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.
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?
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.