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Suppose we have some 2-dimensional bifurcation diagram, say, the following which I found when using google; this is just meant as a general question about how to read such diagrams, I am not dealing with the concrete equations here.

enter image description here

Now my question is how to read such diagrams?

For example focusing on the supercritical hopf bifurcations, i.e. the solid red curve: Does this mean that on the solid red curve we have some stable equilibrium with a purely imaginary pair of eigenvalues and when passing it, this equilibrium gets unstable and a stable limit cycle is born?

How do we know in which direction we have to pass the solid red line in order to get the birth of the stable limit cycle?

mathfemi
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    That's a complicated picture showing an advanced bifurcation situation. There is not much you can do beyond looking over the plot, chewing through the text, looking at the plot again, etc. Your interpretation ofn the red curve is correct. The question about which way you need to cross is also natural and the answer would be somewhere in the article. Note that if you cross the red dashed line, then no stable limit cycle is born. Instead, an unstable limit cycle exists before the crossing. (It's actually funny, I just spoke with one of the authors of that paper 20 minutes ago) – Hans Engler Sep 20 '16 at 17:05
  • But what does "before crossing the red dashed line" mean here? Where is "before" and where is "after"? – mathfemi Sep 20 '16 at 17:10

1 Answers1

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Look at the article, in particular eq. 4a-c. Set $z = 0, \epsilon = 0$ (because you are on the fast system) and choose $s$ anywhere above the red curve, e.g. $s = 0$. Then the origin is stable.

So the Hopf bifurcation occurs (according to this picture) when you cross the solid red parabolic curve from that region, i.e. coming from the top or from the left. The subcritical bifurcation occurs when you cross the dashed red line from below.

Hans Engler
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  • Could we say in general that in a two-dimensional bifurcation diagram when we are on a curve (which belongs to some special type of bifurcation) the bifurcation always happens when we cross this curve while on the curve some "situation in between" occurs? Or does this only hold for Hopf? And are there curves where the bifurcations only happen when we move on the curve itself? In other words: Does the bifurcation always happen when crossing the curve or are there cases where they happen ON the curve? – mathfemi Sep 20 '16 at 17:25
  • Another such case is a locus of a pitchfork bifurcation in a 2-D parameter case. The "situation in between" on the curve is undetermined stability of the single stationary point which is about to plit into three. This could actually mean instability or stability and must be determined separately. A situation where something happens as you move along the curve is the point labelled B where the Hopf bifurcation turns from supercritical to subcritical. – Hans Engler Sep 20 '16 at 17:31
  • That is, so to speak, crossing a bifurcation curve really shows what the bifurcation "is doing" (like e.g. giving birth to a limit cycle or another equilibrium) and moving on a bifurcation curve can only vary the way this bifurcation appears, i.e. its "character" (e.g. from supercritical to subcritical) but does not e.g. stands for a birth of equilibria etc.? – mathfemi Sep 20 '16 at 17:37
  • That is more or less what one would expect. Of course new phenomena may be discovered next week that refute this intuition. – Hans Engler Sep 22 '16 at 16:59