A Novel Fraction-Based Hopfield Neural Networks

  • Jinrong Hu
  • Jiliu Zhou
  • Yifei Pu
  • Yan Liu
  • Yi Zhang
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 158)


In this paper, we propose a novel fractional-based Hopfield Neural Network (FHNN). The capacitors in the standard Hopfield Neural Network (HNN) with traditional integer order derivatives are replaced by fractance components with fractional order derivatives. From this, continues Hopfield net is extended to the fractional-based net in which fractional order equations describe its dynamical structure. We also prove the stability of FHNN through the Lyapunov energy function. In addition, we analyze the performance of FHNN by performing printed number recognition experiments. The simulation results in comparison with the standard HNN, showed some salient advantages in the fractional-based Hopfield Neural Network containing the higher capacity.


Fractional Calculus Hopfield Neural Networks Lyapunov Energy Function Number Recognition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Westerlund, S.: Dead Matter Has Memory! Causal Consulting, Kalmar, Sweden (2002)Google Scholar
  2. 2.
    Wang, J.C.: Realizations of generalized warburg impedance with RC ladder networks and transmission lines. J. of Electrochem. Soc. 134(8), 1915–1920 (1987)CrossRefGoogle Scholar
  3. 3.
    Nakagava, M., Sorimachi, K.: Basic characteristics of a fractance device. IEICE Trans. Fundamentals E75-A(12), 1814–1818 (1992)Google Scholar
  4. 4.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)MATHGoogle Scholar
  5. 5.
    Jenson, V.G., Jeffreys, G.V.: Mathematical Method in Chemical Engineering, 2nd edn. Academic Press, New York (1977)Google Scholar
  6. 6.
    Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Bifurcation and chaos in noninteger order cellular neural networks. International Journal of Bifurcation and Chaos 8(7), 1527–1539 (1998)MATHCrossRefGoogle Scholar
  7. 7.
    Pu, Y.-F.: Implement Any Fractional Order Multilayer Dynamics Associative Neural Network. In: 6th International Conference on ASIC, ASICON 2005, vol. 2, pp. 638–641 (2005)Google Scholar
  8. 8.
    Boroomand, A., Menhaj, M.B.: Fractional-order Hopfield Neural Networks. In: 15th International Conference on Neural Information Processing of the Asia-Pacific Neural Network Assembly, Auckland, New Zealand (November 2008)Google Scholar
  9. 9.
    Oldham, K.B., Spanie, J.: The Fractional Calculus. Academic Press, New York (1974)MATHGoogle Scholar
  10. 10.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)MATHGoogle Scholar
  11. 11.
    Li, C., Chen, G.: Chaos in fractional order Chen system and its control. Chaos, Solutions & Fractals, 305–311 (2004)Google Scholar
  12. 12.
    Cole, K.S.: Electric Conductance of Biological System. In: Proc. Cold Spring Harbor Symp. Quant. Biol., New York, pp. 107–116 (1993)Google Scholar
  13. 13.
    Mandelbrot, B.B.: Some Noises with 1/f Spectrumm, a Bridge Between Direct Current and White Noise. IEEE Tran. on Info. Theory IT-13(2) (1967)Google Scholar
  14. 14.
    Charef, A., Sun, H.M., Tsao, Y.Y., Onaral, B.: Fractional Systems as Represented by Singularity Function. IEEE Trans. on Automatic Control 37(9), 1465–1470 (1992)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Westerland, S.: Capacitor Theory. IEEE Tran. on Dielectrics and Electrical Insulation 1(5), 826–839 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  • Jinrong Hu
    • 1
  • Jiliu Zhou
    • 1
  • Yifei Pu
    • 1
  • Yan Liu
    • 2
  • Yi Zhang
    • 1
  1. 1.College of Computer ScienceSichuan UniversityChengduChina
  2. 2.College of Electronic and Information EngineeringSichuan UniversityChengduChina

Personalised recommendations