Double Bifurcation of Nilpotent Focus

In this paper, an interesting bifurcation phenomenon is investigated -a 3-multiple nilpotent focus of the planar dynamical systems could be broken into two element focuses and an element saddle, and the limit cycles could bifurcate out from two element focuses. As an example, a class of cubic systems with 3-multiple nilpotent focus O(0, 0) is investigated, we prove that nine limit cycles with the scheme 7 superset of (1 boolean OR 1) could bifurcate out from the origin when the origin is a weak focus of order 8. At the end of this paper, the double bifurcations of a class of Z(2) equivalent cubic system with 3-multiple nilpotent focus or center O(0, 0) are investigated.