with angular width $\tfrac{2\pi}{3}$. For example $\mathcal{D}_R = \{ (u,\theta) | \theta \in [0,2\pi/3]\}$. By integrating over these domains and defining time averaged fluctuations: $$\bar{\psi}_r(t) := \int_0^\infty g_m(u) \psi_r(t-u)$$ we obtain the following linearized expressions for $p_R$ and $p_S$: $$\begin{aligned}
\label{pRLin}
p_R(\bar{\psi}_r, \bar{\psi}_s) &= \frac{1}{3} +  \frac{3(2 \sqrt{m \pi} - \sqrt{3})}{8\pi}\bar{\psi}_r  + \frac{3 \sqrt{m}}{2 \sqrt{\pi}} \bar{\psi}_s\\
p_S(\bar{\psi}_r,\bar{\psi}_s) &= \frac{1}{3} - \frac{3 \sqrt{m}}{2 \sqrt{\pi}} \bar{\psi}_r - \frac{3(2 \sqrt{m \pi} +\sqrt{3})}{8 \pi} \bar{\psi}_s.
\label{pSLin}\end{aligned}$$ We can verify the quality of these approximations by comparing the expansion coefficients to the exact values of the derivatives of $p_\omega(r,s)$ evaluated at $(r,s)=(\tfrac{1}{3},\tfrac{1}{3})$, as shown in Figure \[drpR\].

![The derivatives $\frac{\partial p_R}{\partial r}$, $\frac{\partial p_S}{\partial s}$, $\frac{\partial p_R}{\partial s}$, $\frac{\partial p_S}{\partial r}$ evaluated directly from equation (\[pfunc1\]) at $(\bar{r}_m,\bar{s}_m)=(\tfrac{1}{3},\tfrac{1}{3})$. The four derivatives are plotted using open circles, filled circles, open squares, filled squares respectively. Black lines show corresponding expansion coefficients from equations (\[pRLin\]) and (\[pSLin\]). \[drpR\]](dps.eps){width="8cm"}

The linearized delay equations (\[rDelay\]) and (\[sDelay\]) may then be expressed in terms of the expansion coefficients: $$\begin{aligned}
a(m) &:= \frac{3(2 \sqrt{m \pi} - \sqrt{3})}{8\pi}\\
b(m) &:= \frac{3 \sqrt{m}}{2 \sqrt{\pi}}  \\
c(m) &:= \frac{3(2 \sqrt{m \pi} +\sqrt{3})}{8 \pi}\end{aligned}$$ as follows $$\begin{aligned}
\label{linr}
\frac{\dot{\psi}_r(t)}{\epsilon} &= \int_0^\infty \left[a \psi_r(t-u) + b \psi_s(t-u)\right] 