Say the the steady state of system is a stable spiral. The parameters are changed so that it becomes unstable in some way and a bifurcation occurs.
That means there can be two types of bifurcations, a Hopf bifurcation or a saddle node bifurcation. Stable spiral to unstable spiral or node to saddle respectively.
I know how to prove a limit cycle exists from the Poincare-Bendixon theorem when there is a Hopf bifurcation, but how do you prove there is a limit cycle when there is a saddle node bifurcation? I guess my problem is that a stable limit cycle can't surround only a saddle point (it can surround a saddle plus some other type of fixed point). I should specify that there is only one steady state so the latter isn't possible.
This may be of use if you're a bit rusty:
http://www.hasdeu.bz.edu.ro/so...a/Pendul_1/2dflows.gif
Basically, the starting point is up in the stable spiral quadrant and it becomes unstable, meaning it moves right or down.
That means there can be two types of bifurcations, a Hopf bifurcation or a saddle node bifurcation. Stable spiral to unstable spiral or node to saddle respectively.
I know how to prove a limit cycle exists from the Poincare-Bendixon theorem when there is a Hopf bifurcation, but how do you prove there is a limit cycle when there is a saddle node bifurcation? I guess my problem is that a stable limit cycle can't surround only a saddle point (it can surround a saddle plus some other type of fixed point). I should specify that there is only one steady state so the latter isn't possible.
This may be of use if you're a bit rusty:
http://www.hasdeu.bz.edu.ro/so...a/Pendul_1/2dflows.gif
Basically, the starting point is up in the stable spiral quadrant and it becomes unstable, meaning it moves right or down.
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