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Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability


This paper presents a novel fifth-order two-memristor-based Chua’s hyperchaotic circuit, which is synthesized from an active band pass filter-based Chua’s circuit through replacing a nonlinear resistor and a linear resistor with two different memristors. This physical circuit has a plane equilibrium and therefore emerges complex phenomenon of extreme multistability. Based on the mathematical model, stability distributions of three nonzero eigenvalues in the equilibrium plane are exhibited, from which it is observed that four different stability regions with unstable saddle-focus, and stable and unstable node-focus are distributed, thereby leading to coexisting phenomenon of infinitely many attractors. Furthermore, extreme multistability depending on two-memristor initial conditions is investigated by bifurcation diagrams and Lyapunov exponent spectra and coexisting infinitely many attractors’ behavior is revealed by phase portraits and attraction basins. At last, a hardware circuit is fabricated and some experimental observations are captured to verify that extreme multistability indeed exists in the two-memristor-based Chua’s hyperchaotic circuit.

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  1. 1.

    Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S.: The missing memristor found. Nature 453, 80–83 (2008)

  2. 2.

    Fitch, A.L., Yu, D., Iu, H.H., Sreeram, V.: Hyperchaos in a memristor-based modified canonical Chua’s circuit. Int. J. Bifurc. Chaos 22, 1250133 (2012)

  3. 3.

    Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L.V.: A chaotic circuit based on Hewlett-Packard memristor. Chaos 22, 023136 (2012)

  4. 4.

    Li, Q.D., Zeng, H.Z., Li, J.: Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria. Nonlinear Dyn. 79, 2295–2308 (2015)

  5. 5.

    Ma, J., Chen, Z.G., Wang, Z.L., Zhang, Q.: A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium. Nonlinear Dyn. 81, 1275–1288 (2015)

  6. 6.

    Njitacke, Z.T., Kengne, J., Fotsin, H.B., Negou, A.N., Tchiotsop, D.: Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bridge-based Jerk circuit. Chaos Solit. Fractals 91, 180–197 (2016)

  7. 7.

    Chen, M., Li, M.Y., Yu, Q., Bao, B.C., Xu, Q., Wang, J.: Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit. Nonlinear Dyn. 81, 215–226 (2015)

  8. 8.

    Xu, Q., Lin, Y., Bao, B.C., Chen, M.: Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit. Chaos Solit. Fractals 83, 186–200 (2016)

  9. 9.

    Zhou, L., Wang, C.H., Zhou, L.L.: Generating hyperchaotic multi-wing attractor in a 4D memristive circuit. Nonlinear Dyn. 85, 2653–2663 (2016)

  10. 10.

    Pham, V.T., Jafari, S., Vaidyanathan, S., Volos, C.K., Wang, X.: A novel memristive neural network with hidden attractors and its circuitry implementation. Sci. China Tech. Sci. 59, 358–363 (2016)

  11. 11.

    Wu, H.G., Bao, B.C., Liu, Z., Xu, Q., Jiang, P.: Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator. Nonlinear Dyn. 83, 893–903 (2015)

  12. 12.

    Lü, M., Wang, C.N., Tang, J., Ma, J.: Collapse of synchronization in a memristive network. Commun. Theor. Phys. 64, 659–664 (2015)

  13. 13.

    Bao, B.C., Hu, F.W., Liu, Z., Xu, J.P.: Mapping equivalent approach to analysis and realization of memristor-based dynamical circuit. Chin. Phys. B 23, 070503 (2014)

  14. 14.

    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)

  15. 15.

    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)

  16. 16.

    Yuan, F., Wang, G.Y., Wang, X.W.: Extreme multistability in a memristor-based multi-scroll hyperchaotic system. Chaos 26, 073107 (2016)

  17. 17.

    Bao, B.C., Xu, Q., Bao, H., Chen, M.: Extreme multistability in a memristive circuit. Electron. Lett. 52, 1008–1010 (2016)

  18. 18.

    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)

  19. 19.

    Bao, B.C., Bao, H., Wang, N., Chen, M., Xu, Q.: Hidden extreme multistability in memristive hyperchaotic system. Chaos Solit. Fractals 94, 102–111 (2017)

  20. 20.

    Patel, M.S., Patel, U., Sen, A., Sethia, G.C., Hens, C., Dana, S.K.: Experimental observation of extreme multistability in an electronic system of two coupled Rössler oscillators. Phys. Rev. E 89, 022918 (2014)

  21. 21.

    Hens, C., Dana, S.K., Feudel, U.: Extreme multistability: attractors manipulation and robustness. Chaos 25, 053112 (2015)

  22. 22.

    Pisarchik, A.N., Feudel, U.: Control of multistability. Phys. Rep. 540, 167–218 (2014)

  23. 23.

    Li, C.B., Hu, W., Sprott, J.C., Wang, X.: Multistability in symmetric chaotic systems. Eur. Phys. J. Spec. Top. 224, 1493–1506 (2015)

  24. 24.

    Sharma, P.R., Shrimali, M.D., Prasad, A., Kuznetsov, N.V., Leonov, G.A.: Control of multistability in hidden attractors. Eur. Phys. J. Spec. Top. 224, 1485–1491 (2015)

  25. 25.

    Morfu, S., Nofiele, B., Marquié, P.: On the use of multistability for image processing. Phys. Lett. A 367, 192–198 (2007)

  26. 26.

    Bao, B.C., Li, Q.D., Wang, N., Xu, Q.: Multistability in Chua’s circuit with two stable node-foci. Chaos 26, 043111 (2016)

  27. 27.

    Kuznetsov, A.P., Kuznetsov, S.P., Mosekilde, E., Stankevich, N.V.: Co-existing hidden attractors in a radio-physical system. J. Phys. A 48, 125101 (2015)

  28. 28.

    Chua, L.O.: The fourth element. Proc. IEEE 100, 1920–1927 (2012)

  29. 29.

    Bao, B.C., Shi, G.D., Xu, J.P., Liu, Z., Pan, S.H.: Dynamics analysis of chaotic circuit with two memristors. Sci. China Tech. Sci. 54, 2180–2187 (2011)

  30. 30.

    Banerjee, T.: Single amplifier biquad based inductor-free Chua’s circuit. Nonlinear Dyn. 68, 565–573 (2012)

  31. 31.

    Bao, B.C., Wang, N., Chen, M., Xu, Q., Wang, J.: Inductor-free simplified Chua’s circuit only using two op-amps based realization. Nonlinear Dyn. 84, 511–525 (2016)

  32. 32.

    Adhikari, S.P., Sah, M.P., Kim, H., Chua, L.O.: Three fingerprints of memristor. IEEE Trans. Circuits Syst. I(60), 3008–3021 (2013)

  33. 33.

    Dawson, S.P., Grebogi, C., Yorke, J.A., Kan, I., Koçak, H.: Antimonotonicity: inevitable reversals of period-doubling cascades. Phys. Lett. A 162, 249–254 (1992)

  34. 34.

    Kyprianidis, I.M., Stouboulos, I.N., Haralabidis, P.: Antimonotonicity and chaotic dynamics in a forth-order autonomous nonlinear electric circuit. Int. J. Bifurc. Chaos 10, 1903–1915 (2000)

  35. 35.

    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. Simulat. 28, 166–174 (2015)

  36. 36.

    Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)

  37. 37.

    Ma, J., Wu, F.G., Ren, G.D., Tang, J.: A class of initials-dependent dynamical systems. Appl. Math. Comput. 298, 65–76 (2017)

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This work was supported by the grants from the National Natural Science Foundations of China under Grant Nos. 51277017, 61601062, and 51607013 and the Natural Science Foundations of Jiangsu Province, China, under Grant No. BK20160282.

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Correspondence to Bocheng Bao.

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Bao, B., Jiang, T., Wang, G. et al. Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonlinear Dyn 89, 1157–1171 (2017).

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  • Memristive circuit
  • Plane equilibrium
  • Infinitely many attractor
  • Extreme multistability