Connecting orbits for some classes of almost periodic Lagrangian systems
type: Item
instance of: scholarly article
title: Connecting orbits for some classes of almost periodic Lagrangian systems
zbMATH Open document ID: 0913.34040
author: Maria Letizia Bertotti
author: Piero Montecchiari
published in: Journal of Differential Equations
publication date: 1999-05-24T00:00:00Z
review text: For a class of almost-periodic Lagrangian systems the authors prove, by means of variational arguments, the existence of infinitely many heteroclinic solutions connecting possibly degenerate equilibria. The systems dealt with are of the form \[ \ddot q+\alpha (t)\bigtriangledown V(q)=0, \eqno (1) \] where \((\text{ H}_1)\) \(\alpha\) is a continuous almost-periodic function with \(\alpha (t)\geq \underline {\alpha}>0\) for all \(t\in {\mathbb{R}}\), \((\text{ H}_2)\) \(V\in C^1({\mathbb{R}}^N)\) and \(V(x+\zeta)=V(x)\) for all \(x\in {\mathbb{R}}^N\), \(\zeta\in {\mathbb{Z}}^N\), \((\text{ H}_3)\) \(V(x)< V(\zeta)=0\) for any \(\zeta\in {\mathbb{Z}}^N\), \(x\in {\mathbb{R}}^N\setminus {\mathbb{Z}}^N\). Assumptions \((\text{ H}_1) - (\text{ H}_3)\) yield that any \(\zeta\in {\mathbb{Z}}^N\) is an equilibrium and heteroclinic solutions between these equilibria are the subject of studies. The main result is the following. Theorem. If \((\text{ H}_1) - (\text{ H}_3)\) hold, then system (1) admits infinitely many heteroclinic solutions. More precisely, there exist \(K\in {\mathbb{N}}\), \(K\geq 2N\), and \(K\) points \(\zeta _1, \dots ,\zeta _K \in {\mathbb{Z}}^N\setminus \{0\}\) such that \(\{\sum_{i=1}^{K}n_i\zeta _i : n_i\in {\mathbb{N}}\cup \{0\}\}={\mathbb{Z}}^N\) and such that for any fixed \(j\in\{ 1,\dots ,K\}\) there exist infinitely many heteroclinic solutions to (1) connecting \(0\) with \(\zeta _j\). An analogous multiplicity result is established for homoclinic solutions to systems with an almost-periodic singular potential.
reviewed by: Eugene Ershov
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 1206628
zbMATH Keywords: Lagrangian systems
zbMATH Keywords: heteroclinic orbit
zbMATH Keywords: homoclinic orbit
zbMATH Keywords: almost periodicity
zbMATH Keywords: variational arguments
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3415
OpenAlex ID: W2042853823
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DOI: 10.1006/JDEQ.1998.3415
description: scientific article; zbMATH DE number 1206628