the distribution of frequencies resulted from the position dependence of the precession frequency of nuclear magnetic dipoles in an inhomogeneous magnetic field. \[The spin echo forms the basis for modern magnetic resonance imaging (MRI).\] Subsequently, echoes for cyclotron orbits of charged particles in a magnetic field have been studied for the cases in which the distribution in frequency was due to magnetic field inhomogeneity[@gould65], relativistic dependence of the particle mass on its energy[@ott70], and Doppler shifts of the cyclotron frequency[@porkolab68]. Another notable case is that of plasma waves, where the frequency distribution results from the Doppler shift of the wave frequency felt by charged particles with different streaming velocities[@oneil68]. Although echo phenomena are well-known in the above settings, they have so far not received attention in the context of the Kuramoto model and its many related situations. It is our purpose in the present paper to investigate that problem. Two possible motivations for our study of echoes in the Kuramoto model are that they provide increased basic understanding of the model and also that they may be of potential use as a basis for future diagnostic measurements of related systems (see Sec. V).

In what follows, Sec. II will give 