Consider the differential equation on $\mathbb{R}^2$ whose polar-coordinate representation is given by $$ \begin{array}{r c l} \dot{r} & = & r(\alpha - r^2) \\ \dot{\theta} & = & -sin(\theta).\end{array} $$ If $\alpha \leq 0$, then the origin $(0,0)$ is a globally asymptotically stable fixed point.

If $\alpha>0$, then there are three fixed points:

  • the origin $(0,0)$ is an unstable fixed point;
  • the point $(-\sqrt{\alpha},0)$ is a saddle (attracting in the $x$-direction and repelling in the $y$-direction);
  • the point $(\sqrt{\alpha},0)$ is a stable fixed point, whose basin of attraction is $\,\mathbb{R}^2 \setminus \{(x,0):x \leq 0\}$.

Is there a name for this kind of bifurcation scenario?

Remark. There is one heteroclinic connection from $(0,0)$ to $(-\sqrt{\alpha},0)$, namely the set $\{(x,0): -\sqrt{\alpha} \leq x \leq 0\}$. There are two heteroclinic connections from $(-\sqrt{\alpha},0)$ to $(\sqrt{\alpha},0)$, namely, the upper semicircle of radius $\sqrt{\alpha}$ about the origin, and the lower semicircle of radius $\sqrt{\alpha}$ about the origin.

  • $\begingroup$ The everything bagel bifurcation? $\endgroup$ – Anthony Quas Oct 6 '15 at 17:54
  • 2
    $\begingroup$ 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. $\endgroup$ – Michael Renardy Oct 7 '15 at 0:12

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