just before the application of the second delta function stimulus $(t=\tau ^-)$, $$f_1^{(1)}(\omega ,\tau ^-)\cong A(\omega )e^{-(i\omega +\xi )\tau
} \ ,$$ where we have neglected the second term on the right hand side of Eq. (16) on the basis that, due to its more rapid exponential decay, it is small compared to the first term.

Solutions of $D(s)=0$ govern the stability of the state with $R^{(1)}=0$. Let $s=s_0$ denote the solutions of $D(s)=0$ with the largest real part. If $Re(s_0)<0$ the state $R^{(1)}=0$ is stable, and a perturbation away from $R^{(1)}=0$ decays to zero with increasing $t$ at the exponential rate $Re(s_0)$. If $Re(s_0)>0$, then the perturbation grows and $R^{(1)}$ eventually saturates into a sustained nonlinear state of coherent cooperative oscillatory behavior[@kuramoto84; @acebron05]. In general, $Re(s_0)$ is an increasing function of the coupling constant $K$, and $Re(s_0)\stackrel{>}{<}0$ for $K\stackrel{>}{<}K_c$, where $K_c$ is a critical value that depends on $\xi$ and $g(\omega )$. Throughout this paper we shall be considering only the case $K<K_c$ for which $Re(s_0)<0$.

It is instructive to consider $\xi =0$. In that case, the first term in Eq. (16) is of constant magnitude in time, but, as time $t$ increases, it oscillates more and more rapidly 