Crossing-Linear and Self-Quadratic Product Systems

In this chapter, cubic product nonlinear systems with two crossing-linear and self-quadratic products vector fields are presented. The first integral manifolds are developed, and the equilibrium singularity and bifurcations are discussed through the appearing and switching bifurcations. The switching bifurcations are presented through the infinite 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 theory of the two-cubic product systems with crossing-linear and self-quadratic vector field is presented through a theorem and the corresponding proof of the theorem is given.