Inflection Singularity and Bifurcation Dynamics

In this chapter, inflection singularity and bifurcation dynamics of the quadratic and cubic product systems are discussed, and the corresponding phase portraits will be illustrated. The self-linear and crossing-quadratic product vector fields are considered. The inflection-source (sink) flows and parabola-saddles for the appearing bifurcations are discussed first. The networks of the inflection-source (sink) flow, parabola-saddle, hyperbolic (hyperbolic-secant) flows with one of sink, source, and saddle are presented through the first integral manifolds. The parabola-source (sink) infinite-equilibriums are for the switching bifurcations including the hyperbolic and hyperbolic-secant source (sink) and the hyperbolic and circular sink-to-source. The inflection-source (sink) infinite-equilibriums are also for the switching bifurcation, which includes the decreasing-to-increasing inflection-source (sink) and the increasing (decreasing)-inflection source (sink). Finally, the network of connected hyperbolic (hyperbolic-secant) flows, saddles, center, source, and sink are developed. Inflection-source (sink) infinite-equilibriums are for the switching bifurcations, including the parabola-saddles on the infinite-equilibrium, the decreasing-to-increasing inflection-source (sink). The inflection diagonal-saddles exist for the double switching bifurcations.