Saddle-node bifurcation of homoclinic orbits in singular systems
type: Item
instance of: scholarly article
title: Saddle-node bifurcation of homoclinic orbits in singular systems
zbMATH Open document ID: 1023.34034
author: F. Battelli
published in: Discrete and Continuous Dynamical Systems
publication date: 2003-03-16T00:00:00Z
review text: The author studies the following singularly perturbed system \[ \dot \xi=f_0(\xi)+\varepsilon f_1(\xi ,\eta ,\varepsilon),\quad \dot \eta =\varepsilon g(\xi ,\eta ,\varepsilon) ,\tag{1} \] where \(\xi \in \Omega \subset \mathbb{R}^n\), \(\eta \in \mathbb{R}\) and \(\varepsilon \in \mathbb{R}\) is a small parameter. It is assumed that the boundary layer problem associated to (1) of the form \(\dot \xi =f_0(\xi)\) has a nondegenerate heteroclinic solution \(\gamma (t)\) and the associated Melnikov function has a double zero. By using a functional analytic approach, it is shown that if a suitable second-order Melnikov function is not zero, then (1) has two heteroclinic orbits near \(\{\gamma (t)\}\times \mathbb{R}\) for \(\varepsilon \) on one side of \(\varepsilon =0\) and none on the other side. The transversality of those heteroclinic solutions is also investigated. The paper is finished with an example of a 3-dimensional differential system.
reviewed by: Michal Fečkan
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 37C29
zbMATH DE Number: 1882971
zbMATH Keywords: homoclinic orbits
zbMATH Keywords: saddle-node bifurcation
zbMATH Keywords: Melnikov functions
MaRDI profile type: Publication
full work available at URL: dcds.2001.7.203
OpenAlex ID: W2028078983
DOI: 10.3934/DCDS.2001.7.203
description: scientific article; zbMATH DE number 1882971