAA0004-6256
[10] Y. Aizawa and N. Saito, J. Phys. Soc. Jpn. 32, 1636 (1972).JUPSAU0031-9015
[11] John Greene, private communication. Noid, Koszykowski, and Marcus, J. Chem. Phys. 71, 2864 (1979), have shown this system to be separable in parabolic coordinates for the case A = 1, B = 2.JCPSA60021-9606
[12] T. Bountis, H. Segur, and F. Vivaldi, ”Integrable Hamiltonian Systems and the PainlevĂ© Property,” preprint (unpublished).
[13] Y. F. Chang, J. M. Green, M. Tabor, and J. Weiss, ”The Analytic Structure of Dynamical Systems and Self-Similar Natural Boundaries,” LJI Preprint LJI-R-81-152 (November, 1981).
[14] Y. F. Chang and G. Corliss, J. Inst. Math. Appl. 25, 349 (1980).JMTAA80020-2932
[15] Y. F. Chang, M. Tabor, J. Weiss, and G. Corliss, Phys. Lett. A 85, 211 (1981).PYLAAG0375-9601
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