Crossing-Quadratic and Product-Cubic Systems

In this chapter, crossing-quadratic and product-cubic systems are presented, and the corresponding dynamical behaviors for such cubic systems are discussed through two theorems. A self-linear and crossing-quadratic product vector field is discussed first. Based on such a vector field, the appearing bifurcations of double-inflection-saddles and inflection-source (sink) flows exist, and the switching bifurcations are completed through the infinite-equilibriums, including parabola-source (sink), hyperbolic (circular) sink-to-source and source-to-sink, and parabola-saddles. The networks of saddles and centers with paralleled hyperbolic flows are developed. The crossing-linear and self-quadratic vector fields are also discussed. From such a cubic product vector field, the appearing bifurcations includes hyperbolic-to-hyperbolic-secant flows and parabola-saddles. The switching bifurcations include parabola-saddles, hyperbolic (hyperbolic-secant) lower-to-upper saddles, hyperbolic (circular) sink-to-source, and parabola-saddles. For such a cubic product system, a series of center and saddle with a hyperbolic flow can be formed, a series of two separated hyperbolic flows with a paralleled parabola-saddle exists, and a series of paralleled hyperbolic flows with saddle and center is also obtained.