Self-Linear and Crossing-Cubic Vector Fields

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