Crossing-Quadratic and Crossing-Cubic Vector Fields

In this chapter, a cubic dynamical system possessing crossing-quadratic and crossing-cubic vector fields is considered, and the corresponding dynamical behaviors are discussed. As in constant and crossing-cubic system, the third-order parabola flows exist for the appearing bifurcation from a parabola flow to a series of three parabola flows, and the inflection flows are the appearing and switching bifurcations of up-parabola and down parabola flows, respectively. In addition to parabola saddles and third-order saddle sand centers, the third-order parabola-saddle and double inflection-saddles exist. The (2,3)-third-order parabola-saddle are the appearing bifurcations of a 2 × 3 network of saddle and center. The (2,2)-double inflection-saddles are the appearing bifurcations of a 2 × 2 network of saddle and center. The (2,3)-third-order parabola-saddles are also for the switching bifurcations of a double inflection-saddle with saddle (center). The networks of saddles and centers with homoclinic orbits are presented through the first integral manifolds.