Constant and Crossing-Cubic Vector Fields

In this chapter, nonlinear dynamics of dynamical systems with two variable-crossing univariate cubic vector fields is discussed. The dynamical systems with a constant vector field and a variable-crossing univariate cubic vector field is presented first, and the one-dimensional flows are presented. The third-order up-parabola flow is an appearing bifurcation from an up-parabola flow to up-parabola, down-parabola, and up-parabola flows. The third-order down-parabola flow is an appearing bifurcation from a down-parabola flow to down-parabola, down-parabola, and down-parabola flows. The third-order parabola flows are the switching bifurcations of parabola and inflection flows.