Slide 33 of 79
Notes:
Consulting the bifurcation theory we see that this type of behavior is seen at a Takens-Bogdanov bifurcation and that it is accompanied by a third curve of bifurcation’s called saddle loop bifurcations. At a saddle loop bifurcation the period of the limit cycle goes to infinity and the limit cycle disappears as it becomes homoclinic to the saddle point. Running through the diagram: as we cross the bottom curve of saddle nodes a saddle point and node are born. Moving counter clockwise and crossing the curve of Hopfs gives us a limit cycle. As we approach the curve of saddle nodes the period of the limit cycle increases as it gets closer to the saddle point. At the saddle loop bifurcation, the period of the limit cycles goes to infinity as they become homoclinic to the saddle point. This leaves us once again with a saddle and node which are destroyed as we move through the top saddle node bifurcation
