New Double Bifurcation of Nilpotent Focus

In this paper, a new bifurcation phenomenon of nilpotent singular point is analyzed. A nilpotent focus or center of the planar systems with 3-multiplicity can be broken into two complex singular points and a second order elementary weak focus. Then, two more limit cycles enclosing the second order elementary weak focus can bifurcate through the multiple Hopf bifurcation.

Bifurcation of Limit Cycles for Some Liénard Systems with a Nilpotent Singular Point

In this paper, we first present some general theorems on bifurcation of limit cycles in near-Hamiltonian systems with a nilpotent saddle or a nilpotent cusp. Then we apply the theorems to study the number of limit cycles for some polynomial Liénard systems with a nilpotent saddle or a nilpotent cusp, and obtain some new estimations on the number of limit cycles of these systems.