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 formx˙=−y+F(x),y˙=g(x), whereF(x)andg(x)are polynomials such thatdeg(g(x))≤3,g(0)=0, andg′(0)=1,F(0)=F′(0)=0and the system always has a center at(0,0). The set of coefficients ofF(x)andg(x)is split into two strata denoted bySIandSIIand(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))]−1local 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))].