Mean-field behavior in coupled oscillators with attractive and repulsive interactions

Hyunsuk Hong and Steven H. Strogatz
Phys. Rev. E 85, 056210 – Published 23 May 2012

Abstract

We consider a variant of the Kuramoto model of coupled oscillators in which both attractive and repulsive pairwise interactions are allowed. The sign of the coupling is assumed to be a characteristic of a given oscillator. Specifically, some oscillators repel all the others, thus favoring an antiphase relationship with them. Other oscillators attract all the others, thus favoring an in-phase relationship. The Ott-Antonsen ansatz is used to derive the exact low-dimensional dynamics governing the system's long-term macroscopic behavior. The resulting analytical predictions agree with simulations of the full system. We explore the effects of changing various parameters, such as the width of the distribution of natural frequencies and the relative strengths and proportions of the positive and negative interactions. For the particular model studied here we find, unexpectedly, that the mixed interactions produce no new effects. The system exhibits conventional mean-field behavior and displays a second-order phase transition like that found in the original Kuramoto model. In contrast to our recent study of a different model with mixed interactions [Phys. Rev. Lett. 106, 054102 (2011)], the π state and traveling-wave state do not appear for the coupling type considered here.

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  • Received 22 December 2011

DOI:https://doi.org/10.1103/PhysRevE.85.056210

©2012 American Physical Society

Authors & Affiliations

Hyunsuk Hong1 and Steven H. Strogatz2

  • 1Department of Physics and Research Institute of Physics and Chemistry, Chonbuk National University, Jeonju 561-756, Korea
  • 2Department of Mathematics, Cornell University, Ithaca, New York 14853, USA

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Issue

Vol. 85, Iss. 5 — May 2012

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