Saddle-Nodes and Hyperbolic Flow Series

In this chapter, equilibrium series with saddle-nodes and connected hyperbolic and hyperbolic-secant flows are discussed, and the corresponding switching dynamics of such equilibrium series are presented. The saddle-source (sink) is the appearing bifurcation of saddle and source (sink). The up-down and down-up saddle infinite-equilibriums are the switching bifurcations of the saddle-source (sink) and a hyperbolic flow. The inflection-saddles are for the switching bifurcations of saddle with inflection-source and sink flows. The up-down and down-up saddle infinite-equilibriums are the switching bifurcations of saddle-source and saddle-sink with hyperbolic and hyperbolic-secant flows. The increasing and decreasing concave sources and sinks are for the switching bifurcations of the third-order saddle, sink and source with the connected hyperbolic and hyperbolic-secant flows. The inflection-source and sink infinite-equilibriums are the switching bifurcations of a hyperbolic-flow with a saddle and a hyperbolic-secant flow with a sink or sink. In addition, the double switching bifurcations are given by the inflection-source and sink infinite-equilibriums and the up-down and down-up saddle infinite-equilibriums.