Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ELLIPTIC BIRKHOFF'S BILLIARDS WITH $C^2$-GENERIC GLOBAL PERTURBATIONS

  • Kim, Gwang-Il (Department of Mathematics, Gyeongsang National University)
  • Published : 1999.02.01
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Abstract

Tabanov investigated the global symmetric perturbation of the integrable billiard mapping in the ellipse [3]. He showed the nonintegrability of the Birkhoff billiard in the perturbed domain by proving that the principal separatrices splitting angle is not zero.In this paper, using the exact separatrix map of an one-degree-of freedom Hamiltoniam system with time periodic perturbation, we show the existence the stochastic layer including the uniformly hyperbolic invariant set which implies the nonintegrability near the separatrices of a Birkhoff's billiard in the domain bounded by $C^2$ convex simple curve constructed by the generic global perturbation of the ellipse.

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References

  1. Acta Math. v.50 On the periodic motions of the dynamical system G. D. Birkhoff
  2. Amer. Math. Soc. Colloq. Publ. v.9 Dynamical systems G. D. Birkhoff
  3. Chaos v.4 no.4 Separatrices splitting for Birkhoff's billiard in symmetric convex domain closed to an ellipse M. B. Tabanov
  4. Ergd. Th. & Dynam. Sys. v.6 Monotone twist mappings and the calculus of variations J. Moser
  5. Physica D. v.89 Analysis of the separatrix map in Hamiltonian systems T. H. Ahn;G. I. Kim;S. H. Kim
  6. Phys. Rep. v.52 A universal instability of many-dimensional oscillator systems B. V. Chirikov
  7. Ann. Math. v.98 Lagrangian submanifolds and Hamiltonian systems A. Weinstein
  8. C. T. Acad. Sci. Paris v.26 Sur une propie'te topologique des applications globelement canonique de la me'canique classique V. I. Arnold
  9. J. Math. Phys. v.23 Melnikov's method and Arnold diffusion for perturbations of integrable Hamiltonian systems P. Holmes;J. E. Marsden
  10. Am. J. Math. v.96 Homoclinic points of area preserving diffeomorphism R. McGehee;K. Meyer
  11. Physica D. v.56 Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models R. S. MacKay;S. Aubry;C. Baesens