International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 61-64, Pages 3259-3274
doi:10.1155/S0161171204402245

On the critical periods of Liénard systems with cubic restoring forces

Abstract

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))].