Saddle-Node and Hyperbolic-Flow Singular Dynamics

In this chapter, saddle-singularity and corresponding bifurcation dynamics of the quadratic and cubic product systems are discussed. The self-quadratic and crossing-linear product vector fields are considered. The networks of saddle-source (sink), hyperbolic flow with center, hyperbolic-secant flow with saddle, and hyperbolic-to-hyperbolic-secant flows are presented. Second, the inflection-source (sink) infinite-equilibriums for single switching bifurcations are discussed, which include the parabola-saddle and inflection-source (sink). Up-down and down-up upper-saddle (lower-saddle) and the hyperbolic and hyperbolic-secant saddles are discussed for the single switching bifurcation. The down-up (up-down), sink-to-source, and source-to-sink are the double switching bifurcations. Finally, the networks of separated hyperbolic (hyperbolic-secant) flows with saddle (sink or source) with hyperbolic (hyperbolic-secant) flow with center (saddle) will be developed. Inflection-source (sink) infinite-equilibriums for the switching bifurcations are discussed, including the parabola-saddles, the decreasing-to-increasing inflection-source (sink). The diagonal-inflection-saddles for the double switching bifurcations are discussed.