# Hopf bifurcation in a simple system

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 $\lambda = 0$

• Your eigenvalues are real, because you do all the calculations in polar coordinates. – Artem Dec 27 '16 at 16:23

Hint: $-\mu r + r^3=r(-\mu + r^2)$. For the stability of the orbits look at the sign of $-\mu+r^2$.
• 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 $r=\sqrt\mu$. – John B Dec 27 '16 at 10:56
• Oh, okay, so $r=\mu^{1/2}$ is where the limit cycle appears? – Ilya Lapan Dec 28 '16 at 11:44