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Given the system:

$$ \dot{r} = -\mu r + r^3, \\ \dot{\theta} = r $$

There is clearly one single node at $r=0$.

The Jacobian is then: $$ \begin{pmatrix} -\mu + 3r^2 & 0 \\ 1 & 0 \end{pmatrix}$$ Setting $r=0$ and finding the eigenvalues I get: $\lambda = 0 , \lambda = -\mu $. The problem statement says "show that a subcritical Hopf bifurcation occurs at the parameter value $\mu = 0$ ". I don't see how a Hopf bifurcation appears here when all my eigenvalues are all real and I am failing to interpret $\lambda = 0$

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  • $\begingroup$ Your eigenvalues are real, because you do all the calculations in polar coordinates. $\endgroup$
    – Artem
    Commented Dec 27, 2016 at 16:23

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Hint: $-\mu r + r^3=r(-\mu + r^2)$. For the stability of the orbits look at the sign of $-\mu+r^2$.

Don't use the Jacobian, no need for it. Better drawing the orbits based on the former identity.

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  • $\begingroup$ But how do I know the stability of the orbits? $\endgroup$
    – Ilya Lapan
    Commented Dec 27, 2016 at 10:48
  • $\begingroup$ Just look at the sign of $-\mu+r^2$. If it is positive the orbit goes away from the origin, etc. For $\mu>0$ you have a special value $r=\sqrt\mu$. $\endgroup$
    – John B
    Commented Dec 27, 2016 at 10:56
  • $\begingroup$ Oh, okay, so $r=\mu^{1/2}$ is where the limit cycle appears? $\endgroup$
    – Ilya Lapan
    Commented Dec 28, 2016 at 11:44
  • $\begingroup$ Precisely. ---- $\endgroup$
    – John B
    Commented Dec 28, 2016 at 12:51

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