system. Let $\hat t = \Delta t, \hat F = F/\Delta, \hat K = K/\Delta, \hat \sigma = \sigma/\Delta$ and $ \hat \omega_0 = \omega_0/\Delta$. Then the form of stays the same except that $\Delta$ no longer appears (in effect, $\Delta$ has been set to 1 without loss of generality) and all the other parameters now have hats over them. For ease of notation we drop the hats in what follows, remembering that all the parameters are now dimensionless. Also, let $$\label{detuning}
\Omega = \sigma - \omega_0$$ denote the (dimensionless) detuning between the drive frequency and the population’s mean natural frequency. Then if we introduce polar coordinates $$\label{polar}
z = \rho e^{i \phi}$$ and separate into real and imaginary parts, we obtain the dimensionless evolution equations for $\rho$ and $\phi$:

$$\begin{aligned}
\rho' &=& \frac{K}{2} \rho (1 - \rho^2) - \rho + \frac{F}{2}(1- \rho^2) \cos{\phi} \label{eq:phi-rhoa}\\
\phi' &=& -\left[\Omega + \frac{F}{2}\left(\rho + \frac{1}{\rho}\right) \sin{\phi}\right] \label{eq:phi-rhob}\end{aligned}$$

where the prime denotes differentiation with respect to dimensionless time.

Stability diagram and phase portraits
-------------------------------------

Our next goal is to obtain the stability diagram for Eqs. (\[eq:phi-rhoa\])-(\[eq:phi-rhob\]). Before delving into the details, which can become intricate at times, we jump to the final result. Figure 