Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium

Oksana Koltsova; Lev Lerman

doi:10.3934/dcds.2009.25.883

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2009, Volume 25, Issue 3: 883-913. Doi: 10.3934/dcds.2009.25.883

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Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium

Oksana Koltsova^1, and
Lev Lerman^2,

1.
Department of Mathematics, Imperial College London, SW7 2AZ, London
2.
Research Inst. for Applied Math. and Cybernetics, and Dept. of Diff. Equat., Nizhny Novgorod State University, 603950, Nizhny Novgorod

Received: October 2008

Revised: April 2009

Published: September 2009

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Abstract

An orbit behavior of a Hamiltonian system in two degrees of freedom in a neighborhood of a quadratically tangent homoclinic orbit to a saddle-focus equilibrium is studied, the orbit structure within the level of the saddle-focus is described. We show the existence of an intrinsic parameter (a modulus) which determines types of invariant nonuniform hyperbolic subsets. These subsets are described by means of symbolic dynamics with countably many states. Multi-round tangent and transversal homoclinic orbits to the saddle-focus have also shown to exist.

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Mathematics Subject Classification: Primary: 37D05, 37D25; Secondary: 70H07.

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