---
author:
- |
    Lauren M. Childs and Steven H. Strogatz\
    Center for Applied Mathematics,\
    Cornell University, Ithaca, NY 14853 USA\
    \
    `lmchilds@cam.cornell.edu, strogatz@cornell.edu`
title: Stability diagram for the forced Kuramoto model
---

Abbreviated title: Forced Kuramoto model

**The study of synchronization is a classic topic in nonlinear science. Sometimes the concern is with mutual synchronization, as in Huygens’s 1665 discovery of the sympathy of pendulum clocks. In other situations, one is more interested in forced synchronization, as in the injection locking of a laser or the entrainment of circadian rhythms by the daily light-dark cycle. Here we consider a simple mathematical model in which both types of synchronization are present simultaneously, creating a conflict between them. What happens when a network of dissimilar but mutually coupled oscillators is also driven by an external periodic force? For a natural generalization of the Kuramoto model, the interaction of forcing, coupling, and randomness leads to a rich set of collective states and bifurcations. We explain all of these phenomena analytically, using an ansatz recently introduced by Ott and Antonsen.**

Introduction
============

In 1975 Kuramoto proposed an elegant model for an enormous population of coupled biological oscillators \[Kuramoto 1975, 1984\]. Each oscillator was 