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International Journal of Mathematics and Mathematical Sciences
Volume 2004, Issue 61, Pages 3259-3274

On the critical periods of Liénard systems with cubic restoring forces

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

Received 23 February 2004

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study local bifurcations of critical periods in the neighborhood of a nondegenerate center of a Liénard system of the form x˙=y+F(x), y˙=g(x), where F(x) and g(x) are polynomials such that deg(g(x))3, g(0)=0, and g(0)=1, F(0)=F(0)=0 and the system always has a center at (0,0). The set of coefficients of F(x) and g(x) is split into two strata denoted by SI and SII and (0,0) is called weak center of type I and type II, respectively. By using a similar method implemented in previous works which is based on the analysis of the coefficients of the Taylor series of the period function, we show that for a weak center of type I, at most [(1/2)deg(F(x))]1 local critical periods can bifurcate and the maximum number can be reached. For a weak center of type II, the maximum number of local critical periods that can bifurcate is at least [(1/4)deg(F(x))].