This tutorial provides an introduction to the application and non-linear dynamics of globally coupled oscillator systems by considering the popular and well researched Kuramoto model.
The goal of this tutorial is first to present the derivation of the Kuramoto model and therefore justify it's use as an oscillatory model for representing more complex non-linear dynamical systems.
After presenting the derivation of the Kuramoto model, this tutorial then aims to give a brief and accessible analysis of the model in order to introduce the field of Coupled Oscillatory Dynamics. Here we explore some of the more interesting dynamical features of the system that have made it so popular, and also touch upon some of the tools that have been used to analyse this model in research, such as the Order Parameter.
The tutorial includes an interactive Kuramoto Simulator which will allow the reader to fully explore the dynamics of the Kuramoto model (with or without noise) running with between 1 and 100 oscillators. This simulator allows the reader to alter system parameters during the simulation so that they may explore for themselves the different categories of dynamics this model exhibits, especially around certain critical values for the Coupling Parameter.
Finally, the tutorial suggests some future avenues for research and aims to whet the appetite of the reader by giving some examples of where the Kuramoto model has been used in real-life applications.
|Author Institutional Affiliation
Mathematics Research Institute,
School of Engineering, Computer Science and Mathemathics,
University of Exeter. (UK).