Self-Quadratic and Crossing-Cubic Vector Fields

In this chapter, two vector fields in single-variable dynamical systems to be quadratic independent-variable and non-self-univariate cubic vector fields are considered. The singularity, stability, and bifurcations of such dynamical systems are discussed. In addition to the first- and third-order up-parabola and down-parabola flows with increasing-inflection and decreasing-inflection flows, the source and sink with upper-saddle and lower-saddle based on the quadratic independent-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 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 the up-parabola flow to the source, down-parabola, and sink flows. The up-down saddles are the appearing bifurcation from the down-parabola flow to the source, up-parabola, and sink flows. The up-down and down-up saddle infinite-equilibriums are the switching bifurcations for the upper-saddle and lower-saddle flows with up-parabola and down-parabola 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)-parabola upper-saddle and lower-saddle infinite-equilibriums are the appearing bifurcation for the source and sink flows with the up-parabola and down-parabola flows and also are the switching bifurcation for the increasing-inflection and down-inflection flows with the up-parabola and down-parabola flows.