Double-Saddles and Switching Bifurcations

In this chapter, the appearing and switching bifurcations in a crossing-linear and self-quadratic product system with a self-quadratic vector field are discussed, and the corresponding network of equilibriums are presented. The double-saddles with hyperbolic-to-hyperbolic flows are presented first for the appearing bifurcations. The up-down and down-up saddles are the switching bifurcations for the hyperbolic-to-hyperbolic-secant flow with connected double-saddles. The series of hyperbolic-to-hyperbolic-secant flows with connected saddle-source and saddle-sink are presented. The inflection-source (sink) on the infinite-equilibrium is for the switching bifurcations of hyperbolic-to-hyperbolic-secant flows with connected saddle-source (sink). The network of the paralleled hyperbolic and hyperbolic-secant flows with paralleled saddle-source and saddle-sink. The up-down-to-down-up and down-up-to-up-down upper-saddle (lower-saddle) are presented for the switching bifurcation of a hyperbolic (hyperbolic-secant) flow with saddle-source (sink). The networks of paralleled hyperbolic and hyperbolic-secant flows with connected saddle, sink, and source are presented. The connected inflection-sources (sinks) are presented for the paralleled hyperbolic and hyperbolic-secant flows with saddle and source (sink).