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abstract: |
    As exemplified by the Kuramoto model, large systems of coupled oscillators may undergo a transition to phase coherence with increasing coupling strength. It is shown that below the critical coupling strength for this transition such systems may be expected to exhibit ‘echo’ phenomena: a stimulation by two successive pulses separated by a time interval $\tau $ leads to the spontaneous formation of response pulses at a time $\tau$, $2\tau
    $, $3\tau \ldots$, after the second stimulus pulse. Analysis of this phenomenon, as well as illustrative numerical experiments, are presented. The theoretical significance and potential uses of echoes in such systems are discussed.
author:
- 'Edward Ott, John H. Platig, Thomas M. Antonsen and Michelle Girvan'
title: Echo Phenomena in Large Systems of Coupled Oscillators
---

[**Large systems consisting of many coupled oscillators for which the individual natural oscillator frequencies are different naturally occur in a wide variety of interesting applications. As shown by Kuramoto, such systems can undergo a type of dynamical phase transition such that as the coupling strength is raised past a critical value, global synchronous collective behavior results. In this paper we show that another interesting, potentially useful, behavior of these systems also occurs 