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Title:On the maximum number of limit cycles of certain polynomial Lienard equations
Author(s):Yao, Leummim
Doctoral Committee Chair(s):Albrecht, Felix
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
Abstract:We consider the Lienard equation of the form$$\ddot x + \epsilon f(x)\dot x + x\sp{2m+1} = 0,\eqno(1)\cr$$which is equivalent to the planar system$$\eqalignno{\dot x &= y&\enspace\cr \dot{y} &= -x\sp{2m+1} - \epsilon f(x)y,&(2)\cr}$$where m is a nonnegative integer, f is a real polynomial and $\epsilon$ is a small parameter.
We establish a sharp upper bound for the number of limit cycles (nontrivial isolated periodic orbits of system (2) depending on m and the degree of f provided $\epsilon$ is sufficiently small. This is done by investigating the fixed points of the Poincare return map associated with system (2).
This result is an important step in the study of planar polynomial vector fields in connection with the second part of Hilbert's Sixteenth Problem, which is still an open question.
Issue Date:1995
Type:Text
Language:English
URI:http://hdl.handle.net/2142/22869
Rights Information:Copyright 1995 Yao, Leummim
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9624545
OCLC Identifier:(UMI)AAI9624545


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