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Synchronization of a delayed complex dynamical network with free coupling matrix

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Abstract

This paper considers synchronization problem of a delayed complex dynamical network. For the problem, the virtual target node is chosen as one of nodes in the complex network. It should be pointed out that only one connection is needed between a real target node and a virtual target node instead of N connections. Moreover, the proposed synchronization scheme does not require additional conditions for coupling matrix unlike the existing works. Based on Lyapunov stability theory, a new design criterion for an adaptive feedback controller to achieving synchronization between the real target node and all nodes of the delayed complex network is developed. Finally, the proposed method is applied to a numerical example in order to show the effectiveness of our results.

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Abbreviations

ℝn :

is the n-dimensional Euclidean space.

ℝm×n :

denotes the set of m×n real matrices.

X>0:

(respectively, X≥0) means that the matrix X is a real symmetric positive definite matrix (respectively, positive semi-definite).

I n :

denotes the n-dimensional identity matrix.

0 n×m :

denotes the n×m dimensional matrix in which all entries are zero.

1 n×m :

denotes the n×m dimensional matrix in which all entries are one.

∥⋅∥:

refers to the Euclidean vector norm and the induced matrix norm.

diag {⋅⋅⋅}:

denotes the block diagonal matrix.

⋆:

represents the elements below the main diagonal of a symmetric matrix.

⊗:

denotes the Kronecker product.

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Acknowledgements

The work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0009373).

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Authors and Affiliations

  1. Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan, 712-749, Republic of Korea

    Tae H. Lee & Ju H. Park

  2. Department of Information and Communication Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan, 712-749, Republic of Korea

    H. Y. Jung

  3. School of Electronics Engineering, Daegu University, Gyungsan, 712-714, Republic of Korea

    S. M. Lee

  4. School of Electrical Engineering, Chungbuk National University, 52 Naesudong-ro, Cheongju, 361-763, Republic of Korea

    O. M. Kwon

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  1. Tae H. Lee
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  2. Ju H. Park
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Correspondence to Ju H. Park.

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Lee, T.H., Park, J.H., Jung, H.Y. et al. Synchronization of a delayed complex dynamical network with free coupling matrix. Nonlinear Dyn 69, 1081–1090 (2012). https://doi.org/10.1007/s11071-012-0328-z

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  • DOI: https://doi.org/10.1007/s11071-012-0328-z

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