Constant and Crossing-Cubic Vector Fields

In this chapter, two-dimensional, single-variable nonlinear systems with a crossing-cubic vector fields are discussed, and the appearing and switching bifurcations for the one-dimensional flows (e.g., first- and third-order parabola flows, inflection flows) are discussed herein. The third-order parabola flow bifurcations are the switching bifurcations for increasing- and decreasing-inflection flows with a parabola flow. The third-order up-parabola flow bifurcation is an appearing bifurcation from the up-parabola flow to up-parabola, down-parabola, and up-parabola flows. The third-order down-parabola flow bifurcation is an appearing bifurcation for a down-parabola flow to down-parabola, up-parabola, and down-parabola flows.