descriptionPublicationkeyboard_double_arrow_right Article 02 Apr 2004Publisher:IOP PublishingJournal:Nonlinearity, volume 17, pages 1,153-1,177 (issn: 0951-7715, eissn: 1361-6544,

Authors: Rafael Ramírez-Ros; Sergey Bolotin; Sergey Bolotin; Amadeu Delshams;

doi: 10.1088/0951-7715/17/4/002

handle: 2117/874

Persistence of homoclinic orbits for billiards and twist maps

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Abstract

We consider the billiard motion inside a C2-small perturbation of a ndimensional ellipsoid Q with a unique major axis. The diameter of the ellipsoid Q is a hyperbolic two-periodic trajectory whose stable and unstable invariant manifolds are doubled, so that there is a n-dimensional invariant set W of homoclinic orbits for the unperturbed billiard map. The set W is a stratified set with a complicated structure. For the perturbed billiard map the set W generically breaks down into isolated homoclinic orbits. We provide lower bounds for the number of primary homoclinic orbits of the perturbed billiard which are close to unperturbed homoclinic orbits in certain strata of W. The lower bound for the number of persisting primary homoclinic billiard orbits is deduced from a more general lower bound for exact perturbations of twist maps possessing a manifold of homoclinic orbits.

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Keywords

perturbation, Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods, Dynamical aspects of twist maps, Symplectic mappings, fixed points (dynamical systems), :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], Differentiable dynamical systems, Hamilton, and nonholonomic systems, Hamiltonian systems, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems, Lagrangian, homoclinic orbits, :37 Dynamical systems and ergodic theory::37E Low-dimensional dynamical systems [Classificació AMS], :37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [Classificació AMS], Perturbation theories for problems in Hamiltonian and Lagrangian mechanics, Sistemes dinàmics diferenciables, Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory, Sistemes de, Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol'd diffusion, Classificació AMS::37 Dynamical systems and ergodic theory::37E Low-dimensional dynamical systems, Hamilton, Sistemes de, billiards, billiard map, twist maps, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, contact

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