Advertisement

Journal of Nonlinear Science

, Volume 17, Issue 4, pp 309–347 | Cite as

The Spectrum of the Partially Locked State for the Kuramoto Model

  • R. Mirollo
  • S.H. StrogatzEmail author
Article

Abstract

We solve a long-standing stability problem for the Kuramoto model of coupled oscillators. This system has attracted mathematical attention, in part because of its applications in fields ranging from neuroscience to condensed-matter physics, and also because it provides a beautiful connection between nonlinear dynamics and statistical mechanics. The model consists of a large population of phase oscillators with all-to-all sinusoidal coupling. The oscillators' intrinsic frequencies are randomly distributed across the population according to a prescribed probability density, here taken to be unimodal and symmetric about its mean. As the coupling between the oscillators is increased, the system spontaneously synchronizes: The oscillators near the center of the frequency distribution lock their phases together and run at the same frequency, while those in the tails remain unlocked and drift at different frequencies. Although this "partially locked" state has been observed in simulations for decades, its stability has never been analyzed mathematically. Part of the difficulty is in formulating a reasonable infinite-N limit of the model. Here we describe such a continuum limit, and prove that the corresponding partially locked state is, in fact, neutrally stable, contrary to what one might have expected. The possible implications of this result are discussed.

Keywords

Banach Space Characteristic Equation Tangent Space Tangent Vector Continuous Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Mathematics Department, Boston CollegeChestnut Hill, MA 02467USA
  2. 2.Center for Applied Mathematics and Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell UniversityIthaca, NY 14853-1503USA

Personalised recommendations