Weak centers and local criticality on planar Z₂-symmetric cubic differential systems

There exist the necessary and sufficient conditions for planar Z(2)-symmetric cubic differential systems with a bi-center and an isochronous bi-center in the existing literature. Based on these bi-center conditions, we give a complete discussion on the order of weak centers and the local criticality for planar Z(2)-symmetric cubic systems at the origin and the symmetric equilibria (+/- 1, 0) separately. More precisely, we first give the parametric conditions for the origin and the equilibria (+/- 1, 0) being weak centers of exact order separately. Then we prove that the local criticality from the weak centers of finite order at the origin is at most 4, and at each of symmetric equilibria (+/- 1, 0) is at most 3. Finally, we also obtain that the local criticality from the isochronous center at the origin is at most 3, and at each of equilibria (+/- 1, 0) is at least 2. More importantly, these numbers are reachable.