Self and Product Cubic Systems

In this chapter, the nonlinear dynamics and singularity of self and product cubic dynamical systems with a self-linear and crossing-quadratic product vector field are presented, and the corresponding appearing and switching bifurcations are discussed through a theorem. The saddle-source (sink) equilibriums are the appearing bifurcations for saddle and source (sink) equilibriums, and inflection-source (sink) flows are the appearing bifurcations of the connected hyperbolic and hyperbolic-secant flows. The switching bifurcations are obtained by inflection-source and sink infinite-equilibriums, parabola-source and source infinite-equilibriums, and up-down and down-up saddle infinite-equilibriums. The two paralleled double-switching bifurcations are obtained. Equilibrium series with connected hyperbolic and hyperbolic-secant flows are obtained for such cubic systems.