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hal-00368735, version 1

Symplectic invariants near hyperbolic-hyperbolic points

H. R. Dullin 1, San Vu Ngoc () 2, 3

Regular and Chaotic Dynamics 12, 6 (2007) 689-716

Abstract: We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolic-hyperbolic type. These invariants consist in some signs which determine the topology of the critical Lagrangian fibre, together with several Taylor series which can be computed from the dynamics of the system.We show how these series are related to the singular asymptotics of the action integrals at the critical value of the energy-momentum map. This gives general conditions under which the non-degeneracy conditions arising in the KAM theorem (Kolmogorov condition, twist condition) are satisfied. Using this approach, we obtain new asymptotic formulae for the action integrals of the C. Neumann system. As a corollary, we show that the Arnold twist condition holds for generic frequencies of this system

  • 1:  Department of Mathematics
  • Loughborough University
  • 2:  Institut Fourier (IF)
  • CNRS : UMR5582 – Université Joseph Fourier - Grenoble I
  • 3:  Institut de Recherche Mathématique de Rennes (IRMAR)
  • CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
  • Domain : Mathematics/Analysis of PDEs
  • Keywords : completely integrable systems – hyperbolic-hyperbolic point – KAM – isoenergetic nondegeneracy - vanishing twist
 
  • hal-00368735, version 1
  • http://hal.archives-ouvertes.fr/hal-00368735
  • oai:hal.archives-ouvertes.fr:hal-00368735
  • From: Dominique Hervé <>
  • Submitted on: Tuesday, 17 March 2009 14:52:19
  • Updated on: Tuesday, 17 March 2009 14:52:19