Advertisement

Nonlinear Dynamics

, Volume 78, Issue 2, pp 1087–1099 | Cite as

On hyperchaos in a small memristive neural network

  • Qingdu Li
  • Song Tang
  • Hongzheng Zeng
  • Tingting Zhou
Original Paper

Abstract

This paper studies a small Hopfield neural network with a memristive synaptic weight. We show that the previous stable network after one weight replaced by a memristor can exhibit rich complex dynamics, such as quasi-periodic orbits, chaos, and hyperchaos, which suggests that the memristor is crucial to the behaviors of neural networks and may play a significant role. We also prove the existence of a saddle periodic orbit, and then present computer-assisted verification of hyperchaos through a homoclinic intersection of the stable and unstable manifolds, which gives a positive answer to an interesting question that whether a 4D memristive system with a line of equilibria can demonstrate hyperchaos.

Keywords

Hyperchaos Homoclinic intersection Small neural networks Memristor 

Notes

Acknowledgments

We are very grateful to the reviewers for their valuable comments and suggestions. This work is supported in part by the National Natural Science Foundation of China (61104150), Science Fund for Distinguished Young Scholars of Chongqing (cstc2013jcyjjq40001), and the Science and Technology Project of Chongqing Education Commission (KJ130517).

References

  1. 1.
    Hopfield, J.J.: Neurons with graded response have collective computational properties like those of 2-state neurons. Proc. Natl. Acad. Sci. USA 81(10), 3088–3092 (1984)CrossRefGoogle Scholar
  2. 2.
    Chua, L., Sbitnev, V., Kim, H.: Hodgkin-Huxley axon is made of memristors. Int. J. Bifurc. Chaos 22(03), 1230011 (2012)CrossRefGoogle Scholar
  3. 3.
    Kim, H., et al.: Neural synaptic weighting with a pulse-based memristor circuit. IEEE Trans. Circuits Syst. I-Regul. Pap. 59(1), 148–158 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chua, L.O., Kang, S.M.: Memristive devices and systems. Proc. IEEE 64(2), 209–223 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chua, L.O.: Memristor—missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)CrossRefGoogle Scholar
  6. 6.
    Strukov, D.B., et al.: The missing memristor found. Nature 453(7191), 80–83 (2008)CrossRefGoogle Scholar
  7. 7.
    Wu, A.L., Zeng, Z.G.: Anti-synchronization control of a class of memristive recurrent neural networks. Commun. Nonlinear Sci. Numer. Simul. 18(2), 373–385 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wu, A.L., Zhang, J., Zeng, Z.G.: Dynamic behaviors of a class of memristor-based Hopfield networks. Phys. Lett. A 375(15), 1661–1665 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Itoh, M., Chua, L.O.: Autoassociative memory cellular neural networks. Int. J. Bifurc. Chaos 20(10), 3225–3266 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pershin, Y.V., Di Ventra, M.: Experimental demonstration of associative memory with memristive neural networks. Neural Netw. 23(7), 881–886 (2010)CrossRefGoogle Scholar
  11. 11.
    Babloyantz, A., Lourenco, C.: Brain chaos and computation. Int. J. Neural Syst. 7(4), 461–471 (1996)CrossRefGoogle Scholar
  12. 12.
    Guevara, M.R., et al.: Chaos in neurobiology. IEEE Trans. Syst. Man Cybern. 13(5), 790–798 (1983)CrossRefzbMATHGoogle Scholar
  13. 13.
    Yang, X.S., Huang, Y.: Complex dynamics in simple Hopfield neural networks. Chaos 16(3), 033114 (2006)CrossRefGoogle Scholar
  14. 14.
    Li, Q.D., Yang, X.S., Yang, F.Y.: Hyperchaos in Hopfield-type neural networks. Neurocomputing 67, 275–280 (2005)CrossRefGoogle Scholar
  15. 15.
    Aihara, K., Takabe, T., Toyoda, M.: Chaotic neural networks. Phys. Lett. A 144(6), 333–340 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bersini, H., Sener, P.: The connections between the frustrated chaos and the intermittency chaos in small Hopfield networks. Neural Netw. 15(10), 1197–1204 (2002)CrossRefGoogle Scholar
  17. 17.
    Das, P.K., Schieve, W.C., Zeng, Z.J.: Chaos in an effective 4-neuron neural network. Phys. Lett. A 161(1), 60–66 (1991)CrossRefGoogle Scholar
  18. 18.
    Zheng, P., Tang, W., Zhang, J.: Dynamic analysis of unstable Hopfield networks. Nonlinear Dyn. 61(3), 399–406 (2010)Google Scholar
  19. 19.
    Buscarino, A., et al.: Memristive chaotic circuits based on cellular nonlinear networks. Int. J. Bifurc. Chaos 22(3), 1250070 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Riaza, R.: Dynamical properties of electrical circuits with fully nonlinear memristors. Nonlinear Anal.-Real World Appl. 12(6), 3674–3686 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cafagna, D., Grassi, G.: On the simplest fractional-order memristor-based chaotic system. Nonlinear Dyn. 70(2), 1185–1197 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Bao, B.C., et al.: Dynamics analysis of chaotic circuit with two memristors. Sci. China-Technol. Sci. 54(8), 2180–2187 (2011)CrossRefzbMATHGoogle Scholar
  23. 23.
    Messias, M., Nespoli, C., VANESSA, A.B.: Hopf bifurcation from lines of equilibria without parameters in memristor oscillators. Int. J. Bifurc. Chaos 20(02), 437–450 (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, X., Chen, G.: Constructing a chaotic system with any number of equilibria. Nonlinear Dyn. 71(3), 429–436 (2013)CrossRefGoogle Scholar
  25. 25.
    Li, Q., Zeng, H., Yang, X.-S.: On hidden twin attractors and bifurcation in the Chua’s circuit. Nonlinear Dyn. 77 (1-2), 255–266 (2014)Google Scholar
  26. 26.
    Bao, B.C., et al.: A simple memristor chaotic circuit with complex dynamics. Int. J. Bifurc. Chaos 21(9), 2629–2645 (2011)CrossRefzbMATHGoogle Scholar
  27. 27.
    Muthuswamy, B.: Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20(5), 1335–1350 (2010)CrossRefzbMATHGoogle Scholar
  28. 28.
    Itoh, M., Chua, L.O.: Memristor Hamiltonian circuits. Int. J. Bifurc. Chaos 21(9), 2395–2425 (2011)CrossRefzbMATHGoogle Scholar
  29. 29.
    Itoh, M., Chua, L.O.: Memristor oscillators. Int. J. Bifurc. Chaos 18(11), 3183–3206 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    El-Sayed, A., et al.: Dynamical behavior, chaos control and synchronization of a memristor-based ADVP circuit. Commun. Nonlinear Sci. Numer. Simul. 18(1), 148–170 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Muthuswamy, B., Kokate, P.P.: Memristor-based chaotic circuits. IETE Tech. Review 26(6), 417–429 (2009)CrossRefGoogle Scholar
  32. 32.
    Li, Q., et al.: Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation. Int. J. Circuit Theory Appl. (2013). doi: 10.1002/cta.1912
  33. 33.
    Li, Q.: A topological horseshoe in the hyperchaotic Rossler attractor. Phys. Lett. A 372(17), 2989–2994 (2008)Google Scholar
  34. 34.
    Li, Q., Tang, S.: Algorithm for finding horseshoes in three-dimensional hyperchaotic maps and its application. Acta Phys. Sin. 62(2), 020510 (2013)Google Scholar
  35. 35.
    Li, Q., Yang, X.-S.: A simple method for finding topological horseshoes. Int. J. Bifurc. Chaos 20(02), 467–478 (2010)CrossRefzbMATHGoogle Scholar
  36. 36.
    Li, Q., Yang, X.-S.: Two kinds of horseshoes in a hyperchaotic neural network. Int. J. Bifurc. Chaos 22(08), 1250200 (2012)CrossRefGoogle Scholar
  37. 37.
    Li, Q., Yang, X.-S., Chen, S.: Hyperchaos in a spacecraft power system. Int. J. Bifurc. Chaos 21(06), 1719–1726 (2011)CrossRefzbMATHGoogle Scholar
  38. 38.
    Zhou, P., Yang, F.: Hyperchaos, chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points. Nonlinear Dyn. 76(1), 473–480 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Alefeld, G.: Inclusion methods for systems of nonlinear equations—the interval Newton method and modifications. In: Herzberger, J. (ed.) Topics in Validated Computations, pp. 7–26. Elsevier Science Publishers, Amsterdam (1994)Google Scholar
  40. 40.
    Zgliczynski, P.: \(C^{1}\) Lohner algorithm. Found. Comput. Math. 2(4), 429–465 (2008)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wilczak, D., Zgliczyński, P.: \( C^{r}\) -Lohner algorithm. (2007) arXiv, preprint arXiv:0704.0720
  42. 42.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer Verlag, Berlin (2003)zbMATHGoogle Scholar
  43. 43.
    Li, Q.-D., Zhou, L., Zhou, H.-W.: Computation for two-dimension unstable manifold of map. J. Chongqing Univ. Posts Telecommun. (Natural Science) 22(3), 339–344 (2010) Google Scholar
  44. 44.
    Yang, X.-S.: Topological horseshoes and computer assisted verification of chaotic dynamics. Int. J. Bifurc. Chaos 19(4), 1127–1145 (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Qingdu Li
    • 1
  • Song Tang
    • 1
  • Hongzheng Zeng
    • 2
  • Tingting Zhou
    • 1
  1. 1.Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of EducationChongqing University of Posts and TelecommunicationsChongqingChina
  2. 2.Research Center of Analysis and Control for Complex SystemsChongqing University of Posts and TelecommunicationsChongqingChina

Personalised recommendations