The dynamics of some cubic vector fields with a center

摘要

We study planar cubic vector fields[V_H=yP_2frac{partial}{partial x}-x Q_2frac{partial}{partial y}] with a center, having Darboux first integral $H=P_1^2Q_3$. We give the bifurcation diagram of the phase portraits of the vector fields, in 2-dimensional parameter space.