A variational approach to homoclinic orbits in Hamiltonian systems
type: Item
instance of: scholarly article
title: A variational approach to homoclinic orbits in Hamiltonian systems
zbMATH Open document ID: 0731.34050
DOI: 10.1007/BF01444526
author: Vittorio Coti Zelati
author: Ivar Ekeland
author: Éric Séré
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164728
review text: The Hamiltonian system of ordinary differential equations \[ \dot x=J\nabla_ xH(t,x), \tag{1} \] where \(J\) denotes the \(2n\times 2n\) matrix \(J=\begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}\) with \(J^*=J^{-1}=-J\), \(\nabla_ x H(t,x)\) denotes the vector \(\partial H(t,x)/\partial x\), \(1\leq i\leq 2n\), and the map \(H: \mathbb R\times \mathbb R^{2n}\to \mathbb R\) belongs to the class \(\alpha C^ 2\), are considered in this paper. The main purpose is to study the so-called doubly asymptotic solutions. We recall that if \(x\) is a periodic solution of the system (1), another solution \(z\) will be called doubly asymptotic to \(x\) if \(| z(t)-x(t)| \to 0\) when \(t\to\pm\infty\). The author simplifies the problem by applying Floquet's theory to the linearized system around \(x\dot y=JH''(t,x(t))y\) using a variational approach for the solution.
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 4208566
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: doubly asymptotic solutions
zbMATH Keywords: periodic solution
zbMATH Keywords: Floquet's theory
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4208566