Details

The saddle node, subcritical Andronov–Hopf, supercritical Andronov–Hopf, and Bogdanov–Takens systems give bifurcations, not simple equilibrium points. Bifurcations arise from the collision and subsequent interaction between two or more equilibrium points.

The saddle node bifurcation comes from a system like:

,

.

For the case here, .

This saddle node acts as a saddle on the right and as a sink to the left. The resulting equilibrium point has one zero eigenvalue and one negative eigenvalue.

The Andronov–Hopf bifurcation comes from a system like:

,

.

For the supercritical case, and .

For the subcritical case, and .

The subcritical Andronov–Hopf bifurcation is a spiral sink within the limit cycle, so the eigenvalues match those of a spiral sink.

The supercritical Andronov–Hopf bifurcation is a spiral source within the limit cycle, so the eigenvalues match those of a spiral sink.

The Bogdanov–Takens bifurcation comes from a system like:

,

.

For the zero solution, and .

References

[1] J. Guckenheimer and Y. A. Kuznetsov. "Bogdanov–Takens Bifurcation." (Dec 5, 2014) www.scholarpedia.org/article/Bogdanov-Takens_bifurcation.

[2] E. M. Izhikevich. "Equilibrium." (Dec 5, 2014) www.scholarpedia.org/article/Equilibrium.

[3] Y. A. Kuznetsov. "Andronov–Hopf Bifurcation." (Dec 5, 2014) www.scholarpedia.org/article/Andronov-Hopf_bifurcation.

[4] Y. A. Kuznetsov. "Saddle-Node Bifurcation." (Dec 5, 2014) www.scholarpedia.org/article/Saddle-node_bifurcation.