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I am slightly confused about the terminology of a subcritical and supercritical pitchfork bifurcation. Consider the diagrams below:

enter image description here enter image description here

(both pictures from wikipedia, by Claudio Rocchini, CC by 2.5)

I know the LHS one to be a subcritical pitchfork and the RHS one to be a supercritical. But I was wondering how the following pitchforks would be classified:

enter image description here enter image description here

(Pictures as above but flipped)

I would guess that the LHS one would still be a subcritical and the RHS one a supercritical however any definition or description that I have found talks about a subcritical as been one where two unstable fixed points join with a stable as the bifurcation parameter is increased. And likewise for a supercritical, it is described as one where a stable fixed point loses its stability and turns into a unstable fixed point creating two stable once (again as the bifurcation parameter increases). Thus of what type are the latter bifurcations? (a source would be great).

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1 Answer 1

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But I was wondering how the following pitchforks would be classified:

As you answered yourself: the left one is subcritical (two unstable and onestable equilibria collapse to produce one unstable) and the right one is supercritical (two stable and one unstable equilibria collapse to produce one stable one). Compare the terminology with the Poincare-Andronov-Hopf bifurcation.

The reason that "everyone talks about two upper figures" is because this bifurcation is introduced through its normal form $$ \dot x=\mu x\pm x^3, $$ where plus and minus correspond to sub- and supercritical bifurcation respectively. Equally easy you can consider $$ \dot x=-\mu x\pm x^3, $$ which will produce two lower pictures.

What matters is what actually happens in real systems. This depends on several partial defivatives, and all the exact details can be found in the book by Wiggings, Introduction to Applied Nonlinear Dynamical Systems and Chaos.

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