Critical Periods of Planar Revertible Vector Field with Third-Degree Polynomial Functions

In this paper, we consider local critical periods of planar vector field. Particular attention is given to revertible systems with polynomial functions up to third degree. It is assumed that the origin of the system is a center. Symbolic and numerical computations are employed to show that the general cubic revertible systems can have six local critical periods, which is the maximal number of local critical periods that cubic revertible systems may have. This new result corrects that in the literature: general cubic revertible systems can at most have four local critical periods.