Double-Saddles, Third-Order Saddle-Nodes

In this chapter, double-saddle and saddle-nodes (i.e., saddle-source, saddle-sink) with hyperbolic flows singularity exist in self and product cubic systems, and the third-order saddle-source and sink exist. The double-saddles are the appearing bifurcations of (2×2)-network of saddle, source, and sink, and the saddle-source and saddle-sink are the appearing bifurcations of saddle and source and of saddle and sink. The hyperbolic singular flows are the appearing bifurcations of the paralleled hyperbolic and hyperbolic-secant flows. The (2:3)-saddle-sources (sinks) are the appearing and switching bifurcations for (i) a (2:1)-saddle-source (sink) and a (2:2)-double-saddle, (ii) (1×3)-series of three (2:1)-saddle-sink and (2:1)-saddle-source, (iii) (2×1)-series of (1:3)-saddle and (1:3)-source (sink), (iv) (2×3)-network of simple saddles, sink, and source. The corresponding switching bifurcations for such cubic systems are presented through the infinite-equilibriums. The infinite-equilibriums in such a system include inflection-source and sink, up-down and down-up saddles, and third-order concave-source and sink.