Self and Product Cubic Systems

In this chapter, nonlinear dynamics and singularity of self and product cubic systems with a crossing-linear and self-quadratic product vector field are presented and the corresponding switching dynamics of equilibriums in such self and product cubic systems are discussed through a theorem. The double-saddles, saddle-nodes, and hyperbolic singular flows in such a cubic system are for the appearing bifurcations. The (2:3)-saddle-sources (sinks) are the appearing and switching bifurcations. The (2:3)-saddle-sources (sinks) include (US, 3rdSO)-saddle-source, (LS,3rd SO)-saddle-source, (US, 3rdSI)-saddle-sink, and (LS, 3rdSI)-saddle-sink. The corresponding switching bifurcations for such cubic systems are presented through the infinite-equilibriums. The infinite-equilibriums in such a system include inflection-source and sink, up-down and down-up saddles, and third-order concave-source and sink.