Persistence of equilibria as periodic solutions of forced systems
type: Item
instance of: scholarly article
title: Persistence of equilibria as periodic solutions of forced systems
zbMATH Open document ID: 1237.34077
author: Adriana Buică
author: Rafael Ortega
published in: Journal of Differential Equations
publication date: 2012-01-16T00:00:00Z
review text: The authors prove a nice characterization of persistence in dimension two. Let \(X:U\subset{\mathbb R}^2\to{\mathbb R}^2\) be a Lipschitz continuous vector field defined on an open and bounded neighborhood of the origin, and assume that the autonomous system \[ \dot x=X(x) \tag{1} \] has \(x=0\) as its unique equilibrium. Consider the perturbed system \[ \dot x=X(x)+p(t,x,\varepsilon), \tag{2} \] where \(p:{\mathbb R}\times U\times[0,1]\to{\mathbb R}^2\) is continuous, \(T\)-periodic in its first variable and such that \(p(t,x,0)\equiv0\). The equilibrium \(x=0\) is said to persist as a \(T\)-periodic solution if, given any \(p\) as above, there exists a \(T\)-periodic solution \(\varphi_\varepsilon(t)\) of the perturbed system~(2), for small \(\varepsilon>0\), and \[ \lim_{\varepsilon\to0}\|\varphi_\varepsilon(t)\|=0, \] uniformly in \(t\in{\mathbb R}\). The authors prove that \(x=0\) persists as a \(T\)-periodic solution if and only if system~(1) is not \(T\)-isochronous around the origin and deg\((X,U,0)\neq0\).
reviewed by: Alessandro Fonda
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34D10
zbMATH DE Number: 5998049
zbMATH Keywords: perturbation
zbMATH Keywords: diffeotopy
zbMATH Keywords: isochronous
MaRDI profile type: Publication
full work available at URL: j.jde.2011.06.006
OpenAlex ID: W1973397953
cites work: Continuation Theorems for Periodic Perturbations of Autonomous Systems
cites work: Q5727924
cites work: Q4421936
cites work: An exact sequence in differential topology
cites work: Q3336252
cites work: Q3049152
cites work: Q5809022
cites work: Q3952851
DOI: 10.1016/J.JDE.2011.06.006
Recommended article: Periodic solutions of periodically forced non-degenerate systems
Recommended article: On the stability of equilibrium for almost periodic systems
Recommended article: Stability properties of periodic solutions of periodically forced non- degenerate systems
Recommended article: Persistence in periodic and almost periodic Lotka-Volterra systems
Recommended article: Persistence of equilibria for locally asymptotically stable systems
Recommended article: Q3142900
Recommended article: Q3810971
Recommended article: Periodic solutions of a forced Lotka-Volterra equation
Recommended article: Persistence of Periodic Orbits for Perturbed Dissipative Dynamical Systems
Recommended article: Periodic solutions of periodically forced weakly coupled non-degenerate systems∗
description: scientific article; zbMATH DE number 5998049