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Bifurcation Analysis of Delayed Complex-Valued Neural Network with Diffusions

  • Tao Dong
  • Jiaqi Bai
  • Lei Yang
Article

Abstract

In this paper, a class of delayed complex-valued neural network with diffusion under Dirichlet boundary conditions is considered. By using the properties of the Laplacian operator and separating the neural network into real and imaginary parts, the corresponding characteristic equation of neural network is obtained. Then, the dynamical behaviors including the local stability, the existence of Hopf bifurcation of zero equilibrium are investigated. Furthermore, by using the normal form theory and center manifold theorem of the partial differential equation, the explicit formulae which determine the direction of bifurcations and stability of bifurcating periodic solutions are obtained. Finally, a numerical simulation is carried out to illustrate the results.

Keywords

Complex-valued Neural network Diffusion Stability Hopf bifurcation Time delay 

Notes

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 61503310 and 61402381, in part by the supported by the Fundamental Research Funds for the Central Universities under Grant XDJK2017D170 and XDJK2017D183, in part by China Postdoctoral Foundation under Grant 2016M600720, in part by Chongqing Postdoctoral Project under Grant Xm2016003.

References

  1. 1.
    Hirose A (2012) Complex-valued neural networks. Springer, BerlinCrossRefMATHGoogle Scholar
  2. 2.
    Ding X, Cao J, Zhao X et al (2017) Finite-time stability of fractional-order complex-valued neural networks with time delays. Neural Process Lett 46(2):1–20CrossRefGoogle Scholar
  3. 3.
    Lee DL (2006) Improvements of complex-valued Hopfield associative memory by using generalized projection rules. IEEE Trans Neural Netw 17(5):1341–1347CrossRefGoogle Scholar
  4. 4.
    Gong W, Liang J, Zhang C et al (2016) Nonlinear measure approach for the stability analysis of complex-valued neural networks. Neural Process Lett 44(2):539–554CrossRefGoogle Scholar
  5. 5.
    Nitta T (2003) Solving the XOR problem and the detection of symmetry using a single complex-valued neuron. Neural Netw 16(8):1101–1105CrossRefGoogle Scholar
  6. 6.
    Hu J, Wang J (2012) Global stability of complex-valued recurrent neural networks with time-delays. IEEE Trans Neural Netw Learn Syst 23(6):853–865CrossRefGoogle Scholar
  7. 7.
    Dong T, Liao X, Wang A (2015) Stability and Hopf bifurcation of a complex-valued neural network with two time delays. Nonlinear Dyn 82(1–2):173–184MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Wang Z, Huang L (2016) Global stability analysis for delayed complex-valued BAM neural networks. Neurocomputing 173:2083–2089CrossRefGoogle Scholar
  9. 9.
    Song Q, Zhao Z (2016) Stability criterion of complex-valued neural networks with both leakage delay and time-varying delays on time scales. Neurocomputing 171:179–184CrossRefGoogle Scholar
  10. 10.
    Dong T, Liao X (2013) Hopf-Pitchfork bifurcation in a simplified BAM neural network model with multiple delays. J Comput Appl Math 253:222–234MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Li Y, Liao X, Li H (2016) Global attracting sets of non-autonomous and complex-valued neural networks with time-varying delays. Neurocomputing 173:994–1000CrossRefGoogle Scholar
  12. 12.
    Dong T, Liao X (2013) Bogdanov–Takens bifurcation in a tri-neuron BAM neural network model with multiple delays. Nonlinear Dyn 71(3):583–595MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Wang Z, Huang L (2016) Global stability analysis for delayed complex-valued BAM neural networks. Neurocomputing 173:2083–2089CrossRefGoogle Scholar
  14. 14.
    Xu W, Cao J, Xiao M et al (2015) A new framework for analysis on stability and bifurcation in a class of neural networks with discrete and distributed delays. IEEE Trans Cybern 45(10):2224–2236CrossRefGoogle Scholar
  15. 15.
    Li Y (2017) Impulsive synchronization of stochastic neural networks via controlling partial states. Neural Process Lett 46(1):59–69CrossRefGoogle Scholar
  16. 16.
    Lu J, Wang Z, Cao J et al (2012) Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay. Int J Bifurc Chaos 22(7):1250176CrossRefMATHGoogle Scholar
  17. 17.
    Zhao H, Yuan J, Zhang X (2015) Stability and bifurcation analysis of reaction–diffusion neural networks with delays. Neurocomputing 147:280–290CrossRefGoogle Scholar
  18. 18.
    Ren SY, Wu J, Wei PC (2017) Passivity and pinning passivity of coupled delayed reaction–diffusion neural networks with Dirichlet boundary conditions. Neural Process Lett 45(3):869–885CrossRefGoogle Scholar
  19. 19.
    Sheng Y, Zeng Z (2017) Synchronization of stochastic reaction–diffusion neural networks with Dirichlet boundary conditions and unbounded delays. Neural Netw 93:89–98CrossRefGoogle Scholar
  20. 20.
    Lu JG (2008) Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions. Chaos Solitons Fractals 35(1):116–125MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Li R, Cao J (2016) Stability analysis of reaction–diffusion uncertain memristive neural networks with time-varying delays and leakage term. Appl Math Comput 278:54–69MathSciNetGoogle Scholar
  22. 22.
    Zheng CD, Zhang Y, Wang Z (2014) Stability analysis of stochastic reaction–diffusion neural networks with Markovian switching and time delays in the leakage terms. Int J Mach Learn Cybern 5(1):3–12CrossRefGoogle Scholar
  23. 23.
    Tian X, Xu R, Gan Q (2015) Hopf bifurcation analysis of a BAM neural network with multiple time delays and diffusion. Appl Math Comput 266:909–926MathSciNetGoogle Scholar
  24. 24.
    Tian X, Xu R (2017) Stability and Hopf bifurcation of time fractional Cohen–Grossberg neural networks with diffusion and time delays in leakage terms. Neural Process Lett 45(2):593–614CrossRefGoogle Scholar
  25. 25.
    Dong T, Xu W, Liao X (2017) Hopf bifurcation analysis of reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay. Nonlinear Dyn 89(4):2329–2345MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Dong T, Xia L (2017) Stability and Hopf bifurcation of a reaction–diffusion neutral neuron system with time delay. Int J Bifurc Chaos 27(14):1250176MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ruan S, Wei J (2003) On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn Contin Discrete Impuls Syst Series A 10:863–874MathSciNetMATHGoogle Scholar
  28. 28.
    Wu J (2012) Theory and applications of partial functional differential equations. Springer, BerlinGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Electronics and Information EngineeringSouthwest UniversityChongqingPeople’s Republic of China
  2. 2.National Local Joint Engineering Laboratory of Intelligent Transmission and Control TechnologySouthwest UniversityChongqingPeople’s Republic of China
  3. 3.Information CenterChongqing Changan Automobile Company LimitedChongqingPeople’s Republic of China

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