Multiple homoclinic orbits for a class of conservative systems
type: Item
instance of: scholarly article
title: Multiple homoclinic orbits for a class of conservative systems
zbMATH Open document ID: 0806.58018
author: Vittorio Coti Zelati
author: A. Ambrosetti
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1995-02-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108285
review text: A class of autonomous, second order classical hamiltonians \(H = {1\over 2} | p |^ 2 - {1\over 2}| q|^ 2 + W(q)\), \((p,q) \in \mathbb{R}^ n\), is considered, where essentially \(a| q|^ \alpha \leq W(q) \leq b| q|^ \alpha\) for some \(\alpha > 2\). The authors show that given a ``pinching condition'' \(b/a < 2^{(\alpha - 2)/2}\) there exist at least two orbits homoclinic to the origin. Moreover, if \(W\) is even, then there are indeed \(n\) such homoclinics. They also have results for intermediate multiplicities. Since \(H\) is independent of time, this multiplicity has nothing to do with the splitting of stable and unstable manifolds (which would lead to infinitely many homoclinics). The homoclinics here could all be degenerate. The proof is variational. It is possible to project out the radial direction so that it suffices to find critical points of the functional \(F(u) = {1\over 2} | u |^ 2_{H^{1,2}} - \int^ \infty_{- \infty} W(u)\) on the sphere \(\{| u| = 1\}\). Lusternik-Schnirelman category and a comparison argument then yield the minimum number of critical points, respectively homoclinic orbits.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 492573
zbMATH Keywords: hamiltonians
zbMATH Keywords: critical points
zbMATH Keywords: homoclinic orbits
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 492573