Crossing-Linear and Crossing-Cubic Vector Fields

In this chapter, a cubic dynamical system possessing crossing-linear and crossing-cubic vector fields is considered, and the corresponding dynamical behaviors are discussed. The parabola-saddles are for the appearing and switching bifurcations of saddle and center. The third-order centers are the appearing bifurcations from a center to center, saddle, and center. The third-order saddles are the appearing bifurcation from a saddle to saddle, center, and saddle. The third-order saddles and centers are the switching bifurcations of saddle and center with parabola-saddles. The series of saddle and center are presented through the first integral manifold.