Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line
type: Item
instance of: scholarly article
title: Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line
zbMATH Open document ID: 1412.34111
author: Shimin Li
author: Jaume Llibre
published in: Journal of Differential Equations
publication date: 2019-03-21T00:00:00Z
full work available at URL: http://ddd.uab.cat/record/204383
review text: The purpose of this paper is to characterize the global dynamics of the following planar systems related to the Liénard equation \[ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = y- f(x),\\ \frac{\mathrm{d}y}{\mathrm{d}t} =a -x, \end{cases} \tag{1} \] where $a$ is a given constant and $f$ is the continuous piecewise linear function \[ f(x)=\begin{cases} k_1x& \text{ if }x\geq0,\\ k_2x& \text{ otherwise}\end{cases} \] with $k_1$ and $k_2$ nonzero real constants. \par In order to obtain the desired characterization, the possible phase portraits of system (1) in the Poincaré disc are completely classified. Also, necessary and sufficient conditions for existence and uniqueness of limit cycles are given.
reviewed by: Marco Spadini
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 7039972
zbMATH Keywords: phase portraits
zbMATH Keywords: piecewise linear differential system
zbMATH Keywords: limit cycle
MaRDI profile type: Publication
OpenAlex ID: W2908135648
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DOI: 10.1016/J.JDE.2018.12.024
description: scientific article; zbMATH DE number 7039972