Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits
type: Item
instance of: scholarly article
title: Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits
zbMATH Open document ID: 1062.37059
author: Amadeu Delshams
author: Pere Gutiérrez
published in: Discrete and Continuous Dynamical Systems
publication date: 2005-02-28T00:00:00Z
review text: The application of the Poincaré-Melnikov method to the effective detection of transverse homoclinic orbits, associated to an \(n\)-dimensional whiskered torus of a Hamiltonian with \(n+1\) degrees of freedom, needs to be carefully validated in the case of exponentially small splitting of separatrices, since this problem becomes singular with respect to the parameter \(\varepsilon \) of perturbation. Previous results are known only for \(n=1\). The present paper is devoted to an example of an integrable Hamiltonian with three degrees of freedom (i.e., \(n=2\)), possessing a two-dimensional hyperbolic invariant torus with fast frequencies \(\frac{\omega }{\sqrt{\varepsilon }}\) and coincident whiskers, plus a perturbation of order \(\varepsilon ^{p}\). \(\omega \) is chosen as the golden vector and then, the good arithmetic properties of \(\omega \) yield a splitting function with 4 simple zeros (corresponding to the nondegenerate critical points of the splitting potential), giving rise to 4 transverse homoclinic orbits.
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 70H08
zbMATH DE Number: 2139314
zbMATH Keywords: Poincaré-Melnikov method
zbMATH Keywords: transverse homoclinic orbit
zbMATH Keywords: golden ratio
reviewed by: Mircea Crasmareanu
MaRDI profile type: Publication
full work available at URL: dcds.2004.11.757
OpenAlex ID: W2113136084
DOI: 10.3934/DCDS.2004.11.757
Recommended article: Exponentially small splitting of separatrices for whiskered tori in Hamiltonian systems
Recommended article: Existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near-integrable Hamiltonian systems
Recommended article: Exponentially small splitting for whiskered tori in Hamiltonian systems: Flow-box coordinates and upper bounds
Recommended article: Exponentially small splittings in Hamiltonian systems
Recommended article: Erratum to ``Existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near-integrable Hamiltonian systems''
Recommended article: Splitting potential and the Poincaré-Melnikov method for whiskered tori in Hamiltonian systems
Recommended article: Q4407888
Recommended article: A proof of existence of whiskered tori with quasi flat homoclinic intersections in a class of almost integrable hamiltonian systems
Recommended article: Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio
Recommended article: Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing
description: scientific article; zbMATH DE number 2139314