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 Local Bifurcation of Critical Periods in Quadratic-Like Cubic Systems Zhiheng Yu,Zhaoxia Wang Keywords:quadratic-like cubic systems, critical period bifurcation, pseudo division, variety decomposition Abstract: In this paper, we investigate quadratic-like cubic systems having a center at $O$ for the local bifurcation of critical periods. We give an inductive algorithm to compute polynomials of periodic coefficients, find structures of solutions for systems of algebraic equation corresponding to weak centers of finite order and derive conditions on parameters under which the considered equilibrium is a weak center of order $k$, $k=1,2,3,4$. Furthermore, we show that with appropriate perturbations at most $4$ critical periods bifurcate from the weak center of finite order and give conditions under which exactly $k$ critical periods bifurcate from the center $O$ for each integer $k=1,2,3,4$.