Exponentially small splitting of separatrices under fast quasiperiodic forcing
type: Item
instance of: scholarly article
title: Exponentially small splitting of separatrices under fast quasiperiodic forcing
zbMATH Open document ID: 0897.34042
author: Amadeu Delshams
author: V. Gelfreich
author: Àngel Jorba
author: Tere M. Seara
published in: Communications in Mathematical Physics
publication date: 1998-10-06T00:00:00Z
full work available at URL: http://hdl.handle.net/2117/949
review text: The authors consider a Hamiltonian system describing a quasiperiodic high-frequency perturbation of the pendulum and having a Hamiltonian of the form \[ \langle\omega,I\rangle/ \varepsilon+ h(x,y,\theta,\varepsilon), \] where \[ h=y^2/2+\cos x+\varepsilon^pm(\theta_1,\theta_2)\cos x, \] and the symplectic form is \(dx\wedge dy+d\theta_1\wedge dI_1+d\theta_2\wedge dI_2\). It is assumed that \(\varepsilon>0\) is a small parameter, \(p\geq 3\), \( \omega=(1,\gamma)\text{ with }\gamma=(\sqrt{5}+1)/2, \) and the function \(m\) is \(2\pi\)-periodic in \(\theta_1\) and \(\theta_2\). The authors investigate the splitting of invariant manifolds associated with a two-dimensional invariant torus. The main result estimates the difference between the splitting (exponentially small in \(\varepsilon\)) and the corresponding Melnikov function.
Mathematics Subject Classification ID: 70J35
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 34C30
Mathematics Subject Classification ID: 34C45
zbMATH DE Number: 1121188
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: quasiperiodic high-frequency perturbation
zbMATH Keywords: pendulum
zbMATH Keywords: invariant manifolds
reviewed by: Sergei Yu. Pilyugin
MaRDI profile type: Publication
OpenAlex ID: W1993379669
DOI: 10.1007/S002200050190
description: scientific article; zbMATH DE number 1121188