Crossing-Linear and Product-Cubic Systems

In this chapter, nonlinear dynamics and singularity of a crossing-linear and product cubic systems are discussed. For a self-linear and crossing-quadratic product, the inflection-source and sink flows are the appearing bifurcation of the connected hyperbolic and hyperbolic-secant flows, and the parabola-saddles are the appearing bifurcations of saddle and center. The third-order inflection source (sink) are the switching bifurcations for the inflection-source (sink) and parabola-saddles. For a self-crossing and self-quadratic product vector field, the hyperbolic-to-hyperbolic-secant flows are the appearing bifurcations for the separated hyperbolic and hyperbolic-secant flows. The hyperbolic down-up and up-down flows are for the switching bifurcations of saddles and hyperbolic-to-hyperbolic-secant flows. The networks of separated hyperbolic and hyperbolic-secant flows with center and saddle are developed. The up-parabola-saddles on the infinite-equilibriums are the switching bifurcations of hyperbolic flow and center. The down-parabola-saddles on the infinite-equilibriums are the switching bifurcations of hyperbolic-secant flow and saddle.