Advertisement

Science in China Series A: Mathematics

, Volume 44, Issue 2, pp 193–200 | Cite as

Chaos and asymptotical stability in discrete-time recurrent neural networks with generalized input-output function

  • Jinliang Wang
  • Luonan Chen
  • Zhujun Jing
Article

Abstract

We theoretically investigate the asymptotical stability, local bifurcations and chaos of discrete-time recurrent neural networks with the form of
$$u_i \left( {t + 1} \right) = ku_i \left( t \right) + \Delta t\left( {\sum\limits_{j = 1}^n {a_{ij} v_j \left( t \right) + a_i } } \right), i = 1,2, \cdots ,n,$$
, where the input-output function is defined as a generalized sigmoid function, such asv i =2/π arctan(π/2μiμi),\(v_i = \frac{2}{\pi }arctan\left( {\frac{\pi }{2}\mu _i u_i } \right)\) and\(v_i = \frac{1}{{1 + e^{ - u_i /\varepsilon } }},\), etc. Numerical simulations are also provided to demonstrate the theoretical results.

Keywords

chaos asymptotical stability bifurcation neural network snap-back repeller 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hopfield, J., Tank, D., Neural computation of decisions in optimization problems, Biological Cybernetics, 1985, 52: 141.MATHMathSciNetGoogle Scholar
  2. 2.
    Hopfield, J., Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA, 1982, 79: 2554.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen, L., Aihara, K., Chaotic simulated annealing by a neural network model with transient chaos, Neural Networks, 1995. 8(6): 915.CrossRefGoogle Scholar
  4. 4.
    Hopfield, J., Neurons with graded response have collective computational properties like those of two-state neurons, Pmc. Natl. Acad. Sci. USA, 1984, 81: 3088.CrossRefGoogle Scholar
  5. 5.
    Li, J. H., Micheland, A. N., Porod, W., Qualitative analysis and synthesis of a class of neural networks, IEEE Trans. on Circuits and Systems, 1988, 35(8): 976.MATHCrossRefGoogle Scholar
  6. 6.
    Gopalsamy, K., He, X. Z., Stability in a symmetric Hopfield nets with transmission delays, Physica D, 1994, 76: 344.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chen, L., Aihara, K., Chaos and asymptotic stability in discrete-time neural networks, Physica D, 1997, 104: 286.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chen, L., Aihara, K., Global searching ability of chaotic neural networks, IEEE Trans. on Circuits and Systems-I: Fundamental Theory and Applications, 1999, 46(8): 974.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Tong, H., Non-linear Time Series-A Dynarnical System Approach, Oxford: Oxford University Press, 1990.Google Scholar
  10. 10.
    Hoppensteadt, F., Izhikevich, E., Weakly Connected Neural Networks, New York: Springer-Verlag, 1997.Google Scholar
  11. 11.
    Marotto, F. R., Snap-back repellers imply chaos inR η, J. Math. Anal. Appl., 1978, 63: 199.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science in China Press 2001

Authors and Affiliations

  • Jinliang Wang
    • 1
  • Luonan Chen
    • 2
  • Zhujun Jing
    • 1
    • 3
  1. 1.Institute of MathematicsChinese Academy of SciencesBeijingChina
  2. 2.Osaka Sangyo UniversityOsakaJapan
  3. 3.Department of MathematicsHunan Normal UniversityChangshaChina

Personalised recommendations