amplitude equations for all $n$, so long as $\alpha$ evolves according to $$\label{alphaeqn}
\frac{d \alpha}{d t} = \frac{1}{2} (K z + F)^{*} - i (\omega - \sigma) \alpha - \frac{1}{2} (K z + F) \alpha^2$$ and $z$ satisfies $$\label{zintegral}
z(t) = \int_{-\infty}^{\infty} \alpha^{*}(\omega, t) \, g(\omega) \, d \omega .$$ Then, by further assuming that $g(\omega)$ is a Lorentzian, $$\label{lorentzian}
g(\omega)=\frac{\Delta}{\pi \left\{ (\omega-\omega_0)^2+\Delta^2 \right\}},$$ and that $\alpha(\omega, t)$ satisfies certain analyticity conditions in the complex $\omega$-plane, Ott and Antonsen \[2008\] evaluated by contour integration and thereby derived the following exact evolution equation for the order parameter $z$: $$\label{orderparamtimeevol}
\frac{dz}{dt} = \frac{1}{2}\left[ (K z + F) - (K z + F)^* z^2  \right] - \left[ \Delta +i (\sigma - \omega_0)\right] z.$$ The conditions required were that $\alpha(\omega, t)$ can be analytically continued from the real $\omega$-axis into the lower half of the complex $\omega$-plane for all $t \ge 0$; that $|\alpha(\omega, t)| \rightarrow 0$ as $\rm{Im}(\omega) \rightarrow -\infty$; and that $|\alpha(\omega, 0)| \le 1$ for real $\omega$.

Analysis of the reduced equations
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Scaling the equations
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We turn now to the analysis of the two-dimensional system . The first step is to reduce the number of parameters by nondimensionalizing the 