(Fig. 4(a)) the echo is swamped by the noise and is not seen. For $\hat d_0=\hat d_1=1/4$ (Fig. 4(b), same as 2(a)) the echo seems to have appeared, but because of the noise, this conclusion is somewhat questionable. Finally, at the larger driving of $\hat d_0=\hat d_1=1/2$, the echo is clearly present.

Figures 5(a) and 5(b) show the effect of changing $h(\theta )$. In particular, Fig. 5(a) shows numerical results for $\hat d_0=\hat
d_1=1/4$, $N=10^5$, and $h(\theta )=\sin \theta $, with all other parameters the same as before. Since $h_2$ is now zero, Eq. (39) now predicts that there is no echo, in agreement with Fig. 5(a). Figure 5(b) shows numerical results for $\hat d_0=\hat d_1=1/4$, $N=10^5$, and $$h(\theta )=\sin \theta +\sin 2\theta +\sin 3\theta,$$ with all other parameters the same as before. Since $h_1$, $h_2$ and $h_3$ are all nonzero, Eqs. (39) and (40) now predict echoes at both $t\cong 2\tau $ and at $t\cong 3\tau $, and this is confirmed by Fig. 5(b).

![Simulation of $10^5$ oscillators for $\tau =100$, $\hat
d_0=\hat d_1=1/3$, $K=1=\frac{1}{2}K_c$, $h(\theta )=\sin \theta
$. In this case, no echo at $t=2\tau =200$ is observed.](fig4)

![Simulation of $10^5$ oscillators for $\tau =100$, $\hat
d_0=\hat d_1=??$, 