Crossing-Linear and Self-Cubic Vector Fields

In this chapter, consider two vector fields in dynamical systems to be linear variable-crossing and self-univariate cubic vector fields with single variable. The corresponding stability and bifurcations of equilibriums are presented. In addition to the first- and third-order sink and source flows with upper-saddle and lower-saddle flows, the parabola flows based on the linear crossing-variable vector fields are determined. The inflection-sink (source) infinite-equilibriums are for the switching bifurcations of parabola flows and sink (source) flows. The down-up saddles are the appearing bifurcation from an up-parabola flow to source, down-parabola, and sink flows. The up-down saddles are the appearing bifurcation from a down-parabola flow to source, up-parabola, and sink flows. The (3:1)-increasing and decreasing asymptotic source infinite-equilibriums are the switching bifurcations of the up-parabola and down-parabola with the third-order source flow. The (3:1)-increasing and decreasing asymptotic sink infinite-equilibriums are the switching bifurcations of the up-parabola and down-parabola with the third-order sink flow.