Two Single-Variable Cubic Vector Fields

In this chapter, two vector fields in dynamical systems to be two cubic univariate vector fields with a single variable are considered. The singularity, stability, and bifurcations of such dynamical systems will be discussed. In addition to the first- and third-order sink and source flows with upper-saddle and lower-saddle flows, the first- and third-order parabola and inflection flows based on the cubic crossing-variable vector fields are determined. In addition to the (1:3)-up-parabola sink (source) infinite-equilibriums, the down-up and up-down saddles, the (2:2)-inflection upper-saddle infinite-equilibriums, the (3:1)-increasing and decreasing-concave source and sink infinite-equilibriums, the (3:2)-up and down-logarithmic source and sink infinite-equilibriums, the (3:3)-inflection source and sink infinite-equilibrium are for the appearing and switching bifurcations of the source, sink, source flows and the sink, source, and sink flows with up-parabola, down-parabola, and up-parabola flows and the down-parabola, up-parabola, and down-parabola flows. The (3:3)-inflection source and sink infinite-equilibrium are also for the switching bifurcations of the source and sink flows and the upper-saddle and lower-saddle flows with the up-parabola and down-parabola flows and the increasing- and decreasing-inflection flows.