On the maximum number of limit cycles of certain polynomial Lienard equations
Yao, Leummim
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https://hdl.handle.net/2142/22869
Description
Title
On the maximum number of limit cycles of certain polynomial Lienard equations
Author(s)
Yao, Leummim
Issue Date
1995
Doctoral Committee Chair(s)
Albrecht, Felix
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
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