On the critical periods of Liénard systems with cubic restoring forces

We study local bifurcations of critical periods in the neighborhood of a nondegenerate center of a Liénard system of the form ẋ = −y + F(x), ẏ = 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))].