Dynamics of a periodic differential equation with a singular nonlinearity of attractive type
type: Item
instance of: scholarly article
title: Dynamics of a periodic differential equation with a singular nonlinearity of attractive type
zbMATH Open document ID: 0865.34030
author: Pedro J. Torres
author: Pedro Martinez-Amores
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-11-17T00:00:00Z
review text: This paper considers existence and stability of \(T\)-periodic solutions of the forced Liénard equation (1) \(x''+f(x)x'+g(x)=p(t)\), where the functions \(f\), \(g\) and \(p\) are continuous, \(f>0\), \(g\) is decreasing and \(\lim_{x\to0+} g(x)=+\infty\), \(\lim_{x\to+\infty} g(x)=0\). In case \(\overline p={1\over T}\int^T_0 p(s)ds>0\), the authors prove existence of a unique \(T\)-periodic solution of (1) which is unstable. Further, if the variation \(\sup f-\inf f\) is not too large, solutions of (1), defined on \([t_0,+\infty)\), go as \(t\to+\infty\) either to the periodic solution or to infinity. A third type of result concerns the existence of a global stable manifold described by a decreasing function \(h\): if \(x_0=h(v_0)\), solutions with initial conditions \((x_0,v_0)\) go to the periodic one, if \(x_0>h(v_0)\), these solutions go to infinity as \(t\) goes to infinity and, if \(x_0<h(v_0)\), these solutions go to the singularity in a finite time. Similar results are obtained for the global unstable manifold which is described by an increasing function \(x_0=\xi(v_0)\). As a consequence, there are no homoclinic points.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34D35
zbMATH DE Number: 934221
zbMATH Keywords: \(T\)-periodic solutions
zbMATH Keywords: forced Liénard equation
zbMATH Keywords: global stable manifold
zbMATH Keywords: homoclinic points
reviewed by: Patrick Habets
MaRDI profile type: Publication
OpenAlex ID: W1985740010
Wikidata QID: Q115395415
DOI: 10.1006/JMAA.1996.0358
Recommended article: Q4513771
Recommended article: Nonlinear dynamics of differential equations with attractive–repulsive singularities and small time‐dependent coefficients
Recommended article: On periodic solutions of second-order differential equations with attractive-repulsive singularities
Recommended article: Existence and multiplicity of periodic solutions to differential equations with attractive singularities
Recommended article: Periodic solution for second order damped differential equations with attractive-repulsive singularities
Recommended article: Q5096553
Recommended article: Existence and uniqueness of a periodic solution to an indefinite attractive singular equation
Recommended article: Periodic solutions to the Liénard type equations with phase attractive singularities
Recommended article: On periodic solutions of a nonlinear singular differential equation
Recommended article: POSITIVE PERIODIC SOLUTION FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATION WITH SINGULARITY OF ATTRACTIVE TYPE
description: scientific article; zbMATH DE number 934221