Multiple homoclinic solutions for singular differential equations
type: Item
instance of: scholarly article
title: Multiple homoclinic solutions for singular differential equations
zbMATH Open document ID: 1217.34075
author: Guangping Luo
author: Changrong Zhu
author: K. Q. Lan
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2010-06-09T00:00:00Z
review text: Motivated by \textit{J. Gruendler} [Trans. Am. Math. Soc. 350, No.~9, 3797--3814 (1998; Zbl 0902.34038)], singularly perturbed systems \[ \dot x=f(x)+g(x,\epsilon,\epsilon t) \] are considered. Amongst others, the authors assume that the unperturbed system \[ \dot x=f(x) \] has a degenerate homoclinic orbit \(\gamma\) to a hyperbolic equilibrium, i.e., the tangent spaces of the corresponding stable and unstable manifolds have (along \(\gamma\)) a \(d\)-dimensional intersection, \(d>1\). The aim of the paper is to study nearby homoclinic orbits. In the bifurcation analysis, both \(\epsilon\) and the function \(g\) are considered as parameters. The main result states for \(\epsilon\not=0\) the existence of \(d\) bifurcation manifolds in the \(g\)-space. The authors employ a functional analytic approach together with the Liapunov-Schmidt method to derive the bifurcation equations. The spirit of the result and of the procedure resembles pretty much the ones of \textit{C. Zhu} and \textit{W. Zhang} [J. Differ. Equations 240, No.~1, 38--57 (2007; Zbl 1138.34024)].
reviewed by: Jürgen Knobloch
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 5718434
zbMATH Keywords: degenerate homoclinic solution
zbMATH Keywords: Lyapunov-Schmidt reduction
zbMATH Keywords: singular perturbations
MaRDI profile type: Publication
full work available at URL: j.anihpc.2010.01.005
OpenAlex ID: W1996946574
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DOI: 10.1016/J.ANIHPC.2010.01.005
description: scientific article; zbMATH DE number 5718434