Homoclinic orbits to invariant tori near a homoclinic orbit to center-center-saddle equilibrium
type: Item
instance of: scholarly article
title: Homoclinic orbits to invariant tori near a homoclinic orbit to center-center-saddle equilibrium
zbMATH Open document ID: 1080.37068
author: Oksana Koltsova
author: Amadeu Delshams
author: Pere Gutiérrez
author: L. M. Lerman
published in: Physica D
publication date: 2005-05-12T00:00:00Z
review text: The paper deals with the behaviour of the perturbed Hamiltonian \(H= H_0+\varepsilon H_1\) on a neighborhood of one loop in the phase space. First, the authors consider the local map for the perturbed Hamiltonian. With the help of the center manifold theory and KAM theory, the authors introduce new symplectic coordinates such that the persistent hyperbolic invariant tori, as well as their stable and unstable manifolds, become flat. Further, the authors consider the perturbed global map and study the existence of homoclinic intersections between the stable and unstable manifolds associated to persistent hyperbolic tori on the perturbed center manifold. Depending on the preserving (or not preserving) of a homoclinic loop, the authors analyse a number of transverse homoclinic loops. Using the Poincaré-Melnikov method, they prove that a first approximation to the splitting distance between the invariant manifolds can be expressed in terms of a Melnikov potential defined on a torus \(\mathbb{T}^2\). Hence, the transverse homoclinic orbits can be detected as nondegenerate critical points of the Melnikov potential.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37J35
zbMATH DE Number: 2167901
zbMATH Keywords: Homoclinic orbits
zbMATH Keywords: KAM tori
zbMATH Keywords: Center manifold
zbMATH Keywords: Morse theory
zbMATH Keywords: Melnikov potential
MaRDI profile type: Publication
full work available at URL: j.physd.2005.01.002
OpenAlex ID: W2018447484
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DOI: 10.1016/J.PHYSD.2005.01.002
description: scientific article; zbMATH DE number 2167901