Numerical Analysis of a Chaotic Delay Recurrent Neural Network with Four Neurons

  • Haigeng Luo
  • Xiaodong Xu
  • Xiaoxin Liao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3971)


Complex dynamical behavior in a four-neuron recurrent neural network with discrete delays is investigated in this paper. The dissipativity of the system and stability of equilibrium point are studied by mens of Lyapunov theory. Stable fixed point, periodic and quasi-periodic orbits, and chaotic motion are observed in system via numerical calculation. With the change of the slope and threshold of activation function, as well as time delay and synaptic weight, the system passes from stable to periodic and then to chaotic regimes. Interestingly, the system returns to periodic or stable regimes by further changing these parameter values. Furthermore, some numerical evidences, such as phase portraits, bifurcation diagrams, power spectrum density, are given to confirm chaos.


Neural Network Model Phase Portrait Bifurcation Diagram Recurrent Neural Network Synaptic Weight 
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  1. 1.
    Shen, Y., Zhao, G.Y., Jiang, M.H., Mao, X.R.: Stochastic Lotka-Volterra Competitive Systems with Variable Delay. In: Huang, D.-S., Zhang, X.-P., Huang, G.-B. (eds.) ICIC 2005. LNCS, vol. 3645, pp. 238–247. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Zeng, Z.G., Wang, J., Liao, X.X.: Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates. IEEE Trans. Circuits and Systems I 51, 2313–2324 (2004)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Zeng, Z.G., Huang, D.S., Wang, Z.F.: Global Stability of A General Class of Discrete-time Recurrent Neural Networks. Neural Processing Letters 22, 33–47 (2005)CrossRefGoogle Scholar
  4. 4.
    Shen, Y., Zhao, G.Y., Jiang, M.H., Hu, S.G.: Stochastic High-order Hopfield Neural Networks. In: Wang, L., Chen, K., S. Ong, Y. (eds.) ICNC 2005. LNCS, vol. 3610, pp. 740–749. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Shen, Y., Jiang, M.H., Liao, X.X.: Global Exponential Stability of Cohen-Grossberg Neural Networks with Time-varying Delays and Continuously Distributed Delays. In: Wang, J., Liao, X.-F., Yi, Z. (eds.) ISNN 2005. LNCS, vol. 3496, pp. 156–161. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Das, A., das, P., Roy, A.B.: Chaos in Three Dimensional Neural Network. Applied Mathematical Modelling 24, 511–522 (2000)MATHCrossRefGoogle Scholar
  7. 7.
    Das, A., das, P., Roy, A.B.: Chaos in A Three-dimensional Model of Neural Network. Int. J. of Bifur. and Chaos 12, 2271–2281 (2002)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Li, C.G., Yu, J.B., Liao, X.F.: Chaos in A Three-neuron Hysteresis Hopfield-type Neural Network. Physics Letters A 285, 368–372 (2001)MATHCrossRefGoogle Scholar
  9. 9.
    Liao, X.F., Wong, K.W., Leung, C.S., Wu, Z.F.: Hopf Bifurcation and Chaos in A Single Delayed Neuron Equation with Non-monotonic Activation Function. Chaos, Solitons and Fractals 12, 1535–1547 (2001)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Zhou, S.B., Liao, X.F., Yu, J.B., Wong, K.W.: Chaos and Its Synchronization in Two-neuron Systems with Discrete Delays. Chaos, Solitons and Fractals 21, 133–142 (2004)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Haigeng Luo
    • 1
  • Xiaodong Xu
    • 2
  • Xiaoxin Liao
    • 1
  1. 1.Department of Control Science and EngineeringHuazhong University of Science and TechnologyWuhanChina
  2. 2.College of Public AdministrationHuazhong University of Science and TechnologyWuhanChina

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