On Third-Order Nilpotent Critical Points: Integral Factor Method

For third-order nilpotent critical points of a planar dynamical system, the problem of characterizing its center and focus is completely solved in this article by using the integral factor method. The associated quasi-Lyapunov constants are defined and their computation method is given. For a class of cubic systems under small perturbations, it is proved that there exist eight small-amplitude limit cycles created from a nilpotent critical point.