A bifurcation problem to homoclinic orbits for non-autonomous systems
type: Item
instance of: scholarly article
title: A bifurcation problem to homoclinic orbits for non-autonomous systems
zbMATH Open document ID: 0565.34050
DOI: 10.1016/0022-247X(85)90056-3
author: J. Blot
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: Given the nonlinear equation \(x'=F(x)\), \(x\in R^ 3\), with \(x=0\) as a hyperbolic fixed point and a degenerated node-saddle type homoclinic orbit \(\Gamma\), the behaviour in a neighbourhood of \(\Gamma\), of the solutions of the equation \(x'=F(x)+G(\lambda,\mu,t,x)\) is studied when G is a small non-autonomous perturbation that is periodic with respect to t and depends on two parameters. The authors prove under certain generic conditions the existence of two bifurcation curves to homoclinic orbits for a system in \(R^ 3\) (not necessarily Hamiltonian). These curves divide the space of parameters into two regions S,N such that there are homoclinic orbits in S whereas there are not any in N. In a later paper the authors prove that these bifurcation curves can be obtained as the limit of bifurcation curves to subharmonics.
Mathematics Subject Classification ID: 34E10
zbMATH DE Number: 3901218
zbMATH Keywords: hyperbolic fixed point
zbMATH Keywords: degenerated node-saddle type homoclinic orbit
zbMATH Keywords: small non-autonomous perturbation
zbMATH Keywords: bifurcation curves
MaRDI profile type: Publication
cites work: A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam
cites work: Q5508470
cites work: An example of bifurcation to homoclinic orbits
description: scientific article; zbMATH DE number 3901218