Double Saddles and Switching Dynamics

In this chapter, double saddles and hyperbolic flow singularity in cubic systems with two crossing-linear and self-quadratic products vector fields are discussed. The double saddle equilibriums are the appearing bifurcations of the saddle source and saddle sinks, and such double saddle equilibriums are also the appearing bifurcations of the network of saddles, sink, and source. The hyperbolic flow singularity generates the appearing bifurcations of hyperbolic and hyperbolic secant flows. The switching bifurcations are presented through the infinite equilibriums. The up-down and down-up infinite equilibriums are the switching bifurcation. The hyperbolic lower-to-upper saddle is for the switching of positive (negative) saddle with paralleled hyperbolic secant-to-hyperbolic flow. The hyperbolic secant lower-to-upper saddle is for the switching of a center with paralleled hyperbolic-to-hyperbolic secant flow. The up-down (down-up) saddles are for the switching bifurcation of double saddles with a connected hyperbolic-to-hyperbolic flow.