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Dynamical Behavior of Complex-Valued Hopfield Neural Networks with Discontinuous Activation Functions

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Abstract

This paper presents some theoretical results on dynamical behavior of complex-valued neural networks with discontinuous neuron activations. Firstly, we introduce the Filippov differential inclusions to complex-valued differential equations with discontinuous right-hand side and give the definition of Filippov solution for discontinuous complex-valued neural networks. Secondly, by separating complex-valued neural networks into real and imaginary part, we study the existence of equilibria of the neural networks according to Leray–Schauder alternative theorem of set-valued maps. Thirdly, by constructing appropriate Lyapunov function, we derive the sufficient condition to ensure global asymptotic stability of the equilibria and convergence in finite time. Numerical examples are given to show the effectiveness and merits of the obtained results.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (1573003, 11601143), Natural Science Foundation of Hunan Province of China (13JJ4111, 14JJ3141), a key Project supported by Scientific Research Fund of Hunan Provincial Education Department (15k026, 15A038) and Aid Program for Science and Technology Innovative Research Team in Higher Educational Institution of Hunan Province.

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Authors and Affiliations

  1. Department of Mathematics, Hunan First Normal University, Changsha, 410205, China

    Zengyun Wang

  2. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China

    Zhenyuan Guo & Lihong Huang

  3. Changsha University of Science and Technology, Changsha, 410114, China

    Lihong Huang

  4. Department of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Zengyun Wang & Xinzhi Liu

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  1. Zengyun Wang
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  2. Zhenyuan Guo
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  4. Xinzhi Liu
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Correspondence to Zhenyuan Guo.

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Wang, Z., Guo, Z., Huang, L. et al. Dynamical Behavior of Complex-Valued Hopfield Neural Networks with Discontinuous Activation Functions. Neural Process Lett 45, 1039–1061 (2017). https://doi.org/10.1007/s11063-016-9563-5

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  • DOI: https://doi.org/10.1007/s11063-016-9563-5

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