Parabola-Saddles and Bifurcation Dynamics

In this chapter, parabola-saddles and hyperbolic-to-hyperbolic-secant flows with the corresponding bifurcation dynamics are discussed. The hyperbolic-to-hyperbolic-secant flows are presented for the appearing bifurcations of paralleled hyperbolic and hyperbolic-secant flows. The parabola-saddles are illustrated for the appearing bifurcations of center and saddle. The hyperbola-saddles on the inflection upper-saddle and lower-saddle infinite equilibriums are the switching bifurcation of a hyperbolic flow with a paralleled parabola-saddle. A series of center and saddle with a hyperbolic flow is presented. The hyperbolic and hyperbolic-secant lower-to-upper saddles are the switching bifurcations of hyperbolic-to-hyperbolic-secant flows with paralleled saddle or center. A line series of two separated hyperbolic flows with a paralleled parabola-saddle are presented. The hyperbolic (circular) sink-to-source is the switching bifurcation of a hyperbolic (hyperbolic-secant) flow and a paralleled parabola-saddle. A line series of paralleled hyperbolic flows with saddle and center is presented. The paralleled parabola-saddles are for the switching bifurcations of a hyperbolic flow with a paralleled saddle or center.