Bifurcation of local critical periods in the generalized Loud's system

We study the bifurcation of local critical periods in the differential system { ẋ = −y +Bxn−1y, ẏ = x+Dx + Fxn−2y2, where B,D,F ∈ R and n > 3 is a fixed natural number. Here by “local” we mean in a neighbourhood of the center at the origin. For n even we show that at most two local critical periods bifurcate from a weak center of finite order or from the linear isochrone, and at most one local critical period from a nonlinear isochrone. For n odd we prove that at most one local critical period bifurcates from the weak centers of finite or infinite order. In addition, we show that the upper bound is sharp in all the cases. For n = 2 this was proved by Chicone and Jacobs in [Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433–486] and our proof strongly relies on their general results about the issue.