distribution function $g(\omega )$ but for the case where the coupling between oscillators is small. That is, $K\ll \Delta $, where $\Delta $ denotes the frequency width of $g(\omega )$ about its mean value $\omega =\bar \omega $. In this case a good approximation is provided by setting $K=0$. Thus $D[-(i\omega +\xi )]\cong 1$ and Eq. (33) yields $$f^{(2)}_{1,e}(\omega ,\tau ^+)=d_0d_1\sum ^\infty
_{n=1}nh^*_nh_{n+1}\exp [-(-in\omega +n^2\xi )\tau ] \ ,$$ where we have replaced $n$ by $-n$ and used $h_n=h^*_{-n}$. Inserting Eq. (42) into Eq. (32) we obtain $$R_e^{(2)*}(t)=\sum ^\infty _{n=2}(n-1)d_0d_1h^*_{n-1}h_n\tilde
g(t-n\tau )e^{-[(n^{2}-1)\tau +t]\xi } \ ,$$ where $\tilde g(t)$ is defined by $$\tilde g(t)=\int ^{+\infty}_{-\infty}d\omega e^{-i\omega
t}g(\omega ) \ .$$ Thus, for $K\ll \Delta $, the shape of the echoes at $t=2\tau
,3\tau ,\ldots$ is directly given by the Fourier transform (44) of the frequency distribution function $g(\omega )$. Another point is that with $K\rightarrow 0$, Eq. (1) shows that the oscillators do not interact, and the nonlinearity needed to produce the echo phenomenon comes entirely from the stimulus function $h(\theta )$.

IV. Simulations
===============

We have performed direct numerical simulations of the system (1) with a Lorentzian oscillator distribution (see Eq. (35)), $\bar
\omega =0$, $\Delta =1$ (corresponding to 