Local bifurcation of critical periods in quadratic-like cubic systems

In this paper, we investigate quadratic-like cubic systems having a center at $O$ for the local bifurcation of critical periods. We provide an inductive algorithm to compute polynomials of periodic coefficients, find structures of solutions for systems of algebraic equations corresponding to weak centers of finite order, and derive conditions on parameters under which the considered equilibrium is a weak center of order $k$, $k=0,1,2,3,4$. Furthermore, we show that with appropriate perturbations, at most four critical periods bifurcate from the weak center of finite order, and we give conditions under which exactly $k$ critical periods bifurcate from the center $O$ for each integer $k=1,2,3,4$.