Double-Inflection Saddles with Switching Dynamics

In this chapter, double-inflection diagonal-saddles are discussed in a product-cubic system possessing two self-linear and crossing-quadratic product vector fields. The inflection-source and sink flows exist for the appearing bifurcation of connected hyperbolic and hyperbolic-secant flows. The double-inflection diagonal-saddles are for the appearing bifurcations of two-connected parabola-saddles and also for the appearing bifurcations of the positive and negative saddles with the counter-clockwise and clockwise centers. The networks of double-inflection saddles and inflection-source (sink) with saddle, source, and sink are presented, and the corresponding switching bifurcations for such networks are presented through the infinite-equilibriums. The parabola-sink and sources in the vertical and horizontal directions are obtained for the switching bifurcation. The double-switching bifurcations are hyperbola source and sink with circle source-sink.