Narrow Results

We give a new and short proof of the characterization of monodromic nilpotent critical points. We also calculate the first generalized Lyapunov constants in order to solve the stability problem. We apply the results to several families of planar systems obtaining necessary and sufficient conditions for having a center. Our method also allows us to generate limit cycles from the origin.

Bifurcations of limit cycles created from a multiple critical point of planar dynamical systems are studied. It is different from the usual Hopf bifurcations of limit cycles created from an elementary critical point. This bifurcation phenomena depends on the stability of the multiple critical point and the multiple number of the critical point. As an example, a cubic system which can created four small amplitude limit cycles from the origin (a multiple critical point) is given.

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.

For third-order nilpotent critical points of a planar dynamical system, the analytic center problem is completely solved in this article by using the integrating factor method. The associated quasi-Lyapunov constants are defined and their computation method is given. For a class of cubic-order systems under small perturbations, sufficient and necessary conditions for an analytic center are obtained.

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 ⊃ (1 ∪ 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 Z₂ equivalent cubic system with 3-multiple nilpotent focus or center O(0, 0) are investigated.