Self-Linear and Crossing-Quadratic Product Systems

In this chapter, nonlinear dynamics and singularity of a cubic dynamical system possessing two self-linear and crossing-quadratic product cubic vector fields are discussed, and the corresponding appearing and switching bifurcations are presented through a theorem. The double-inflection saddles and two inflection-source and sink flows are for the appearing bifurcations in such product cubic systems. The connected parabola-saddle bifurcations exist in such product cubic systems, and the connected hyperbolic and hyperbolic-secant flows are obtained through the inflection-source (sink) flows. The switching bifurcations in such product cubic systems include parabola-source (sink), increasing (decreasing)-inflection diagonal-source (sink), inflection-source (sink), hyperbolic and hyperbolic-secant-source (sink), circular source-sink (sink-source), hyperbolic source-sink (sink-source), and inflection diagonal-saddles. The equilibrium networks with two connected hyperbolic and hyperbolic-secant flows are presented, and the corresponding inflection source and sink infinite-equilibriums exist.