Nonlinear Dynamics

, Volume 93, Issue 2, pp 611–627 | Cite as

Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with discontinuous memductance function

  • Yingjie Fan
  • Xia Huang
  • Zhen Wang
  • Yuxia Li
Original Paper


In this paper, a simplified memristor-based fractional-order neural network (MFNN) with discontinuous memductance function is proposed. It is essentially a switched system with irregular switching laws and consists of eight fractional-order neural network (FNN) subsystems. The nonlinear dynamics of the simplified MFNN including equilibrium points and their stability, bifurcation and chaos is investigated analytically and numerically. In particular, the effect of the switching jump on the dynamics of the simplified MFNN is explored for the first time. Taking the fractional order, the memristive connection weight or the switching jump as the bifurcation parameter, the dynamics such as chaotic motion, tangent bifurcation and intermittent chaos is identified over a wide range of some specified parameter. It can be seen that the incorporation of the memristors greatly improves and enriches the dynamics of the corresponding FNN. Different from the period-doubling route to chaos, this paper reveals that the mechanism behind the emergence of chaos for the simplified MFNN is the intermittency route to chaos. In particular, for some typical parameter, the existence of chaotic attractors is verified with the phase portraits, bifurcation diagrams, Poincaré sections and maximum Lyapunov exponents, respectively. This paper not only provides a way of designing chaotic MFNN with discontinuous memductance function but also suggests a possible method of generating more complicated chaotic attractors, such as multi-scroll or multi-wing attractors.


Memristor Fractional-order neural networks Stability Nonlinear dynamics Chaos Switching jump 


  1. 1.
    Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)zbMATHGoogle Scholar
  2. 2.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives and Some of Their Applications. Naukai Technika, Minsk (1987)zbMATHGoogle Scholar
  3. 3.
    Abdelouahab, M.S., Lozi, R., Chua, L.: Memfractance: a mathematical paradigm for circuit elements with memory. Int. J. Bifurcat. Chaos 24(09), 1430023 (2014)CrossRefzbMATHGoogle Scholar
  4. 4.
    Westerlund, S., Ekstam, L.: Capacitor theory. IEEE Trans. Dielectr. Electr. Insul. 1(5), 826–839 (1994)CrossRefGoogle Scholar
  5. 5.
    Boroomand, A., Menhaj, M.: Fractional-order Hopfield neural networks. Lect. Notes Comput. Sci. 5506, 883–890 (2009)CrossRefGoogle Scholar
  6. 6.
    Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci. 11(11), 1335–1342 (2008)CrossRefGoogle Scholar
  7. 7.
    Huang, X., Zhao, Z., Wang, Z., Li, Y.: Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing 94, 13–21 (2012)CrossRefGoogle Scholar
  8. 8.
    Wang, Z., Wang, X., Li, Y., Huang, X.: Stability and Hopf bifurcation of fractional-order complex-valued single neuron model with time delay. Int. J. Bifurcat. Chaos 27(13), 1750209 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Netw. 32, 245–256 (2012)CrossRefzbMATHGoogle Scholar
  10. 10.
    Tao, B., Xiao, M., Sun, Q., Cao, J.: Hopf bifurcation analysis of a delayed fractional-order genetic regulatory network model. Neurocomputing 275, 677–686 (2018)CrossRefGoogle Scholar
  11. 11.
    Xiao, M., Zheng, W.X., Jiang, G., Cao, J.: Undamped oscillations generated by hopf bifurcations in fractional-order recurrent neural networks with caputo derivative. IEEE Trans. Neural Netw. Learn. Syst. 26(12), 3201–3214 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chua, L.O.: Memristor-the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)CrossRefGoogle Scholar
  13. 13.
    Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S.: The missing memristor found. Nature 453(7191), 80–83 (2008)CrossRefGoogle Scholar
  14. 14.
    Kvatinsky, S., Friedman, E.G., Kolodny, A., Weiser, U.C.: TEAM: threshold adaptive memristor model. IEEE Trans. Circuits Syst. I Regul. Pap. 60(1), 211–221 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kvatinsky, S., Ramadan, M., Friedman, E.G., Kolodny, A.: VTEAM: a general model for voltage-controlled memristors. IEEE Trans. Circuits Syst. II Express Briefs 62(8), 786–790 (2015)CrossRefGoogle Scholar
  16. 16.
    Pickett, M.D., Strukov, D.B., Borghetti, J.L., Yang, J.J., Snider, G.S., Stewart, D.R., Williams, R.S.: Switching dynamics in titanium dioxide memristive devices. J. Appl. Phys. 106(7), 074508 (2009)CrossRefGoogle Scholar
  17. 17.
    Itoh, M., Chua, L.O.: Memristor oscillators. Int. J. Bifurcat. Chaos 18(11), 3183–3206 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chen, M., Li, M., Yu, Q., Bao, B., Xu, Q., Wang, J.: Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit. Nonlinear Dyn. 81(1–2), 215–226 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Muthuswamy, B., Chua, L.O.: Simplest chaotic circuit. Int. J. Bifurcat. Chaos 20(05), 1567–1580 (2010)CrossRefGoogle Scholar
  20. 20.
    Kengne, J., Tabekoueng, Z.N., Tamba, V.K., Negou, A.N.: Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit. Chaos 25(10), 103126 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kengne, J., Negou, A.N., Tchiotsop, D.: Antimonotonicity, chaos and multiple attractors in a novel autonomous memristor-based jerk circuit. Nonlinear Dyn. 88(4), 2589–2608 (2017)CrossRefGoogle Scholar
  22. 22.
    Li, Y., Huang, X., Song, Y., Lin, J.: A new fourth-order memristive chaotic system and its generation. Int. J. Bifurcat. Chaos 25(11), 1550151 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Teng, L., Iu, H.H., Wang, X., Wang, X.: Chaotic behavior in fractional-order memristor-based simplest chaotic circuit using fourth degree polynomial. Nonlinear Dyn. 77(1–2), 231–241 (2014)CrossRefGoogle Scholar
  24. 24.
    Huang, X., Jia, J., Li, Y., Wang, Z.: Complex nonlinear dynamics in fractional and integer order memristor-based systems. Neurocomputing 218, 296–306 (2016)CrossRefGoogle Scholar
  25. 25.
    Rajagopal, K., Karthikeyan, A., Srinivasan, A.: Dynamical analysis and FPGA implementation of a chaotic oscillator with fractional-order memristor components. Nonlinear Dyn. 91(3), 1491–1512 (2018)CrossRefGoogle Scholar
  26. 26.
    Si, G., Diao, L., Zhu, J.: Fractional-order charge-controlled memristor: theoretical analysis and simulation. Nonlinear Dyn. 87(4), 2625–2634 (2017)CrossRefGoogle Scholar
  27. 27.
    Hu, W., Ding, D., Zhang, Y., Wang, N., Liang, D.: Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system. Optik 130, 189–200 (2017)CrossRefGoogle Scholar
  28. 28.
    Cafagna, D., Grassi, G.: On the simplest fractional-order memristor-based chaotic system. Nonlinear Dyn. 70(2), 1185–1197 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Petras, I.: Fractional-order memristor-based Chua’s circuit. IEEE Trans. Circuits Syst. II Express Briefs 57(12), 975–979 (2010)CrossRefGoogle Scholar
  30. 30.
    Jo, S.H., Chang, T., Ebong, I., Bhadviya, B.B., Mazumder, P., Lu, W.: Nanoscale memristor device as synapse in neuromorphic systems. Nano Lett. 10(4), 1297–1301 (2010)CrossRefGoogle Scholar
  31. 31.
    Thomas, A.: Memristor-based neural networks. J. Phys. D Appl. Phys. 46(9), 093001 (2013)CrossRefGoogle Scholar
  32. 32.
    Li, L., Wang, Z., Li, Y., Lu, J., Shen, H.: Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays. Appl. Math. Comput. (2018).
  33. 33.
    Li, Q., Tang, S., Zeng, H., Zhou, T.: On hyperchaos in a small memristive neural network. Nonlinear Dyn. 78(2), 1087–1099 (2014)CrossRefzbMATHGoogle Scholar
  34. 34.
    Pham, V.T., Jafari, S., Vaidyanathan, S., Volos, C., Wang, X.: A novel memristive neural network with hidden attractors and its circuitry implementation. Sci. China Technol. Sci. 59(3), 358–363 (2016)CrossRefGoogle Scholar
  35. 35.
    Jiang, P., Zeng, Z., Chen, J.: On the periodic dynamics of memristor-based neural networks with leakage and time-varying delays. Neurocomputing 219, 163–173 (2017)CrossRefGoogle Scholar
  36. 36.
    Wen, S., Zeng, Z., Huang, T.: Exponential stability analysis of memristor-based recurrent neural networks with time-varying delays. Neurocomputing 97, 233–240 (2012)CrossRefGoogle Scholar
  37. 37.
    Hu, J., Wang, J.: Global uniform asymptotic stability of memristor-based recurrent neural networks with time delays. In: The 2010 International Joint Conference on Neural Networks (IJCNN), pp. 1–8 (2010)Google Scholar
  38. 38.
    Wen, S., Huang, T., Zeng, Z., Chen, Y., Li, P.: Circuit design and exponential stabilization of memristive neural networks. Neural Netw. 63, 48–56 (2015)CrossRefzbMATHGoogle Scholar
  39. 39.
    Abdurahman, A., Jiang, H., Teng, Z.: Finite-time synchronization for memristor-based neural networks with time-varying delays. Neural Netw. 69, 20–28 (2015)CrossRefGoogle Scholar
  40. 40.
    Li, N., Cao, J.: New synchronization criteria for memristor-based networks: adaptive control and feedback control schemes. Neural Netw. 61, 1–9 (2015)CrossRefzbMATHGoogle Scholar
  41. 41.
    Zhang, G., Shen, Y., Wang, L.: Global anti-synchronization of a class of chaotic memristive neural networks with time-varying delays. Neural Netw. 46, 1–8 (2013)CrossRefzbMATHGoogle Scholar
  42. 42.
    Cao, J., Li, R.: Fixed-time synchronization of delayed memristor-based recurrent neural networks. Sci. China Inf. Sci. 60(3), 032201 (2017)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Guo, Z., Wang, J., Yan, Z.: Attractivity analysis of memristor-based cellular neural networks with time-varying delays. IEEE Trans. Neural Netw. Learn. Syst. 25(4), 704–717 (2014)CrossRefGoogle Scholar
  44. 44.
    Wen, S., Zeng, Z., Huang, T., Chen, Y.: Passivity analysis of memristor-based recurrent neural networks with time-varying delays. J. Franklin Inst. 350(8), 2354–2370 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Chen, J., Zeng, Z., Jiang, P.: Global Mittag–Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw. 51, 1–8 (2014)CrossRefzbMATHGoogle Scholar
  46. 46.
    Chen, J., Chen, B., Zeng, Z.: Global uniform asymptotic fixed deviation stability and stability for delayed fractional-order memristive neural networks with generic memductance. Neural Netw. 98, 65–75 (2018)CrossRefGoogle Scholar
  47. 47.
    Velmurugan, G., Rakkiyappan, R.: Hybrid projective synchronization of fractional-order memristor-based neural networks with time delays. Nonlinear Dyn. 83(1), 419–432 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Bao, H., Park, J.H., Cao, J.: Adaptive synchronization of fractional-order memristor-based neural networks with time delay. Nonlinear Dyn. 82(3), 1343–1354 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Zhang, L., Yang, Y., Wang, F.: Lag synchronization for fractional-order memristive neural networks via period intermittent control. Nonlinear Dyn (2017). zbMATHGoogle Scholar
  50. 50.
    Choi, H.D., Ahn, C.K., Karimi, H.R., Lim, M.T.: Filtering of discrete-time switched neural networks ensuring exponential dissipative and \(l_{2}-l_{\infty }\) performances. IEEE Trans. Cybern. 47(10), 3195–3207 (2017)CrossRefGoogle Scholar
  51. 51.
    Chua, L.O., Komuro, M., Matsumoto, T.: The double scroll family. IEEE Trans. Circuits Syst. 33(11), 1072–1118 (1986)CrossRefzbMATHGoogle Scholar
  52. 52.
    Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, vol. 2, pp. 963–968. Lille (1996)Google Scholar
  53. 53.
    Chua, L.O.: Resistance switching memories are memristors. Appl. Phys. A 102(4), 765–783 (2011)CrossRefzbMATHGoogle Scholar
  54. 54.
    Wu, A., Zhang, J., Zeng, Z.: Dynamic behaviors of a class of memristor-based Hopfield networks. Phys. Lett. A 375(15), 1661–1665 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Wu, A., Zeng, Z.: Anti-synchronization control of a class of memristive recurrent neural networks. Commun. Nonlinear Sci. Numer. Simul. 18(2), 373–385 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Yu, J., Hu, C., Jiang, H., Fan, X.: Projective synchronization for fractional neural networks. Neural Netw. 49, 87–95 (2014)CrossRefzbMATHGoogle Scholar
  57. 57.
    Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Kim, H., Eykholt, R., Salas, J.D.: Nonlinear dynamics, delay times, and embedding windows. Physica D 127(1), 48–60 (1999)CrossRefzbMATHGoogle Scholar
  60. 60.
    Yalcin, M.E., Suykens, J.A., Vandewalle, J.: True random bit generation from a double-scroll attractor. IEEE Trans. Circuits Syst. I Regul. Pap. 51(7), 1395–1404 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Wang, Z., Huang, X., Shi, G.: Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. Comput. Math. Appl. 62(3), 1531–1539 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    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

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Electrical Engineering and AutomationShandong University of Science and TechnologyQingdaoChina
  2. 2.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina

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