Synchronizing large number of nonidentical oscillators with small coupling

¹ School of Science, Beijing University of Posts and Telecommunications - Beijing 100876, China
² State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications - Beijing 100876, China
³ Department of Physics, Beijing Normal University - Beijing 100875, China
⁴ State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences - Wuhan 430071, China

^a zhanmeng@wipm.ac.cn

Received: 4 November 2011
Accepted: 11 January 2012

Abstract

The topic of synchronization of oscillators has attracted great and persistent interest, and all previous conclusions and intuitions have convinced that large coupling is required for synchronizing a large number of coupled nonidentical oscillators. Here the influences of different spatial frequency distributions on the efficiency of frequency synchronization are investigated by studying arrays of coupled oscillators with diverse natural frequency distributions. A universal log-normal distribution of critical coupling strength K_c for synchronization irrespective of the initial natural frequency is found. In particular, a physical quantity “roughness” R of spatial frequency configuration is defined, and it is found that the efficiency of synchronization increases monotonously with R. For large R we can reach full synchronization of arrays with a large number of oscillators at finite K_c. Two typical kinds of synchronization, the “multiple-clustering” one and the “single-center-clustering” one, are identified for small and large R's, respectively. The mechanism of the latter type is the key reason for synchronizing long arrays with finite K_c.