Complex dynamics from a novel memristive 6D hyperchaotic autonomous system

  • Brice Anicet Mezatio
  • Marceline Motchongom Tingue
  • Romanic KengneEmail author
  • Aurelle Tchagna Kouanou
  • Theophile Fozin Fonzin
  • Robert Tchitnga


A simple 5D hyperchaotic system recently introduced in the literature is modified by using a charge-controlled memristor model and striking behaviors are uncovered. The resulting system is a 6D hyperchaotic system, which generates hidden attractors with the unusual feature of having plan and line equilibrium under different parameter conditions. Its dynamical behaviors are characterized through bifurcation diagrams, Lyapunov exponents, phase portraits, Poincaré sections and time series. Rich nonlinear dynamics such as limit cycles, quasi-periodicity, chaos, hyperchaos, bursting and hidden extreme multistability are found for appropriate sets of parameter values. The high complexity of the system is confirmed by its Kaplan–yorke dimension (greater than five). Additionally, an electronic circuit is designed to implement the novel system and PSpice simulation results are in good accordance with the numerical investigations. To the best of our knowledge, this system is the first with higher order presenting all those phenomena.


6D hyperchaotic system Hidden extreme multistability Bursting oscillations Offset boosting 


  1. 1.
    Chua L (1971) Memristor-the missing circuit element. IEEE Trans Circuit Theory 18(5):507–519CrossRefGoogle Scholar
  2. 2.
    Strukov DB, Snider GS, Stewart DR, Williams RS (2008) The missing memristor found. Nature 453(7191):80CrossRefGoogle Scholar
  3. 3.
    Yang JJ, Pickett MD, Li X, Ohlberg DA, Stewart DR, Williams RS (2008) Memristive switching mechanism for metal/oxide/metal nanodevices. Nature Nanotechnol 3(7):429CrossRefGoogle Scholar
  4. 4.
    Chan M, Zhang T, Ho V, Lee P (2008) Resistive switching effects of HFO2 high-k dielectric. Microelectron Eng 85(12):2420–2424CrossRefGoogle Scholar
  5. 5.
    Kim TH, Jang EY, Lee NJ, Choi DJ, Lee K-J, Jang J-T, Choi J-S, Moon SH, Cheon J (2009) Nanoparticle assemblies as memristors. Nano Lett 9(6):2229–2233CrossRefGoogle Scholar
  6. 6.
    Almeida S, Aguirre B, Marquez N, McClure J, Zubia D (2011) Resistive switching of SnO2 thin films on glass substrates. Integr Ferroelectr 126(1):117–124CrossRefGoogle Scholar
  7. 7.
    Pershin YV, Di Ventra M (2008) Spin memristive systems: spin memory effects in semiconductor spintronics. Phys Rev B 78(11):113309CrossRefGoogle Scholar
  8. 8.
    Jo SH, Chang T, Ebong I, Bhadviya BB, Mazumder P, Lu W (2010) Nanoscale memristor device as synapse in neuromorphic systems. Nano Lett 10(4):1297–1301CrossRefGoogle Scholar
  9. 9.
    Buscarino A, Fortuna L, Frasca M, Gambuzza LV (2013) A gallery of chaotic oscillators based on HP memristor. Int J Bifurc Chaos 23(05):1330015MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Xu Q, Lin Y, Bao B, Chen M (2016) Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit. Chaos Solitons Fractals 83:186–200MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Alombah NH, Fotsin H, Romanic K (2017) Coexistence of multiple attractors, metastable chaos and bursting oscillations in a multiscroll memristive chaotic circuit. Int J Bifurc Chaos 27(05):1750067MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fonzin TF, Kengne J, Pelap FB (2018) Dynamical analysis and multistability in autonomous hyperchaotic oscillator with experimental verification. Nonlinear Dyn 93(2):653–669CrossRefGoogle Scholar
  13. 13.
    Fonzin TF, Srinivasan K, Kengne J, Pelap F (2018) Coexisting bifurcations in a memristive hyperchaotic oscillator. AEU Int J Electron Commun 90:110–122CrossRefGoogle Scholar
  14. 14.
    Kengne J, Negou AN, Tchiotsop D, Tamba VK, Kom G (2018) On the dynamics of chaotic systems with multiple attractors: a case study, In: Recent advances in nonlinear dynamics and synchronization. Springer, pp 17–32Google Scholar
  15. 15.
    Kengne J, Negou AN, Njitacke Z (2017) Antimonotonicity, chaos and multiple attractors in a novel autonomous Jerk circuit. Int J Bifurc Chaos 27(07):1750100MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bao B, Jiang T, Xu Q, Chen M, Wu H, Hu Y (2016) Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dyn 86(3):1711–1723CrossRefGoogle Scholar
  17. 17.
    Bao B, Bao H, Wang N, Chen M, Xu Q (2017) Hidden extreme multistability in memristive hyperchaotic system. Chaos Solitons Fractals 94:102–111MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Njitacke Z, Kengne J, Tapche RW, Pelap F (2018) Uncertain destination dynamics of a novel memristive 4D autonomous system. Chaos Solitons Fractals 107:177–185MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Leonov G, Kuznetsov N, Vagaitsev V (2011) Localization of hidden Chua’s attractors. Phys Lett A 375(23):2230–2233MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Leonov GA, Kuznetsov NV (2013) Hidden attractors in dynamical systems from hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int J Bifurc Chaos 23(01):1330002MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Leonov G, Kuznetsov N, Mokaev T (2015) Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur Phys J Spec Top 224(8):1421–1458CrossRefGoogle Scholar
  22. 22.
    Singh JP, Roy BK (2018) Second order adaptive time varying sliding mode control for synchronization of hidden chaotic orbits in a new uncertain 4-D conservative chaotic system. Trans Inst Meas Control 40(13):3573–3586CrossRefGoogle Scholar
  23. 23.
    Pham V-T, Vaidyanathan S, Volos C, Jafari S (2015) Hidden attractors in a chaotic system with an exponential nonlinear term. Eur Phys J Spec Top 224(8):1507–1517CrossRefGoogle Scholar
  24. 24.
    Pham V-T, Jafari S, Kapitaniak T, Volos C, Kingni ST (2017) Generating a chaotic system with one stable equilibrium. Int J Bifurc Chaos 27(04):1750053MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhou P, Huang K, Yang Cd (2013) A fractional-order chaotic system with an infinite number of equilibrium points. Discrete Dyn Nat Soc. MathSciNetzbMATHGoogle Scholar
  26. 26.
    Jafari S, Sprott J (2013) Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 57:79–84MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Li C, Thio WJ-C, Iu HH, Lu T (2018) A memristive chaotic oscillator with increasing amplitude and frequency. IEEE Access 6:12945–12950CrossRefGoogle Scholar
  28. 28.
    Pham V-T, Jafari S, Volos C, Vaidyanathan S, Kapitaniak T (2016) A chaotic system with infinite equilibria located on a piecewise linear curve. Optik Int J Light Electron Opt 127(20):9111–9117CrossRefGoogle Scholar
  29. 29.
    Jafari S, Ahmadi A, Khalaf AJM, Abdolmohammadi HR, Pham V-T, Alsaadi FE (2018) A new hidden chaotic attractor with extreme multi-stability. AEU Int J Electron Commun 89:131–135CrossRefGoogle Scholar
  30. 30.
    Yang Q, Bai M (2017) A new 5D hyperchaotic system based on modified generalized Lorenz system. Nonlinear Dyn 88(1):189–221CrossRefzbMATHGoogle Scholar
  31. 31.
    Reiterer P, Lainscsek C, Schürrer F, Letellier C, Maquet J (1998) A nine-dimensional Lorenz system to study high-dimensional chaos. J Phys A Math Gen 31(34):7121CrossRefzbMATHGoogle Scholar
  32. 32.
    Xiao-Yu D, Chun-Biao L, Bo-Cheng B, Hua-Gan W (2015) Complex transient dynamics of hidden attractors in a simple 4D system. Chin Phys B 24(5):050503CrossRefGoogle Scholar
  33. 33.
    Bao H, Wang N, Bao B, Chen M, Jin P, Wang G (2018) Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria. Commun Nonlinear Sci Numer Simul 57:264–275MathSciNetCrossRefGoogle Scholar
  34. 34.
    Singh JP, Roy B (2017) The simplest 4-D chaotic system with line of equilibria, chaotic 2-torus and 3-torus behaviour. Nonlinear Dyn 89(3):1845–1862MathSciNetCrossRefGoogle Scholar
  35. 35.
    Pham V-T, Volos C, Jafari S, Wei Z, Wang X (2014) Constructing a novel no-equilibrium chaotic system. Int J Bifurc Chaos 24(05):1450073MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Pham V-T, Volos C, Gambuzza LV (2014) A memristive hyperchaotic system without equilibrium. Sci World J. Google Scholar
  37. 37.
    Pham V-T, Volos C, Jafari S, Kapitaniak T (2017) Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn 87(3):2001–2010CrossRefGoogle Scholar
  38. 38.
    Pham V-T, Akgul A, Volos C, Jafari S, Kapitaniak T (2017) Dynamics and circuit realization of a no-equilibrium chaotic system with a boostable variable. AEU Int J Electron Commun 78:134–140CrossRefGoogle Scholar
  39. 39.
    Ren S, Panahi S, Rajagopal K, Akgul A, Pham V-T, Jafari S (2018) A new chaotic flow with hidden attractor: the first hyperjerk system with no equilibrium. Zeitschrift für Naturforschung A 73(3):239–249CrossRefGoogle Scholar
  40. 40.
    Pham V-T, Wang X, Jafari S, Volos C, Kapitaniak T (2017) From Wang–Chen system with only one stable equilibrium to a new chaotic system without equilibrium. Int J Bifurc Chaos 27(06):1750097MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Jafari S, Sprott JC, Molaie M (2016) A simple chaotic flow with a plane of equilibria. Int J Bifurc Chaos 26(06):1650098MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Bao B, Jiang T, Wang G, Jin P, Bao H, Chen M (2017) Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonlinear Dyn 89(2):1157–1171CrossRefGoogle Scholar
  43. 43.
    Singh JP, Roy B (2018) Hidden attractors in a new complex generalised Lorenz hyperchaotic system, its synchronisation using adaptive contraction theory, circuit validation and application. Nonlinear Dyn 92(2):373–394CrossRefzbMATHGoogle Scholar
  44. 44.
    Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys D Nonlinear Phenom 16(3):285–317MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Kengne J, Kenmogne F (2014) On the modeling and nonlinear dynamics of autonomous Silva–Young type chaotic oscillators with flat power spectrum. Chaos Interdiscip J Nonlinear Sci 24(4):043134MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Tchitnga R, Zebaze Nanfa’a R, Pelap FB, Louodop P, Woafo P (2017) A novel high-frequency interpretation of a general purpose Op-Amp-based negative resistance for chaotic vibrations in a simple a priori nonchaotic circuit. J Vib Control 23(5):744–751CrossRefGoogle Scholar
  47. 47.
    Tchitnga R, Mezatio B, Fozin TF, Kengne R, Fotso PL, Fomethe A (2019) A novel hyperchaotic three-component oscillator operating at high frequency. Chaos Solitons Fractals 118:166–180CrossRefGoogle Scholar
  48. 48.
    Lai Q, Wang L (2016) Chaos, bifurcation, coexisting attractors and circuit design of a three-dimensional continuous autonomous system. Optik Int J Light Electron Opt 127(13):5400–5406CrossRefGoogle Scholar
  49. 49.
    Lai Q, Akgul A, Zhao X-W, Pei H (2017) Various types of coexisting attractors in a new 4D autonomous chaotic system. Int J Bifurc Chaos 27(09):1750142MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Li C, Sprott JC (2016) Variable-boostable chaotic flows. Optik Int J Light Electron Opt 127(22):10389–10398CrossRefGoogle Scholar
  51. 51.
    Leutcho GD, Kengne J, Kengne R (2019) Remerging feigenbaum trees, and multiple coexisting bifurcations in a novel hybrid diode based hyper jerk circuit with offset boosting. Int J Dyn Control 7:61–82MathSciNetCrossRefGoogle Scholar
  52. 52.
    Ando H, Suetani H, Kurths J, Aihara K (2012) Chaotic phase synchronization in bursting-neuron models driven by a weak periodic force. Phys Rev E 86(1):016205CrossRefGoogle Scholar
  53. 53.
    Fallah H (2016) Symmetric fold/super-Hopf bursting, chaos and mixed-mode oscillations in Pernarowski model of pancreatic beta-cells. Int J Bifurc Chaos 26(09):1630022MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Kingston SL, Thamilmaran K (2017) Bursting oscillations and mixed-mode oscillations in driven Liénard system. Int J Bifurc Chaos 27(07):1730025CrossRefzbMATHGoogle Scholar
  55. 55.
    Izhikevich EM (2000) Neural excitability, spiking and bursting. Int J Bifurc Chaos 10(06):1171–1266MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Bao B, Wu P, Bao H, Wu H, Zhang X, Chen M (2018) Symmetric periodic bursting behavior and bifurcation mechanism in a third-order memristive diode bridge-based oscillator. Chaos Solitons Fractals 109:146–153CrossRefGoogle Scholar
  57. 57.
    Babacan Y, Kaçar F, Gürkan K (2016) A spiking and bursting neuron circuit based on memristor. Neurocomputing 203:86–91CrossRefGoogle Scholar
  58. 58.
    Hu M, Chen Y, Yang JJ, Wang Y, Li HH (2017) A compact memristor-based dynamic synapse for spiking neural networks. IEEE Trans Comput Aided Des Integr Circuits Syst 36(8):1353–1366CrossRefGoogle Scholar
  59. 59.
    Feali MS, Ahmadi A (2017) Transient response characteristic of memristor circuits and biological-like current spikes. Neural Comput Appl 28(11):3295–3305CrossRefGoogle Scholar
  60. 60.
    Kingni ST, Nana B, Ngueuteu GM, Woafo P, Danckaert J (2015) Bursting oscillations in a 3D system with asymmetrically distributed equilibria: mechanism, electronic implementation and fractional derivation effect. Chaos Solitons Fractals 71:29–40MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Wang H, Wang Q, Lu Q (2011) Bursting oscillations, bifurcation and synchronization in neuronal systems. Chaos Solitons Fractals 44(8):667–675CrossRefzbMATHGoogle Scholar
  62. 62.
    Singh JP, Lochan K, Kuznetsov NV, Roy B (2017) Coexistence of single-and multi-scroll chaotic orbits in a single-link flexible joint robot manipulator with stable spiral and index-4 spiral repellor types of equilibria. Nonlinear Dyn 90(2):1277–1299CrossRefGoogle Scholar
  63. 63.
    Singh JP, Roy B (2016) Crisis and inverse crisis route to chaos in a new 3D chaotic system with saddle, saddle foci and stable node foci nature of equilibria. Optik Int J Light Electron Opt 127(24):11982–12002CrossRefGoogle Scholar
  64. 64.
    Pelap F, Tanekou G, Fogang C, Kengne R (2018) Fractional-order stability analysis of earthquake dynamics. J Geophys Eng 15(4):1673CrossRefGoogle Scholar
  65. 65.
    Kengne R, Tchitnga R, Mabekou S, Tekam BRW, Soh GB, Fomethe A (2018) On the relay coupling of three fractional-order oscillators with time-delay consideration: global and cluster synchronizations. Chaos Solitons Fractals 111:6–17MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Tanekou G, Fogang C, Kengne R, Pelap F (2018) Lubrication pressure and fractional viscous damping effects on the spring-block model of earthquakes. Eur Phys J Plus 133(4):150CrossRefGoogle Scholar
  67. 67.
    Kengne R, Tchitnga R, Mezatio A, Fomethe A, Litak G (2017) Finite-time synchronization of fractional-order simplest two-component chaotic oscillators. Eur Phys J B 90(5):88MathSciNetCrossRefGoogle Scholar
  68. 68.
    Takougang Kingni S, Fautso Kuiate G, Kengne R, Tchitnga R, Woafo P (2017) Analysis of a no equilibrium linear resistive-capacitive inductance shunted junction model, dynamics, synchronization, and application to digital cryptography in its fractional-order form. Complexity. MathSciNetzbMATHGoogle Scholar
  69. 69.
    Kengne R, Tchitnga R, Kouanou AT, Fomethe A (2013) Dynamical properties and chaos synchronization in a fractional-order two-stage colpitts oscillator. J Eng Sci Technol Rev 6(4):24–32CrossRefGoogle Scholar
  70. 70.
    Kengne R, Tchitnga R, Fomethe A, Hammouch Z (2017) Generalized finite-time function projective synchronization of two fractional-order chaotic systems via a modified fractional nonsingular sliding mode surface. Commun Numer Anal 2017:233–248MathSciNetCrossRefGoogle Scholar
  71. 71.
    Usama M, Zakaria N (2017) Chaos-based simultaneous compression and encryption for hadoop. PLoS ONE 12(1):e0168207CrossRefGoogle Scholar
  72. 72.
    Kouanou AT, Tchiotsop D, Kengne R, Tansaa ZD, Adele NM, Tchinda R (2018) An optimal big data workflow for biomedical image analysis. Inf Med Unlocked 11:68–74CrossRefGoogle Scholar
  73. 73.
    Kouanou AT, Tchiotsop D, Tchinda R, Tchapga CT, Telem ANK, Kengne R (2018) A machine learning algorithm for biomedical images compression using orthogonal transforms. Int J Image Graph Signal Process 11:38–53CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Brice Anicet Mezatio
    • 1
    • 2
  • Marceline Motchongom Tingue
    • 3
  • Romanic Kengne
    • 1
    • 2
    Email author
  • Aurelle Tchagna Kouanou
    • 2
  • Theophile Fozin Fonzin
    • 2
  • Robert Tchitnga
    • 1
    • 2
  1. 1.Research Group on Experimental and Applied Physics for Sustainable Development, Department of Physics, Faculty of ScienceUniversity of DschangDschangCameroon
  2. 2.Unité de Recherche de Matière Condensée d’Electronique et de Traitement du Signal (URMACETS), Department of Physics, Faculty of ScienceUniversity of DschangDschangCameroon
  3. 3.Higher Technical Teachers Training CollegeUniversity of BamendaBambiliCameroon

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