Transverse bounded solutions to saddle-centers in periodically perturbed ordinary differential equations
type: Item
instance of: scholarly article
title: Transverse bounded solutions to saddle-centers in periodically perturbed ordinary differential equations
zbMATH Open document ID: 1120.34031
author: Joseph Gruendler
author: Michal Fečkan
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-05-15T00:00:00Z
review text: A periodically perturbed system of ordinary differential equation is considered consisting of two equations with nonlinear coupling where the linear parts of the two equations have equilibria which are, respectively, a saddle and a center. Perturbation terms included in the system correspond to damping and forcing. It is assumed that (1) the reduced equation which is obtained from the hyperbolic equation by setting to zero the variable from the center equation has a homoclinic solution; (2) the center equation is resonant in the equilibrium. Under certain additional assumptions, the Melnikov theory is used to prove the existence of a transverse bounded solution of the whole system. The techniques make use of exponential dichotomies and an averaging procedure.
reviewed by: Eugene Ershov
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 5152791
zbMATH Keywords: homoclinic solutions
zbMATH Keywords: bifurcations
zbMATH Keywords: center manifold
zbMATH Keywords: averaging method
zbMATH Keywords: exponential dichotomies
MaRDI profile type: Publication
full work available at URL: j.na.2006.05.008
OpenAlex ID: W1997514569
cites work: Exponential dichotomies, heteroclinic orbits, and Melnikov functions
cites work: Q4146134
cites work: On the Nature of the Spectrum of Singular Second Order Linear Differential Equations
cites work: Dichotomies in stability theory
cites work: Homoclinic saddle-node bifurcations in singularly perturbed systems
cites work: Periodic motions
cites work: Q4490310
cites work: The existence of chaos for ordinary differential equations with a center manifold
cites work: Homoclinic Solutions for Autonomous Dynamical Systems in Arbitrary Dimension
cites work: Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations
cites work: The existence of transverse homoclinic solutions for higher order equations
cites work: Q5580482
cites work: Q3997135
cites work: Exponential dichotomies and transversal homoclinic points
cites work: Averaging methods in nonlinear dynamical systems
cites work: Differentiable dynamical systems
cites work: Global bifurcations and chaos. Analytical methods
DOI: 10.1016/J.NA.2006.05.008
description: scientific article; zbMATH DE number 5152791