Advertisement

The European Physical Journal Special Topics

, Volume 227, Issue 7–9, pp 811–820 | Cite as

A Hopfield neural network with multiple attractors and its FPGA design

  • Karthikeyan Rajagopal
  • Jesus M. Munoz-Pacheco
  • Viet-Thanh Pham
  • Duy Vo Hoang
  • Fawaz E. Alsaadi
  • Fuad E. Alsaadi
Regular Article
  • 39 Downloads
Part of the following topical collections:
  1. Nonlinear Effects in Life Sciences

Abstract

Neural network is important for a wide range of applications. Especially, a small neural network can display various complex behaviors. In this work, the investigations of a Hopfield neural network and its field programmable gate array (FPGA) implementation have been reported. The considered Hopfield neural network is simple because it includes only three neurons. It is interesting that we observed chaos and numerous coexisting attractors in such a network. In addition, the network has been implemented via an FPGA platform to verify its feasibility.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C.M. Bishop et al., Neural Network for Pattern Recognition (Clarendon Press, Oxford, 1995) Google Scholar
  2. 2.
    S. Haykin et al., Neural Network: A Comprehensive Foundation (Prentice Hall, New Jersey, 1998) Google Scholar
  3. 3.
    W. Yu et al., Inf. Sci. 158, 131 (2004) CrossRefGoogle Scholar
  4. 4.
    J. Rubio, W. Yu, Neurocomputing 70, 2460 (2007) CrossRefGoogle Scholar
  5. 5.
    Q. Wang, Z.Y.J. Ma, Chaos Soliton. Fract. 70, 19 (2013) ADSGoogle Scholar
  6. 6.
    T. Brosch, H. Neumann, Neural Comput. 26, 2735 (2014) MathSciNetCrossRefGoogle Scholar
  7. 7.
    H.X. Qin, J. Ma, W.Y. Jin, C.N. Wang, Sci. China Tech. Sci. 57, 936 (2014) CrossRefGoogle Scholar
  8. 8.
    L. Laskowski et al., Neural Comput. Appl. 23, 2435 (2013) CrossRefGoogle Scholar
  9. 9.
    M. Itoh, L.O. Chua, Int. J. Bifurcat. Chaos 20, 3225 (2010) CrossRefGoogle Scholar
  10. 10.
    S. Wen, Z. Zeng, T. Huang, Q. Meng, W. Yao, IEEE Trans. Neural Netw. Learn. Syst. 26, 1493 (2015) MathSciNetCrossRefGoogle Scholar
  11. 11.
    J.J. Hopfield et al., Proc. Natl. Acad. Sci. USA 81, 3088 (1984) ADSCrossRefGoogle Scholar
  12. 12.
    X.S. Yang, Y. Huang, Chaos 16, 033114 (2006) ADSCrossRefGoogle Scholar
  13. 13.
    Y. Huang, W.S. Yang, Neurocomputing 69, 1787 (2006) CrossRefGoogle Scholar
  14. 14.
    W.Z. Huang, Y. Huang, Appl. Math. Comp. 206, 1 (2008) CrossRefGoogle Scholar
  15. 15.
    W.Z. Huang, Y. Huang, Int. J. Bifurcat. Chaos 21, 885 (2011) CrossRefGoogle Scholar
  16. 16.
    M. Akhmet, M.O. Fen, Neurocomputing 145, 230 (2014) CrossRefGoogle Scholar
  17. 17.
    G. Pajeras, J.M. Cruz, J. Aranda, Pattern Recognit. 31, 561 (1998) CrossRefGoogle Scholar
  18. 18.
    J. Yang, L.D. Wang, Y. Wang, T.T. Guo, Neurocomputing 227, 142 (2016) CrossRefGoogle Scholar
  19. 19.
    H. Bersini et al., Neural Netw. 11, 1017 (1998) CrossRefGoogle Scholar
  20. 20.
    Q. Yuan, Q.D. Li, X.S. Yang, Chaos Soliton. Fract. 39, 1522 (2009) ADSCrossRefGoogle Scholar
  21. 21.
    J. Li, F. Liu, Z.H. Guan, T. Li, Neurocomputing 117, 33 (2013) CrossRefGoogle Scholar
  22. 22.
    P.C. Rech et al., Int. J. Mach. Learn. Cybern. 6, 1 (2015) CrossRefGoogle Scholar
  23. 23.
    H. Bersini, P. Sener, Neural Netw. 15, 1197 (2002) CrossRefGoogle Scholar
  24. 24.
    X.S. Yang, Q. Yuan, Neurocomputing 69, 232 (2005) CrossRefGoogle Scholar
  25. 25.
    Q. Li, X.S. Yang, F. Yang, Neurocomputing 67, 275 (2005) CrossRefGoogle Scholar
  26. 26.
    P.S. Zheng, W.S. Tang, J.X. Zhang, Neurocomputing 73, 2280 (2005) CrossRefGoogle Scholar
  27. 27.
    Q.D. Li, S. Tang, H.Z. Zeng, T.T. Zhou, Nonlinear Dyn. 78, 1087 (2014) CrossRefGoogle Scholar
  28. 28.
    V.T. Pham, S. Jafari, S. Vaidyanathan, C.K. Volos, X. Wang, Sci. China Tech. Sci. 59, 358 (2016) CrossRefGoogle Scholar
  29. 29.
    B. Bao, H. Qian, Q. Xu, M. Chen, J. Wang, Y. Yu, Front. Comput. Neurosci. 11, 81 (2017) CrossRefGoogle Scholar
  30. 30.
    B. Bao, H. Qian, J. Wang, Q. Xu, M. Chen, H. Wu, Y. Yu, Nonlinear Dyn. 90, 2359 (2017) CrossRefGoogle Scholar
  31. 31.
    A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16, 285 (1985) ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    C. Hens, S.K. Dana, U. Feudel, Chaos 25, 053112 (2015) ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Z. Zeng, T. Huang, W. Zheng, IEEE Trans. Neural Netw. 21, 1371 (2010) CrossRefGoogle Scholar
  34. 34.
    J. Kengne, J.C. Chedjou, M. Kom, K. Kyamakya, V.K. Tamba, Nonlinear Dyn. 76, 1119 (2014) CrossRefGoogle Scholar
  35. 35.
    Z.T. Zhusubaliyev, E. Mosekilde, A.N. Churilov, A. Medvedev, Eur. Phys. J. Special Topics 224, 1519 (2015) ADSCrossRefGoogle Scholar
  36. 36.
    V. Vaithianathan, J. Veijun, IEEE Trans. Circuits Syst. I 46, 405 (1999) CrossRefGoogle Scholar
  37. 37.
    J. Kengne et al., Int. J. Bifurcat. Chaos 25, 1550052 (2015) MathSciNetCrossRefGoogle Scholar
  38. 38.
    A. Chudzid, P. Perlikowski, A. Stefanski, T. Kapitaniak, Int. J. Bifurcat. Chaos 21, 1907 (2011) CrossRefGoogle Scholar
  39. 39.
    Z.T. Zhusubaliyev, E. Mosekilde, Math. Comput. Simul. 109, 32 (2015) CrossRefGoogle Scholar
  40. 40.
    T. Malashchenko, A. Shilnikov, G. Cymbalyuk, PLoS One 6, e21782 (2011) ADSCrossRefGoogle Scholar
  41. 41.
    T. Malashchenko, A. Shilnikov, G. Cymbalyuk, Phys. Rev. E 84, 041910 (2011) ADSCrossRefGoogle Scholar
  42. 42.
    N. Stankevich, E. Mosekilde, Chaos 27, 123101 (2017) ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    A. Buscarino, L. Fortuna, M. Frasca, L.V. Gambuzza, Chaos 22, 023136 (2012) ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    B. Muthuswamy, S. Banerjee, A Route to Chaos Using FPGAs (Springer, Berlin, 2015) Google Scholar
  45. 45.
    A. Senouci, H. Bouhedjeur, K. Tourche, A. Boukabou, AEU Int. J. Electron. Commun. 82, 211 (2017) CrossRefGoogle Scholar
  46. 46.
    E. Tlelo-Cuautle, A.D. Pano-Azucena, J.J. Rangel-Magdaleno, Nonlinear Dyn. 85, 2143 (2016) CrossRefGoogle Scholar
  47. 47.
    E. Dong, Z. Liang, S. Du, Z. Chen, Nonlinear Dyn. 83, 623 (2016) CrossRefGoogle Scholar
  48. 48.
    E. Tlelo-Cuautle, J.J. Rangel-Magdaleno, A.D. Pano-Azucena, P.J. Obeso-Rodelo, J.C. Nunez-Perez, Commun. Nonlinear Sci. Numer. Simul. 27, 66 (2015) ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    K. Rajagopal, A. Akgul, S. Jafari, A. Karthikeyan, I. Koyuncu, Chaos Soliton. Fract. 103, 476 (2017) ADSCrossRefGoogle Scholar
  50. 50.
    E. Tlelo-Cuautle, L.G. de la Fraga, V.T. Pham, C. Volos, S. Jafari, A. de Jesus Quintas-Valles, Nonlinear Dyn. 89, 1129 (2017) CrossRefGoogle Scholar
  51. 51.
    J.M. Bahi, X. Fang, C. Guyeux, L. Larger, Appl. Math. Inf. Sci. 7, 2175 (2013) MathSciNetCrossRefGoogle Scholar
  52. 52.
    A. Akgul, H. Calgan, I. Koyuncu, I. Pehlivan, A. Istanbullu, Nonlinear Dyn. 84, 481 (2016) CrossRefGoogle Scholar
  53. 53.
    Y.M. Xu, L.D. Wang, S.K. Duan, Acta Phys. Sin. 65, 120503 (2016) Google Scholar
  54. 54.
    W. Guang-Yi, B. Xu-Lei, W. Zhong-Lin, Chin. Phys. B 17, 3596 (2008) ADSCrossRefGoogle Scholar
  55. 55.
    K. Rajagopal, A. Karthikeyan, P. Duraisamy, Complexity 2017, 1 (2017) Google Scholar
  56. 56.
    D. Valli, B. Muthuswamy, S. Banerjee, M.R.K. Ariffin, A.W.A. Wahad, K. Ganesan, C.K. Subramaniam, J. Kurths, Eur. Phys. J. Special Topics 223, 1465 (2014) ADSCrossRefGoogle Scholar
  57. 57.
    K. Rajagopal, L. Guessas, S. Vaidyanathan, A. Karthikeyan, A. Srinivasan, Math. Probl. Eng. 2017, 1 (2017) CrossRefGoogle Scholar
  58. 58.
    K. Rajagopal, L. Guessas, A. Karthikeyan, A. Srinivasan, G. Adam, Complexity 2017, 1 (2017) Google Scholar
  59. 59.
    K. Rajagopal, A. Karthikeyan, A. Srinivasan, Nonlinear Dyn. 87, 2281 (2017) CrossRefGoogle Scholar
  60. 60.
    K. Rajagopal, G. Laarem, A. Karthikeyan, A. Srinivasan, Adv. Differ. Equ. 2017, 273 (2017) CrossRefGoogle Scholar
  61. 61.
    A. Karthikeyan, K. Rajagopal, Complexity 2017, 1 (2017) MathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Karthikeyan Rajagopal
    • 1
  • Jesus M. Munoz-Pacheco
    • 2
  • Viet-Thanh Pham
    • 3
  • Duy Vo Hoang
    • 3
  • Fawaz E. Alsaadi
    • 4
  • Fuad E. Alsaadi
    • 5
  1. 1.Center for Nonlinear Dynamics, College of Engineering, Defence UniversityBishoftuEthiopia
  2. 2.Faculty of Electronics Sciences, Autonomous University of PueblaPueblaMexico
  3. 3.Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical & Electronics Engineering, Ton Duc Thang UniversityHo Chi Minh CityVietnam
  4. 4.Department of Information Technology, Faculty of Computing and ITKing Abdulaziz UniversityJeddahSaudi Arabia
  5. 5.Department of Electrical and Computer Engineering, Faculty of EngineeringKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations