Cubic Lienard equation with quadratic damping. I
type: Item
instance of: scholarly article
title: Cubic Lienard equation with quadratic damping. I
zbMATH Open document ID: 0956.34020
DOI: 10.1007/BF02670963
author: Yuquan Wang
author: Zhujun Jing
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 2001-02-26T00:00:00Z
review text: The authors study the general cubic Lienard system \[ \dot x=y-a_1x-a_2x^2-a_3x^3,\quad \dot y=-b_1x-b_2x^2-b_3x^3.\tag{1} \] It is assumed that \(a_2^2-3a_1a_3>0.\) By using the Zhang Zhifen theorem and the Poincaré-Bendixson theorem, conditions of existence and uniqueness of one limit cycle when \(a_1a_3<0\) for system (1) with one equilibrium are given. Applying Hopf bifurcation theory and qualitative analysis, sufficient conditions are obtained that system (1) has at least two limit cycles when \(a_1a_3>0\). Using a simulation program, three examples with one limit cycle and two limit cycles are presented to demonstrate the theoretical analysis.
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 1462493
zbMATH Keywords: cubic Lienard equation
zbMATH Keywords: limit cycles
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: stability
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: qualitative analysis
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02670963
OpenAlex ID: W74946010
description: scientific article; zbMATH DE number 1462493