Small-amplitude limit cycles of some Liénard-type systems
type: Item
instance of: scholarly article
title: Small-amplitude limit cycles of some Liénard-type systems
zbMATH Open document ID: 1279.34046
author: Jiao Jiang
author: Maoan Han
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-12-18T00:00:00Z
review text: Consider the Liénard systems \[ \dot x= y-{\sum^n_{i=0} a_i x^i\over 1+\sum^m_{i=1} b_i x^i},\quad \dot y= -g(x),\tag{1} \] \[ \dot x= y- {\sum^n_{j=0} a_j x^{2j}\over \sum^m_{j=0} b_j x^{2j}},\quad \dot y= -x(x^2- 1).\tag{2} \] The authors prove the following results: (i) Let \(g\) be a smooth function satisfying \(g(0)= 0\), \(g'(0)> 0\). Then the cyclicity of the equilibrium point \((0,a_0)\) of (1) is \([{n+m-1\over 2}]\). (ii) Let \(\sum^m_{j=0} b_j\neq 0\). Then system (2) has at most \(2[{n+m-1\over 2}]\) small amplitude limit cycles near the equilibrium \((0,a_0)\), where this number can be reached.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 5649974
zbMATH Keywords: Liénard equation
zbMATH Keywords: limit cycle
zbMATH Keywords: Hopf bifurcation
MaRDI profile type: Publication
full work available at URL: j.na.2009.09.011
OpenAlex ID: W2079444102
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DOI: 10.1016/J.NA.2009.09.011
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description: scientific article; zbMATH DE number 5649974