Nonlinear Dynamics

, Volume 96, Issue 3, pp 1879–1894 | Cite as

Hidden extreme multistability and dimensionality reduction analysis for an improved non-autonomous memristive FitzHugh–Nagumo circuit

  • Han BaoEmail author
  • Wenbo Liu
  • Mo Chen
Original Paper


Due to the introduction of ideal memristors, extreme multistability has been found in many autonomous memristive circuits. However, such extreme multistability has not yet been reported in a non-autonomous memristive circuit. To this end, this paper presents an improved non-autonomous memristive FitzHugh–Nagumo circuit that possesses a smooth hyperbolic tangent memductance nonlinearity, from which coexisting infinitely many attractors are obtained. By utilizing voltage–current circuit model, a three-dimensional non-autonomous dynamical model is established, based on which the initial-dependent dynamics is explored by numerical plots and extreme multistability is thereby exhibited. To confirm that the improved non-autonomous memristive circuit operates in hidden oscillating patterns, an accurate two-dimensional non-autonomous dimensionality reduction model with initial-related parameters is further built by using incremental integral transformation, upon which stability analysis and bifurcation behaviors are elaborated. Because the equilibrium state of the dimensionality reduction model is always a stable node-focus, hidden extreme multistability with coexisting infinitely many attractors is truly confirmed. Finally, PSIM circuit simulations validate the initial-related hidden dynamical behaviors.


FitzHugh–Nagumo circuit Ideal memristor Hidden extreme multistability Dimensionality reduction model 



This work was supported by the grants from the National Natural Science Foundations of China under 51777016, 61471191, and 61601062, and the Aeronautical Science Foundation of China under 20152052026.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest.


  1. 1.
    Eshraghian, K., Kavehei, O., Cho, K.R., Chappell, J.M., Iqbal, A., Al-Sarawi, S.F., Abbott, D.: Memristive device fundamentals and modeling: applications to circuits and systems simulation. Proc. IEEE 100(6), 1991–2007 (2012)CrossRefGoogle Scholar
  2. 2.
    Kumar, S., Strachan, J.P., Williams, R.S.: Chaotic dynamics in nanoscale \(\text{ NbO }_{{2}}\) Mott memristor for analogue computing. Nature 548(7667), 318–321 (2017)CrossRefGoogle Scholar
  3. 3.
    Wang, Z., Joshi, S., Savel’Ev, S.E., Jiang, H., Rivu, M., Lin, P., Hu, M., Ge, N., Strachan, J.P., Li, Z., Wu, Q., Barnell, M., Li, G.L., Xin, H.L., Williams, R.S., Xia, Q., Yang, J.J.: Memristors with diffusive dynamics as synaptic emulators for neuromorphic computing. Nat. Mater. 16(1), 101–108 (2017)CrossRefGoogle Scholar
  4. 4.
    Duan, S., Hu, X., Dong, Z., Wang, L., Mazumder, P.: Memristor-based cellular nonlinear/neural network: design, analysis, and applications. IEEE Trans. Neural Netw. Learn. Syst. 26(6), 1202–1213 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Serb, A., Bill, J., Khiat, A., Berdan, R., Legenstein, R., Prodromakis, T.: Unsupervised learning in probabilistic neural networks with multi-state metal-oxide memristive synapses. Nat. Commun. 7, 12611 (2016)CrossRefGoogle Scholar
  6. 6.
    Yakopcic, C., Hasan, R., Taha, T.M., Mclean, M., Palmer, D.: Memristor-based neuron circuit and method for applying learning algorithm in SPICE? Electron. Lett. 50(7), 492–494 (2014)CrossRefGoogle Scholar
  7. 7.
    Babacan, Y., Kaçar, F., Gürkan, K.: A spiking and bursting neuron circuit based on memristor. Neurocomputing 203, 86–91 (2016)CrossRefGoogle Scholar
  8. 8.
    Bao, B.C., Jiang, T., Wang, G.Y., Jin, P.P., Bao, H., Chen, M.: Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonlinear Dyn. 89(2), 1157–1171 (2017)CrossRefGoogle Scholar
  9. 9.
    Bao, H., Wang, N., Bao, B.C., Chen, M., Jin, P.P., Wang, G.Y.: Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria. Commun. Nonlinear Sci. Numer. Simul. 57, 264–275 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Adhikari, S.P., Sah, M.P., Kim, H., Chua, L.O.: Three fingerprints of memristor. IEEE Trans. Circuits Syst. I(60), 3008–3021 (2013)CrossRefGoogle Scholar
  11. 11.
    Bao, H.B., Park, J.H., Cao, J.D.: Exponential synchronization of coupled stochastic memristor-based neural networks with time-varying probabilistic delay coupling and impulsive delay. IEEE Trans. Neural Netw. Learn. Syst. 27(1), 190–201 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zhang, J.H., Liao, X.F.: Synchronization and chaos in coupled memristor-based FitzHugh–Nagumo circuits with memristor synapse. AEÜ Int. J. Electron. Commun. 75, 82–90 (2017)CrossRefGoogle Scholar
  13. 13.
    Bao, B.C., Hu, A.H., Bao, H., Xu, Q., Chen, M., Wu, H.G.: Three-dimensional memristive Hindmarsh–Rose neuron model with hidden coexisting asymmetric behaviors. Complexity 2018, 3872573 (2018)Google Scholar
  14. 14.
    Bao, B.C., Xu, Q., Bao, H., Chen, M.: Extreme multistability in a memristive circuit. Electron. Lett. 52, 1008–1010 (2016)CrossRefGoogle Scholar
  15. 15.
    Jafari, S., Ahmadi, A., Panahi, S., Rajagopal, K.: Extreme multi-stability: when imperfection changes quality. Chaos Solitons Fractals 108, 182–186 (2018)CrossRefGoogle Scholar
  16. 16.
    Yuan, F., Wang, G.Y., Wang, X.W.: Extreme multistability in a memristor-based multi-scroll hyperchaotic system. Chaos 26, 073107 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bao, B.C., Jiang, T., Xu, Q., Chen, M., Hu, H.G., Hu, Y.H.: Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dyn. 86, 1711–1723 (2016)CrossRefGoogle Scholar
  18. 18.
    Bao, B.C., Bao, H., Wang, N., Chen, M., Xu, Q.: Hidden extreme multistability in memristive hyperchaotic system. Chaos Solitons Fractals 94, 102–111 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Tan, Q.W., Zeng, Y.C., Li, Z.J.: A simple inductor-free memristive circuit with three line equilibria. Nonlinear Dyn. 94, 1585–1602 (2018)CrossRefGoogle Scholar
  20. 20.
    Patel, M.S., Patel, U., Sen, A., Sethia, G.C., Hens, C., Dana, S.K., Feudel, U., Showalter, K., Ngonghala, C.N., Amritkar, R.E.: Experimental observation of extreme multistability in an electronic system of two coupled Rössler oscillators. Phys. Rev. E 89(2), 022918 (2014)CrossRefGoogle Scholar
  21. 21.
    Li, C.B., Sprott, J.C., Hu, W., Xu, Y.J.: Infinite multistability in a self-reproducing chaotic system. Int. J. Bifurc. Chaos 27(10), 1750160 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ma, J., Wu, F.G., Ren, G.D., Tang, J.: A class of initials-dependent dynamical systems. Appl. Math. Comput. 298, 65–76 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Jafari, S., Ahmadi, A., Khalaf, A.J.M., Abdolmohammadi, H.R., Pham, V.T., Alsaadi, F.E.: A new hidden chaotic attractor with extreme multi-stability. AEÜ Int. J. Electron. Commun. 89, 131–135 (2018)CrossRefGoogle Scholar
  24. 24.
    Kim, H., Sah, M.P., Yang, C., Roska, T., Chua, L.O.: Neural synaptic weighting with a pulse-based memristor circuit. IEEE Trans. Circuits Syst. I 59(1), 148–158 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Prezioso, M., Merrikh-Bayat, F., Hoskins, B.D., Adam, G.C., Likharev, K.K., Strukov, D.B.: Training and operation of an integrated neuromorphic network based on metal-oxide memristors. Nature 521(7550), 61–64 (2015)CrossRefGoogle Scholar
  26. 26.
    Wu, J., Xu, Y., Ma, J.: Lévy noise improves the electrical activity in a neuron under electromagnetic radiation. PLoS ONE 12, e0174330 (2017)CrossRefGoogle Scholar
  27. 27.
    Lv, M., Wang, C., Ren, G., Ma, J., Song, X.: Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn. 85(3), 1479–1490 (2016)CrossRefGoogle Scholar
  28. 28.
    Wang, Y., Ma, J., Xu, Y., Wu, F., Zhou, P.: The electrical activity of neurons subject to electromagnetic induction and Gaussian white noise. Int. J. Bifurc. Chaos 27(2), 1750030 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Wu, F.Q., Wang, C.N., Jin, W.Y., Ma, J.: Dynamical responses in a new neuron model subjected to electromagnetic induction and phase noise. Physica A 469, 81–88 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Xu, Y., Jia, Y., Ma, J., Alsaedi, A., Ahmad, B.: Synchronization between neurons coupled by memristor. Chaos Solitons Fractals 104, 435–442 (2017)CrossRefGoogle Scholar
  31. 31.
    Ma, J., Lv, M., Zhou, P., Xu, Y., Hayat, T.: Phase synchronization between two neurons induced by coupling of electromagnetic field. Appl. Math. Comput. 307, 321–328 (2017)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Ren, G.D., Xu, Y., Wang, C.N.: Synchronization behavior of coupled neuron circuits composed of memristors. Nonlinear Dyn. 88(2), 893–901 (2017)CrossRefGoogle Scholar
  33. 33.
    Xu, F., Zhang, J., Fang, T., Huang, S., Wang, M.: Synchronous dynamics in neural system coupled with memristive synapse. Nonlinear Dyn. 92(3), 1395–1402 (2018)CrossRefGoogle Scholar
  34. 34.
    Bao, H., Liu, W.B., Hu, A.H.: Coexisting multiple firing patterns in two adjacent neurons coupled by memristive electromagnetic induction. Nonlinear Dyn. 95(1), 43–56 (2019)CrossRefGoogle Scholar
  35. 35.
    Zhang, G., Wang, C., Alzahrani, F., Wu, F., An, X.: Investigation of dynamical behaviors of neurons driven by memristive synapse. Chaos Solitons Fractals 108, 15–24 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Takmbo, C.N., Mvogo, A., Ekobena Fouda, H.P., Kofane, T.C.: Localized modulated wave solution of diffusive FitzHugh–Nagumo cardiac networks under magnetic flow effect. Nonlinear Dyn. (2018)
  37. 37.
    Ahamed, A.I., Lakshmanan, M.: Nonsmooth bifurcations, transient hyperchaos and hyperchaotic beats in a memristive Murali–Lakshmanan–Chua circuit. Int. J. Bifurc. Chaos 23, 1350098 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50(10), 2061–2070 (1962)CrossRefGoogle Scholar
  39. 39.
    Jiang, W., Deng, B., Tsang, K.M.: Chaotic synchronization of neurons coupled with gap junction under external electrical stimulation. Chaos Solitons Fractals 22(2), 469–476 (2004)zbMATHCrossRefGoogle Scholar
  40. 40.
    Handa, H., Sharma, B.B.: Synchronization of a set of coupled chaotic FitzHugh–Nagumo and Hindmarsh–Rose neurons with external electrical stimulation. Nonlinear Dyn. 85(3), 1517–1532 (2016)zbMATHCrossRefGoogle Scholar
  41. 41.
    El-Sayed, A.M.A., Elsaid, A., Nour, H.M., Elsonbaty, A.: Dynamical behavior, chaos control and synchronization of a memristor-based ADVP circuit. Commun. Nonlinear Sci. Numer. Simul. 18(1), 148–170 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Zhang, J.H., Liao, X.F.: Effects of initial conditions on the synchronization of the coupled memristor neural circuits. Nonlinear Dyn. (2018)
  43. 43.
    Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Localization of hidden Chua’s attractors. Phys. Lett. A 375, 2230–2233 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Leonov, G.A., Kuznetsov, N.V., Mokaev, T.N.: Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity. Commu. Nonlinear Sci. Numer. Simul. 28(1–3), 166–174 (2015)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Li, Q.D., Zeng, H.Z., Yang, X.S.: On hidden twin attractors and bifurcation in the Chua’s circuit. Nonlinear Dyn. 77, 255–266 (2014)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Bao, B.C., Jiang, P., Xu, Q., Chen, M.: Hidden attractors in a practical Chua’s circuit based on a modified Chua’s diode. Electron. Lett. 52(1), 23–25 (2016)CrossRefGoogle Scholar
  47. 47.
    Sprott, J.C., Jafari, S., Pham, V.T., Hosseini, Z.S.: A chaotic system with a single unstable node. Phys. Lett. A 379(36), 2030–2036 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Jafari, S., Sprott, J.C., Golpayegani, S.M.R.H.: Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A 377(9), 699–702 (2013)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Wei, Z.C., Yang, R., Liu, A.: A new finding of the existence of hidden hyperchaotic attractors with no equilibria. Math. Comput. Simul. 100(1), 13–23 (2014)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Zhou, W., Wang, G.Y., Shen, Y.R., Yuan, F., Yu, S.M.: Hidden coexisting attractors in a chaotic system without equilibrium point. Int. J. Bifurc. Chaos 28(10), 1830033 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Xu, Q., Zhang, Q., Bao, B.C., Hu, Y.H.: Non-autonomous second-order memristive chaotic circuit. IEEE Access 5, 21039–21045 (2017)CrossRefGoogle Scholar
  52. 52.
    Bao, B.C., Jiang, P., Wu, H.G., Hu, F.W.: Complex transient dynamics in periodically forced memristive Chua’s circuit. Nonlinear Dyn. 79, 2333–2343 (2015)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Chen, M., Sun, M.X., Bao, B.C., Wu, H.G., Xu, Q., Wang, J.: Controlling extreme multistability of memristor emulator-based dynamical circuit in flux-charge domain. Nonlinear Dyn. 91(2), 1395–1412 (2018)CrossRefGoogle Scholar
  54. 54.
    Chen, M., Bao, B.C., Jiang, T., Bao, H., Xu, Q., Wu, H.G., Wang, J.: Flux-charge analysis of initial state-dependent dynamical behaviors in a memristor emulator-based Chua’s circuit. Int. J. Bifurc. Chaos 28(10), 1850120 (2018)zbMATHCrossRefGoogle Scholar
  55. 55.
    Bao, H., Jiang, T., Chu, K.B., Chen, M., Xu, Q., Bao, B.C.: Memristor-based canonical Chua’s circuit: extreme multi-stability in voltage–current domain and its controllability in flux-charge domain. Complexity 2018, 5935637 (2018)zbMATHGoogle Scholar
  56. 56.
    Chen, M., Feng, Y., Bao, H., Bao, B.C., Yu, Y.J., Wu, H.G., Xu, Q., Wang, J.: State variable mapping method for studying initial-dependent dynamics in memristive hyper-jerk system with line equilibrium. Chaos Solitons Fractals 115, 313–324 (2018)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Hu, X.Y., Liu, C.X., Liu, L., Ni, J.K., Li, S.L.: An electronic implementation for Morris–Lecar neuron model. Nonlinear Dyn. 84(4), 2317–2332 (2016)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Bao, B.C., Qian, H., Xu, Q., Chen, M., Wang, J., Yu, Y.J.: Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network. Front. Comput. Neurosci. 11(81), 1–14 (2017)Google Scholar
  59. 59.
    Ratas, I., Pyragas, K.: Effect of high-frequency stimulation on nerve pulse propagation in the FitzHugh–Nagumo model. Nonlinear Dyn. 67(4), 2899–2908 (2012)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Leutcho, G.D., Kengne, J., Kengne, L.K.: Dynamical analysis of a novel autonomous 4-D hyperjerk circuit with hyperbolic sine nonlinearity: chaos, antimonotonicity and a plethora of coexisting attractors. Chaos Solitons Fractals 107, 67–87 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    He, S.B., Sun, K.H., Banerjee, S.: Dynamical properties and complexity in fractional-order diffusionless Lorenz system. Eur. Phys. J. Plus 131(8), 254 (2016)CrossRefGoogle Scholar
  62. 62.
    Ye, X.L., Mou, J., Luo, C.F., Wang, Z.S.: Dynamics analysis of Wien-bridge hyperchaotic memristive circuit system. Nonlinear Dyn. 92(3), 923–933 (2018)CrossRefGoogle Scholar
  63. 63.
    Hayati, M., Nouri, M., Haghiri, S., Abbott, D.: Digital multiplierless realization of two coupled biological Morris–Lecar neuron model. IEEE Trans. Circuits Syst. I 62(7), 1805–1814 (2015)CrossRefGoogle Scholar
  64. 64.
    Bao, B.C., Ma, Z.H., Xu, J.P., Liu, Z., Xu, Q.: A simple memristor chaotic circuit with complex dynamics. Int. J. Bifurc. Chaos 21(9), 2629–2645 (2011)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of Automation EngineeringNanjing University of Aeronautics and AstronauticsNanjingChina
  2. 2.School of Information Science and EngineeringChangzhou UniversityChangzhouChina

Personalised recommendations