Draw the orbits in the {$\rho,\theta$}-plane and describe the evolution in each cases, classfying subcritical and supercritical if appropriate. Given 
$$\frac{dA}{d\tau}=\sigma A-\beta A|A|^2,
$$
where $\sigma=\sigma_r+i\sigma_i$, $\beta$ is real and $A(\tau)=\rho(\tau)\exp(i\theta(\tau))$.
Draw the orbits in the {$\rho,\theta$}-plane and describe the evolution in each cases($\sigma_r$ and $\beta$ positive and negative), classfying subcritical and supercritical if appropriate.
Here is what I have done so far: Substitute all into the ODE we have two equations:
$$\rho_\tau=\sigma_r \rho-\beta\rho^3
$$
$$\theta_\tau=\sigma_i.$$
The first one looks like a pitchfork bifurcation with parameter $\sigma_r/\beta$. I am not sure what to do next. Can anyone help?
 A: You will have to study all cases. For example, when $\beta>0$, we consider $\sigma_r\rho-\beta\rho^3=0$. So $\rho(\sigma_r-\beta\rho^2)=0$. This is a cubic curve which goes to $\infty$ on the left and $-\infty$ on the right. It always passes through $0$. The other root depends on the sign of $\sigma_r$. If $\sigma_r<0$, there's no other roots. If $\sigma_r>0$, the other two roots are $\pm\sqrt{\frac{\sigma_r}{\beta}}$.
Draw the bifurcation diagram as follows:

The arrows indicate the direction of flow of $\rho$ on either sides of the fixed points. The solid dot means stable fixed point, and hollow dot means unstable fixed point.
You can see this is the case of supercritical bifurcation.
Now to draw the orbit in $\rho,\theta$ plane, again separate these three different cases. In the first case where $\beta>0,\sigma_r<0$, there is one stable fixed point for $\rho$, which is zero. Notice that there is no fixed point for $\theta$, and it has a constant rate. As in your edit, the orbits should be like spirals toward the origin.
You can try all the other cases in a similar way. Be careful when there are more than one fixed points. The orbits would look different.
