Quadratic and Cubic Product Systems

In this chapter, a theory for nonlinear dynamics and singularity of cubic dynamical systems possessing product-cubic and product-quadratic vector fields are developed. The self-linear and crossing-quadratic product vector fields in such cubic dynamical systems are discussed first. For such systems, the parabola-saddle equilibriums and inflection-source (sink) flows exist for the appearing bifurcations. The inflection-source (sink) infinite-equilibriums are for switching bifurcations of the connected hyperbolic flows with saddle, sink, and source equilibrium. Secondly, the crossing-linear and self-quadratic product vector fields are discussed. The saddle-source (sink) and the hyperbolic-to-hyperbolic-secant flows exist for appearing bifurcations. The switching bifurcations for such a system are the up-down and down-up upper-saddle (lower-saddle) and the hyperbolic and hyperbolic-secant saddles. The parabola-saddles on the inflection-source (sink) infinite-equilibriums are the switching bifurcations for the hyperbolic (hyperbolic-secant) flows with paralleled center (saddle). The inflection-source (sink) infinite-equilibriums are for the switching bifurcations, and the inflection diagonal-saddles are for the double switching bifurcations.