Self-Quadratic and Product-Cubic Systems

In this chapter, nonlinear dynamics and singularity of self-quadratic and product-cubic systems are discussed. The equilibrium singularity and bifurcations are discussed through the appearing and switching bifurcations. The saddle-sources (sink) equilibriums are the appearing bifurcations for saddle and source (sink) equilibriums. 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 infinite-equilibriums for the switching bifurcations exist in such cubic nonlinear systems. Inflection upper-saddle and lower-saddle infinite-equilibriums are the switching bifurcation of saddle-sink and saddle-source with inflection-sink and source flows. Hyperbolic and hyperbolic-secant, sink and source infinite-equilibriums are for the switching bifurcations of saddle, source, and sink with inflection-sink and source flows. Up-down and down-up lower-saddle and upper-saddle infinite-equilibriums are the switching bifurcations of saddle-source and saddle-sink with hyperbolic and hyperbolic-secant flows.