中国物理B ›› 2009, Vol. 18 ›› Issue (8): 3337-3346.doi: 10.1088/1674-1056/18/8/038

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Pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions

邹艳丽1, 陈关荣2   

  1. (1)College of Physics and Electronic Engineering, Guangxi Normal University, Guilin 541004, China;Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China; (2)Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China
  • 收稿日期:2008-12-26 修回日期:2009-01-23 出版日期:2009-08-20 发布日期:2009-08-20
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant No 10647001), the Guangxi Natural Science Foundation (Grant No 0728042), the Program for Excellent Talents in Guangxi Higher Education Institutions (Grant No RC2007006) and the NSFC-HK Joint Research Scheme (Grant No N-CityU107/07).

Pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions

Zou Yan-Li(邹艳丽)a)b)† and Chen Guan-Rong(陈关荣)b)   

  1. a College of Physics and Electronic Engineering, Guangxi Normal University, Guilin 541004, China; b Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China
  • Received:2008-12-26 Revised:2009-01-23 Online:2009-08-20 Published:2009-08-20
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant No 10647001), the Guangxi Natural Science Foundation (Grant No 0728042), the Program for Excellent Talents in Guangxi Higher Education Institutions (Grant No RC2007006) and the NSFC-HK Joint Research Scheme (Grant No N-CityU107/07).

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摘要： This paper studies pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions. To study a state-feedback pinning-controlled network with N nodes, it first converts the controlled network to an extended network of N+1 nodes without controls. It is shown that the controlled synchronizability of the given network is determined by the real part of the smallest nonzero eigenvalue of the coupling matrix of its extended network when the synchronized region is unbounded; but it is determined by the ratio of the real parts of the largest and the smallest nonzero eigenvalues of the coupling matrix when the synchronized region is bounded. Both theoretical analysis and numerical simulation show that the portion of controlled nodes has no critical values when the synchronized region is unbounded, but it has a critical value when the synchronized region is bounded. In the former case, therefore, it is possible to control the network to achieve synchronization by pinning only one node. In the latter case, the network can achieve controlled synchronization only when the portion of controlled nodes is larger than the critical value.

Abstract: This paper studies pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions. To study a state-feedback pinning-controlled network with N nodes, it first converts the controlled network to an extended network of N+1 nodes without controls. It is shown that the controlled synchronizability of the given network is determined by the real part of the smallest nonzero eigenvalue of the coupling matrix of its extended network when the synchronized region is unbounded; but it is determined by the ratio of the real parts of the largest and the smallest nonzero eigenvalues of the coupling matrix when the synchronized region is bounded. Both theoretical analysis and numerical simulation show that the portion of controlled nodes has no critical values when the synchronized region is unbounded, but it has a critical value when the synchronized region is bounded. In the former case, therefore, it is possible to control the network to achieve synchronization by pinning only one node. In the latter case, the network can achieve controlled synchronization only when the portion of controlled nodes is larger than the critical value.

Key words: synchronization, pinning control, complex network, synchronized region

中图分类号:  (Computer science and technology)

  • 89.20.Ff
05.45.Xt (Synchronization; coupled oscillators) 89.75.Hc (Networks and genealogical trees)