Nonlinear Dynamics

, Volume 90, Issue 3, pp 1681–1694 | Cite as

Hyperchaotic memristive system with hidden attractors and its adaptive control scheme

  • Dimitrios A. ProusalisEmail author
  • Christos K. Volos
  • Ioannis N. Stouboulos
  • Ioannis M. Kyprianidis
Original Paper


In this research work a novel 4-D memristive system is presented. The proposed system belongs to the category of dynamical systems with hidden attractors as it displays a line of equilibrium points. Also, it has an hyperchaotic dynamical behavior in a particular range of its parameters space. System’s behavior is investigated through numerical simulations, by using well-known tools of nonlinear theory, such as phase portrait, bifurcation diagram, Lyapunov exponents and Poincaré map. Next, the case of chaos control of the system with unknown parameters using adaptive control method is investigated. Finally, an electronic circuit realization of the novel hyperchaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model.


Memristive system Hidden attractor Chaos control Hyperchaotic system Adaptive control Equilibrium line 


  1. 1.
    Chua, L.O.: Memristors-the missing circuit element. IEEE Trans. Circuit Theory. 18, 507–519 (1971)CrossRefGoogle Scholar
  2. 2.
    Chua, L.O., Kang, S.M.: Memristive devices and systems. Proc. IEEE 375(23), 209–223 (1976)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Strukov, D.B., Snider, G.S., Stewart, G.R., Williams, R.S.: The missing memristor found. Nature 453, 80–83 (2008)CrossRefGoogle Scholar
  4. 4.
    Ventra, M.D., Pershin, Y.V., Chua, L.O.: Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc. IEEE 97(10), 1717–1724 (2009)CrossRefGoogle Scholar
  5. 5.
    Chua, L.O.: Resistances switching memories are memristors. Appl. Phys. A 102(4), 765–783 (2011)CrossRefGoogle Scholar
  6. 6.
    Kim, K.H., Gaba, S., Wheeler, D., Cruz-Albrecht, J.M., Hussain, T., Srinivasa, N., Lu, W.: A functional hybrid memristor crossbar-array/CMOS system for data storage and neuromorphic applications. Nano Lett. 12(1), 389–395 (2011)CrossRefGoogle Scholar
  7. 7.
    Kuekes, P.J., Snider, G.S., Williams, R.S.: Crossbar nanocomputers. Sci. Am. 293(5), 72–80 (2005)CrossRefGoogle Scholar
  8. 8.
    Strukov, D.B., Likharev, K.K.: Defect-tolerant architectures for nanoelectronic crossbar memories. J. Nanosci. Nanotechnol. 7(1), 151167 (2007)Google Scholar
  9. 9.
    Greenlee, J.D., Calley, W.L., Moseley, M.W., Doolittle, W.A.: Comparison of interfacial and bulk ionic motion in analog memristors. IEEE Trans. Electron Devices 60(1), 427–432 (2013)CrossRefGoogle Scholar
  10. 10.
    Chang, T., Jo, S.H., Kim, K.H., Sheridan, P., Gaba, S., Lu, W.: Synaptic behaviors and modeling of a metal oxide memristive device. Appl. Phys. A 102(4), 857–863 (2011)CrossRefGoogle Scholar
  11. 11.
    Prez-Carrasco, J.A., Zamarreno-Ramos, C., Serrano-Gotarredona, T., Linares-Barranco, B.: On neuromorphic spiking architectures for asynchronous STDP memristive systems. In: Proceedings of IEEE International Symposium on Circuits and Systems, pp. 1659–1662 (2010)Google Scholar
  12. 12.
    Serrano-Gotarredona, T., Prodromakis, T., Linares-Barranco, B.: A proposal for hybrid memristor-CMOS spiking neuromorphic learning systems. IEEE Circuits Syst. Mag. 13(2), 74–88 (2013)CrossRefGoogle Scholar
  13. 13.
    Itoh, M., Chua, L.O.: Memristor oscillators. Int. J. Bifurc. Chaos 18, 3183–3206 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zhao, Y.B., Tse, C.K., Feng, J.C., Guo, Y.C.: Application of memristor-based controller for loop filter design in charge-pump phase-locked loop. Circuits Syst. Signal Process. 32, 1013–1023 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wu, A.L., Zhang, J.N., Zeng, Z.G.: Dynamical behaviors of a class of memristor-based Hopfield networks. Phys. Lett. A 375, 1661–1665 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sabarathinam, S., Volos, C.K., Thamilmaran, K.: Implementation and study of the nonlinear dynamics of a memristor-based Duffing oscillator. Nonlinear Dyn. 87(1), 37–49 (2017)CrossRefGoogle Scholar
  17. 17.
    Chen, M., Li, M., Yu, Q., Bao, B., Xu, Q., Wang, J.: Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chuas circuit. Nonlinear Dyn. 81(1–2), 215–226 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bao, B., Jiang, T., Xu, Q., Chen, M., Wu, H., Hu, Y.: Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dyn. 86(3), 1711–1723 (2016)CrossRefGoogle Scholar
  19. 19.
    Wu, H., Bao, B., Liu, Z., Xu, Q., Jiang, P.: Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator. Nonlinear Dyn. 83(1–2), 893–903 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Biswas, D., Banerjee, T.: A simple chaotic and hyperchaotic time-delay system: design and electronic circuit implementation. Nonlinear Dyn. 83(4), 2331–2347 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ponomarenko, V.I., Prokhorov, M.D., Karavaev, A.S., Kulminskiy, D.D.: An experimental digital communication scheme based on chaotic time-delay system. Nonlinear Dyn. 74(4), 1013–1020 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Özkaynak, F., Yavuz, S.: Designing chaotic S-boxes based on time-delay chaotic system. Nonlinear Dyn. 74(3), 551–557 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ye, G., Wong, K.W.: An image encryption scheme based on time-delay and hyperchaotic system. Nonlinear Dyn. 71(1–2), 259–267 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Banerjee, T., Biswas, D., Sarkar, B.C.: Design and analysis of a first order time-delayed chaotic system. Nonlinear Dyn. 70(1), 721–734 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Banerjee, T., Biswas, D., Sarkar, B.C.: Design of chaotic and hyperchaotic time-delayed electronic circuit. Bonfring Int. J. Power Syst. Integr. Circuits 2(4), 13 (2012)CrossRefGoogle Scholar
  26. 26.
    Banerjee, T., Biswas, D.: Theory and experiment of a first-order chaotic delay dynamical system. Int. J. Bifurc. Chaos 23(06), 1330020 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N.: The Memristor as an electric synapse synchronization phenomena. In: Proceedings of International Conference DSP2011, pp. 1–6. Confu, Greece 2011Google Scholar
  28. 28.
    Yang, J.J., Strukov, D.B., Stewart, D.R.: Memristive devices for computing. Nat. Nanotechnol. 8, 13–24 (2013)CrossRefGoogle Scholar
  29. 29.
    Driscoll, T., Quinn, J., Klein, S., Kim, H.T., Kim, B.J., Pershin, Y.V., Di Ventra, M., Basov, D.N.: Memristive adaptive filters. Appl. Phys. Lett. 97, 093502 (2010)CrossRefGoogle Scholar
  30. 30.
    Wang, L., Zhang, C., Chen, L., Lai, J., Tong, J.: A novel memristor-based rSRAM structure for multiple-bit upsets immunity. IEICE Electron. Express 9, 861–867 (2012)CrossRefGoogle Scholar
  31. 31.
    Shang, Y., Fei, W., Yu, H.: Analysis and modeling of internal state variables for dynamic effects of nonvolatile memory devices. IEEE Trans. Circuits Syst. I Regul. Pap. 59, 1906–1918 (2012)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Shin, S., Kim, K., Kang, S.M.: Memristor applications for programmable analog ICs. IEEE Trans. Nanotechnol. 10, 266–274 (2011)CrossRefGoogle Scholar
  33. 33.
    Cepisca, C., Bardis, N.G.: Measurement and Control in Street Lighting. Electra publication, Paris (2011)Google Scholar
  34. 34.
    Bogdan, M., Buga, M., Medianu, R., Cepisca, C., Bardis, N.: Obtaining a model of photovoltaic cell with optimized quantum efficiency. Sci. Bull. Electr. Eng. Fac. 2(16), 1843–6188 (2011)Google Scholar
  35. 35.
    Cepisca, C., Grigorescu, S.D., Ganatsios, S., Bardis, N.G.: Passive and active compensations for current transformers. Metrologie 4, 5–10 (2008)Google Scholar
  36. 36.
    Jafari, S., Sprott, J.C.: Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 57, 79–84 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kuznetsov, N.V., Leonov, G.A., Vagaitsev, V.I.: Analytical-numerical method for attractor localization of generalized Chuas system. IFAC Proc. Vol. 43(11), 29–33 (2010). (IFAC-Papers Online) CrossRefGoogle Scholar
  38. 38.
    Leonov, G.A., Kuznetsov, N.V., Vagaytsev, V.I.: Localization of hidden Chuas attractors. Phys. Lett. A 375(23), 2230–2233 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Chua, L.O., Lin, G.N.: Canonical realization of Chuas circuit family. IEEE Trans. Circuits Syst. 37(4), 885902 (1990)MathSciNetGoogle Scholar
  40. 40.
    Li, C., Pehlivan, I., Sprott, J.C., Agkul, A.: A novel four-wing strange attractor born in bistability. IEICE Electron. Express 12(4), 2011116–20141116 (2015)Google Scholar
  41. 41.
    Li, Q., Hu, S., Tang, S., Zeng, G.: Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation. Int. J. Circuit Theory Appl. 42(11), 1172–1188 (2014)CrossRefGoogle Scholar
  42. 42.
    Li, Q., Zeng, H., Li, J.: Hyperchaos in a 4D memristive circuit with innitely many stable equilibria. Nonlinear Dyn. 79, 22952308 (2015)Google Scholar
  43. 43.
    Pham, V.T., Volos, C., Gambuzza, L.V.: A memristive hyperchaotic system without equilibrium. Sci. World J. 2014, 9 (2014)CrossRefGoogle Scholar
  44. 44.
    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)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Pham, V.T., Vaidyanathan, S., Volos, C.K., Hoang, T.M., Van Yem, V.: Dynamics, synchronization and SPICE implementation of a memristive system with hidden hyperchaotic attractor. In: Azar, A., Vaidyanathan, S. (eds.) Advances in Chaos Theory and Intelligent Control. Studies in Fuzziness and Soft Computing, vol. 337, pp. 35–52. Springer, Cham (2016)Google Scholar
  46. 46.
    Sundarapandian, V.: Output regulation of the Lorenz attractor. Far East J. Math. Sci. 42(2), 289299 (2010)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Vaidyanathan, S.: Output regulation of Arneodo–Coullet chaotic system. Commun. Comput. Inf. Sci. 133, 98107 (2011)Google Scholar
  48. 48.
    Vaidyanathan, S.: Output regulation of the unified chaotic system. Commun. Comput. Inf. Sci. 198, 19 (2011)Google Scholar
  49. 49.
    Vaidyanathan, S.: Analysis, control and synchronization of a novel 4-D highly hyperchaotic system with hidden attractors. In: Azar, A., Vaidyanathan, S. (eds.) Advances in Chaos Theory and Intelligent Control. Studies in Fuzziness and Soft Computing, vol. 337, pp. 529–552. Springer, Cham (2016)Google Scholar
  50. 50.
    Sundarapandian, V.: Adaptive control and synchronization design for the Lu-Xiao chaotic system. Lect. Notes Electr. Eng. 131, 319327 (2013)Google Scholar
  51. 51.
    Vaidyanathan, S.: Adaptive controller and syncrhonizer design for the Qi-Chen chaotic system. Adv. Comput. Sci. Inf. Technol. Comput. Sci. Eng. 84, 7382 (2012)Google Scholar
  52. 52.
    Vaidyanathan, S.: A ten-term novel 4-D hyperchaotic system with three quadratic non-linearities and its control. Int. J. Control Theory Appl. 6(2), 97109 (2013)MathSciNetGoogle Scholar
  53. 53.
    Vaidyanathan, S.: Analysis, control and synchronization of hyperchaotic Zhou system via adaptive control. Adv. Intell. Syst. Comput. 177, 110 (2013)Google Scholar
  54. 54.
    Vaidyanathan, S.: Qualitative analysis and control of an eleven-term novel 4-D hyperchaotic system with two quadratic nonlinearities. Int. J. Control. Theory Appl. 7, 3547 (2014)Google Scholar
  55. 55.
    Azar, A.T., Vaidyanathan, S.: Analysis and control of a 4-D novel hyperchaotic system. Stud. Comput. Intell. 581, 317 (2015)MathSciNetGoogle Scholar
  56. 56.
    Vaidyanathan, S.: Global chaos control of hyperchaotic Liu system via sliding control method. Int. J. Control Theory Appl. 5(2), 117123 (2012)MathSciNetGoogle Scholar
  57. 57.
    Vaidyanathan, S.: Sliding mode control based global chaos control of Liu-Liu–Liu-Su chaotic system. Int. J. Control Theory Appl. 5(1), 1520 (2012)Google Scholar
  58. 58.
    Njah, A.N., Sunday, O.D.: Generalization on the chaos control of 4-D chaotic systems using recursive backstepping nonlinear controller. Chaos Solitons Fractals 41(5), 23712376 (2009)CrossRefzbMATHGoogle Scholar
  59. 59.
    Vincent, U.E., Njah, A.N., Laoye, J.A.: Controlling chaos and deterministic directed transport in inertia ratchets using backstepping control. Physica D 231(2), 130136 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Vaidyanathan, S., Volos, C.: Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system. Arch. Control Sci. 25(3), 333–353 (2015)MathSciNetGoogle Scholar
  61. 61.
    Wei, Z., Moroz, I., Liu, A.: Degenerate Hopf bifurcations, hidden attractors, and control in the extended Sprott E system with only one stable equilibrium. Turk. J. Math. 38(4), 672–687 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Cao, C., Ma, L., Xu, Y.: Adaptive control theory and applications. J. Control Sci. Eng. 2012, 2 (2012)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Vaidyanathan, S.: Adaptive control of Rikitake two-disk dynamo system. Int. J. ChemTech Res. 8(8), 121–133 (2015)MathSciNetGoogle Scholar
  64. 64.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.: Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, New Jersey (2001)Google Scholar
  66. 66.
    Barakat, M., Mansingka, A., Radwan, A.G., Salama, K.N.: Generalized hardware post processing technique for chaosbased pseudorandom number generators. ETRI J. 35, 448–458 (2013)CrossRefGoogle Scholar
  67. 67.
    Gamez-Guzman, L., Cruz-Hernandez, C., Lopez-Gutierrez, R., Garcia-Guerrero, E.E.: Synchronization of Chuas circuits with multiscroll attractors: application to communication. Commun. Nonlinear Sci. Numer. Simul. 14, 2765–2775 (2009)CrossRefGoogle Scholar
  68. 68.
    Sadoudi, S., Tanougast, C., Azzaz, M.S., Dandache, A.: Design and FPGA implementation of a wireless hyperchaotic communication system for secure realtime image transmission. EURASIP J. Image Video Process. 943, 1–18 (2013)Google Scholar
  69. 69.
    Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N.: A chaotic path planning generator for autonomous mobile robots. Robot. Auto. Syst. 60, 651–656 (2012)CrossRefGoogle Scholar
  70. 70.
    Yalcin, M.E., Suykens, J.A.K., Vandewalle, J.: True random bit generation from a double-scroll attractor. IEEE Trans. Circuits Syst. I Regul. Pap. 51, 1395–1404 (2004)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Buscarino, A., Fortuna, L., Frasca, M.: Experimental robust synchronization of hyperchaotic circuits. Physica D 238, 1917–1922 (2009)CrossRefzbMATHGoogle Scholar
  72. 72.
    Sundarapandian, V., Pehlivan, I.: Analysis, control, synchronization, and circuit design of a novel chaotic system. Math. Comput. Model. 55, 1904–1915 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Bouali, S., Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L.V.: Emulating complex business cycles by using an electronic analogue. Nonlinear Anal. Real World Appl. 13, 2459–2465 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Fortuna, L., Frasca, M., Xibilia, M.G.: Chuas Circuit Implementation: Yesterday, Today and Tomorrow. World Scientific, Singapore (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Dimitrios A. Prousalis
    • 1
    Email author
  • Christos K. Volos
    • 1
  • Ioannis N. Stouboulos
    • 1
  • Ioannis M. Kyprianidis
    • 1
  1. 1.Department of PhysicsAristotle University of ThessalonikiThessalonikiGreece

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