Crossing-Quadratic and Self-Cubic Vector Fields

In this chapter, consider two vector fields in single-variable dynamical systems to be quadratic crossing-variable and self-univariate cubic vector fields. The singularity, stability, and bifurcations of such dynamical systems are discussed. In addition to the first- and third-order sink and source flows with upper-saddle and lower-saddle flows, the parabola and inflection flows based on the quadratic crossing-variable vector fields are determined. The up-parabola sink (source) infinite-equilibriums are for the switching bifurcations of increasing-inflection flows and sink (source) flows. The down-parabola sink (source) infinite-equilibriums are for the switching bifurcations of decreasing-inflection 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 (2:2)-inflection upper-saddle infinite-equilibriums are the switching bifurcations for sink and source flows with up-parabola and down-parabola flows. The (3:2)-up and down-logarithmic source infinite-equilibriums are the switching bifurcations of the increasing- and decreasing-inflection flows with a third-order source flow. The (3:2)-up and down-logarithmic sink infinite-equilibriums are the switching bifurcations of the increasing- and decreasing-inflection flows with a third-order sink flow.