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Oct 7, 2015 at 0:12 comment added Michael Renardy This example is not smooth at the origin if you transform back to Cartesian coordinates, but you can fix that by changing $\sin\theta$ to $r\sin\theta$. This does not change the nature of the fixed points. The bifurcation is then an example of a bifurcation with a double zero eigenvalue, and there is quite a bit of literature on those.
Oct 6, 2015 at 17:54 comment added Anthony Quas The everything bagel bifurcation?
Oct 6, 2015 at 16:40 history asked Julian Newman CC BY-SA 3.0