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Stability and Hopf bifurcation of a complex-valued neural network with two time delays

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Abstract

In this paper, a class of complex-valued neural networks with two time delays is considered. By considering that the activation function can be expressed by separating into its real and imaginary part and regarding the sum of time delays as a bifurcating parameter, the dynamical behaviors that include local asymptotical stability and local Hopf bifurcation are investigated. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the sum of time delays passes through a sequence of critical value. The linearized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

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Acknowledgments

This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant XDJK2014C017, in part by China Postdoctoral Foundation 2014M552319, in part by the Science and technology project of Chongqing Education Committee under Grant KJ130519, and in part by the Natural Science Foundation project of CQCSTC under Grant ctsc2014cyjA40053.

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Authors and Affiliations

  1. College of Electronics and Information Engineering, Southwest University, Chongqing, 400715, People’s Republic of China

    Tao Dong, Xiaofeng Liao & Aijuan Wang

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  1. Tao Dong
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  2. Xiaofeng Liao
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  3. Aijuan Wang
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Correspondence to Tao Dong or Xiaofeng Liao.

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Dong, T., Liao, X. & Wang, A. Stability and Hopf bifurcation of a complex-valued neural network with two time delays. Nonlinear Dyn 82, 173–184 (2015). https://doi.org/10.1007/s11071-015-2147-5

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  • DOI: https://doi.org/10.1007/s11071-015-2147-5

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