Nonlinear Dynamics

, Volume 88, Issue 4, pp 2589–2608 | Cite as

Antimonotonicity, chaos and multiple attractors in a novel autonomous memristor-based jerk circuit

  • J. Kengne
  • A. Nguomkam Negou
  • D. Tchiotsop
Original Paper


A novel memristor-based oscillator derived from the autonomous jerk circuit (Sprott in IEEE Trans Circuits Syst II Express Briefs 58:240–243, 2011) is proposed. A first-order memristive diode bridge replaces the semiconductor diode of the original circuit. The complex behavior of the oscillator is investigated in terms of equilibria and stability, phase space trajectories plots, bifurcation diagrams, graphs of Lyapunov exponents, as well as frequency spectra. Antimonotonicity (i.e. concurrent creation and destruction of periodic orbits), chaos, periodic windows and crises are reported. More interestingly, one of the main features of the novel memristive jerk circuit is the presence of a region in the parameters’ space in which the model develops hysteretic behavior. This later phenomenon is marked by the coexistence of four different (periodic and chaotic) attractors for the same values of system parameters, depending solely on the choice of initial conditions. Basins of attractions of various competing attractors display complex basin boundaries thus suggesting possible jumps between coexisting solutions in experiment. Compared to previously published jerk circuits with similar behavior, the novel system distinguishes by the presence of a single equilibrium point and a relatively simpler structure (only off-the-shelf electronic components are involved). Results of theoretical analyses are perfectly traced by laboratory experimental measurements.


Memristive jerk circuit Bifurcation analysis Antimonotonicity Coexistence of multiple attractors Experimental study 


  1. 1.
    Chua, L.O.: Memristor—the missing circuit element. IEEE Trans. Circuits Theory 18(5), 507–519 (1971)CrossRefGoogle Scholar
  2. 2.
    Strukov, D.B., Snider, G.S., Stewart, G.R., Williams, R.S.: The missing memristor found. Nature 453, 80–83 (2008)CrossRefGoogle Scholar
  3. 3.
    Adhikari, S.P., Sah, M.P., Kim, H., Chua, L.O.: Three fingerprints of memristor. IEEE Trans. Circuit Syst. I 60(11), 3008–3021 (2013)CrossRefGoogle Scholar
  4. 4.
    Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L.V.: A chaotic circuit based on Hewlett–Packard memristor. Chaos 22(2), 023136 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bao, B., Zhong, L., Xu, J.-P.: Transient chaos in smooth memristor oscillator. Chin. Phys. B 19(3), 030510 (2010)CrossRefGoogle Scholar
  6. 6.
    Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L.V.: A gallery of chaotic oscillators based on HP memristor. Int. J. Bifurc. Chaos 23(5), 1330015 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Budhathoki, R.K., Sah, M.P., Yang, D., Kim, H., Chua, L.O.: Transient behavior of multiple memristor circuits based on flux charge relationship. Int. J. Bifurc. Chaos 24(2), 1430006 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bao, B., Zou, X., Liu, Z., Hu, F.: Generalized memory element and chaotic memory system. Int. J. Bifurc. Chaos 23(8), 1350135 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Muthuswamy, B., Chua, L.O.: Simplest chaotic circuit. Int. J. Bifurc. Chaos 20(5), 1567–1580 (2010)CrossRefGoogle Scholar
  10. 10.
    Wang, G.Y., He, J.L., Yuan, F., Peng, C.J.: Dynamical behaviour of a TiO\(_2\) memristor oscillator. Chin. Phys. Lett. 30, 110506 (2013)CrossRefGoogle Scholar
  11. 11.
    Itoh, M., Chua, L.O.: Memristor oscillators. Int. J. Bifurc. Chaos 18, 3183–3206 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Muthuswamy, B.: Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20, 1335–1350 (2010)CrossRefMATHGoogle Scholar
  13. 13.
    Bao, B., Xu, J.P., Zhou, G.H., Ma, Z.H., Zou, L.: Chaotic memristive circuit: equivalent circuit realization and dynamical analysis. Chin. Phys. B 20, 120502 (2011)CrossRefGoogle Scholar
  14. 14.
    Bao, B., Yu, J., Hu, F., Liu, Z.: Generalized memristor consisting of diode bridge with first order parallel RC filter. Int. J. Bifurc. Chaos 24(11), 1450143 (2014)CrossRefMATHGoogle Scholar
  15. 15.
    Chen, M., Li, M., Yu, Q., Bao, B., Xu, Q., Wang, J.: Dynamics of self-excited and hidden attractors in generalized memristor-based Chua’s circuit. Nonlinear Dyn. 81(1–2), 215–226 (2015)Google Scholar
  16. 16.
    Chen, M., Yu, J., Xu, Q., Li, C., Bao, B.: A memristive diode bridge-based canonical Chua’s circuit. Entropy 16, 6464–6476 (2014)CrossRefGoogle Scholar
  17. 17.
    Sprott, J.C.: A new chaotic jerk circuit. IEEE Trans. Circuits Syst. II Express Briefs 58, 240–243 (2011)CrossRefGoogle Scholar
  18. 18.
    Li, C., Hu, W., Sprott, J.C., Wang, X.: Multistability in symmetric chaotic systems. Eur. Phys. J. Spec. Top. 224, 1493–1506 (2015)CrossRefGoogle Scholar
  19. 19.
    Sprott, J.C.: Some simple Jerk functions. Am. J. Phys. 65, 537–543 (1997)CrossRefGoogle Scholar
  20. 20.
    Sprott, J.C.: Simplest dissipative chaotic flow. Phys. Lett. A 228, 271–274 (1997)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Sprott, J.C.: Simple chaotic systems and circuits. Am. J. Phys. 68, 758–763 (2000)CrossRefGoogle Scholar
  22. 22.
    Sprott, J.C.: Elegant Chaos: Algebraically Simple Flow. World Scientific Publishing, Singapore (2010)CrossRefMATHGoogle Scholar
  23. 23.
    Kengne, J., Njitacke, Z.T., Fotsin, H.B.: Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn. 83, 751–765 (2016)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kengne, J., Njitacke, Z.T., Nguomkam Negou, A., Fouodji Tsotsop, M., Fotsin, H.B.: Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit. Int. J. Bifurc. Chaos 25(4), 1550052 (2015)CrossRefMATHGoogle Scholar
  25. 25.
    Njitacke, Z.T., kengne, J., Fotsin, H.B., Nguomkam Negou, A., Tchiotsop, D.: Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bidge-based Jerk circuit. Chaos Solitons Fract. 91, 180–197 (2016)CrossRefMATHGoogle Scholar
  26. 26.
    Louodop, P., Kountchou, M., Fotsin, H., Bowong, S.: Practical finite-time synchronization of jerk systems: theory and experiment. Nonlinear Dyn. 78, 597–607 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Leipnik, R.B., Newton, T.A.: Double strange attractors in rigid body motion with linear feedback control. Phys. Lett. A 86, 63–87 (1981)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Letellier, C., Gilmore, R.: Symmetry groups for 3D dynamical systems. J. Phys. A.: Math. Theor. 40, 5597–5620 (2007)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Hanias, M.P., Giannaris, G., Spyridakis, A.R.: Time series analysis in chaotic diode resonator circuit. Chaos Chaos Solitons Fract. 27, 569–573 (2006)CrossRefMATHGoogle Scholar
  30. 30.
    Kengne, J., Chedjou, J.C., Fonzin Fozin, T., Kyamakya, K., Kenne, G.: On the analysis of semiconductor diode based chaotic and hyperchaotic chaotic generators—a case study. Nonlinear Dyn. 77, 373–386 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos. Addison-Wesley, Reading (1994)Google Scholar
  32. 32.
    Argyris, J., Faust, G., Haase, M.: An Exploration of Chaos. North-Holland, Amsterdam (1994)MATHGoogle Scholar
  33. 33.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. Wiley, New York (1995)CrossRefMATHGoogle Scholar
  34. 34.
    Kuznetsov, N., Leonov, G., Vagaitsev, V.: Analytical–numerical method for attractor localization of generalized Chua’s system. IFAC Proc. Vol. 4(1), 29–33 (2010)CrossRefGoogle Scholar
  35. 35.
    Leonov, G., Kuznetsov, N., Vagaitsev, V.: Localization of hidden Chua’s attractors. Phys. Lett. A 375(23), 2230–2233 (2011)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Leonov, G., Kuznetsov, N., Vagaitsev, V.: Hidden attractor in smooth Chua systems. Phys. D: Nonlinear Phenom. 241(18), 1482–1486 (2012)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Leonov, G.A., Kuznetsov, N.V.: 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), 1330002 (2013)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Leonov, G.A., Kuznetsov, N.V.: 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(1), 1330002 (2013)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Leonov, G.A., Kuznetsov, N.V., Mokaev, T.N.: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Top. 224, 1421–1458 (2015)CrossRefGoogle Scholar
  40. 40.
    Pham, V.T., Volos, C., Jafari, S., Vaidyanathan, S.: Hidden hyperchaotic attractor in a novel simple memristive neural network. Optoelectron. Adv. Mater. Rapid Commun. 8(11–12), 1157–1163 (2014)Google Scholar
  41. 41.
    Pham, V.T., Jafari, S., Vaidyanathan, S., 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
  42. 42.
    Pham, V.T., Vaidyanathan, S., Volos, C.K., Jafari, S., Kuznetsov, N.V., Hoang, T.M.: A novel memristive time-delay chaotic system without equilibrium points. Eur. Phys. J. Spec. Top. 225(1), 127–136 (2016)CrossRefGoogle Scholar
  43. 43.
    Pham, V.-T., Vaidyanathan, S., Volos, C.K., Jafari, S., Wang, X.: A Chaotic Hyperjerk System Based on Memristive Device, in Advances and Applications in Chaotic Systems, pp. 39–58. Springer, Berlin (2016)Google Scholar
  44. 44.
    Wolf, A., Swift, J.B., Swinney, H.L., Wastano, J.A.: Determining Lyapunov exponents from time series. Phys. D 16, 285–317 (1985)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Swathy, P.S., Thamilmaran, K.: An experimental study on SC-CNN based canonical Chua’s circuit. Nonlinear Dyn. 71, 505–514 (2013)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Wang, X., Chen, G.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17(3), 1264–1272 (2012)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Huan, S., Li, Q., Yang, X.S.: Horseshoes in a chaotic system with only one stable equilibrium. Int. J. Bifurc. Chaos 23(01), 1350002 (2013)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Molaie, M., Jafari, S.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23(11), 1350188 (2013)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Kingni, S.T., Jafari, S., Simo, H., Woafo, P.: Three-dimensional chaotic autonomous system with only one stable equilibrium: analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form. Eur. Phys. J. Plus 129(5), 1–16 (2014)CrossRefGoogle Scholar
  50. 50.
    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)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Wei, Z., Zhang, W.: Hidden hyperchaotic attractors in a modified Lorenz–Stenflo system with only one stable equilibrium. Int. J. Bifurc. Chaos 24(10), 1450127 (2014)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Masoller, C.: Coexistence of attractors in a laser diode with optical feedback from a large external cavity. Phys. Rev. A 50, 2569–2578 (1994)CrossRefGoogle Scholar
  53. 53.
    Cushing, J.M., Henson, S.M.: Blackburn: multiple mixed attractors in a competition model. J. Biol. Dyn. 1, 347–362 (2007)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Upadhyay, R.K.: Multiple attractors and crisis route to chaos in a model of food-chain. Chaos Chaos Solitons Fract. 16, 737–747 (2003)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Li, C., Sprott, J.C.: Coexisting hidden attractors in a 4-D simplified Lorenz system. Int. J. Bifurc. Chaos 24, 1450034 (2014)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Vaithianathan, V., Veijun, J.: Coexistence of four different attractors in a fundamental power system model. IEEE Trans. Circuits Syst I 46, 405–409 (1999)CrossRefGoogle Scholar
  57. 57.
    Kengne, J.: On the dynamics of Chua’s oscillator with a smooth cubic nonlinearity: occurrence of multiple attractors. Nonlinear Dyn (2016). doi: 10.1007/s11071-016-3047-z
  58. 58.
    Pivka, L., Wu, C.W., Huang, A.: Chua’s oscillator: a compendium of chaotic phenomena. J. Frankl. Inst. 331B(6), 705–741 (1994)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Kuznetsov, A.P., Kuznetsov, S.P., Mosekilde, E., Stankevich, N.V.: Co-existing hidden attractors in a radio-physical oscillator. J. Phys. A: Math. Theor. 48, 125101 (2015)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Lai, Q., Chen, S.: Generating multiple chaotic attractors from Sprott B system. Int. J. Bifurc. Chaos 26(11), 1650177 (2016)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Lai, Q., Chen, S.: Coexisting attractors generated from a new 4D smooth chaotic system. Int. J. Control Autom. Syst. 14(4), 1124–1131 (2016)CrossRefGoogle Scholar
  62. 62.
    Lai, Q., Chen, S.: Research on a new 3D autonomous chaotic system with coexisting attractors. Optik 127(5), 3000–3004 (2016)CrossRefGoogle Scholar
  63. 63.
    Hens, C., Dana, S.K., Feudel, U.: Extreme multistability: attractors manipulation and robustness. Chaos 25, 053112 (2015)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Bao, B.C., Xu, B., Bao, H., Chen, M.: Extreme multistability in a memristive circuit. Electron. Lett. 52(12), 1008–1010 (2016)CrossRefGoogle Scholar
  65. 65.
    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
  66. 66.
    Luo, X., Small, M.: On a dynamical system with multiple chaotic attractors. Int. J. Bifurc. Chaos 17(9), 3235–3251 (2007)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Pisarchik, A.N., Feudel, U.: Control of multistability. Phys. Rep. 540(4), 167–218 (2014)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    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)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Parlitz, U., Lauterborn, W.: Superstructure in the bifurcation set of the Duffing equation \(\ddot{{\rm x}}+ \text{ d }\dot{{\rm x}}+ \text{ x }+ \text{ x }3\)= f cos (\(\upomega \)t). Phys. Lett. A 107, 351–355 (1985)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Parlitz, U., Lauterborn, W.: Period-doubling cascades and devil’s staircases of the driven van der Pol oscillator. Phys. Rev. A 36, 1428 (1987)CrossRefGoogle Scholar
  71. 71.
    Kocarev, L., Halle, K., Eckert, K., Chua, L.: Experimental observation of antimonotonicity in Chua’s circuit. Int. J. Bifurc. Chaos 3, 1051–1055 (1993)CrossRefMATHGoogle Scholar
  72. 72.
    Ogawa, T.: Quasiperiodic instability and chaos in the bad-cavity laser with modulated inversion: numerical analysis of a Toda oscillator system. Phys. Rev. A 37, 4286–4302 (1988)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Kyprianidis, I., Stouboulos, I., Haralabidis, P., Bountis, T.: Antimonotonicity and chaotic dynamics in a fourth-order autonomous nonlinear electric circuit. Int. J. Bifurc. Chaos 10, 1903–1915 (2000)Google Scholar
  74. 74.
    Manimehan, I., Philominathan, P.: Composite dynamical behaviors in a simple series–parallel LC circuit. Chaos Solitons Fract. 45, 1501–1509 (2012)CrossRefGoogle Scholar
  75. 75.
    Bier, M., Bountis, T.C.: Remerging Feigenbaum trees in dynamical systems. Phys. Lett. A 104, 239–244 (1984)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Kiers, K., Schmidt, D.: Precision measurement of a simple chaotic circuit. Am. J. Phys. 76(4), 503–509 (2004)CrossRefGoogle Scholar
  77. 77.
    Kingni, S.T., Keuninckx, L., Woafo, P., van der Sande, G., Danckaert, J.: Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation. Nonlinear Dyn. 73, 1111–1123 (2013)MathSciNetCrossRefMATHGoogle Scholar
  78. 78.
    Kingni, S.T., Jafari, S., Simo, H., Woafo, P.: Three-dimensional chaotic autonomous system with only one stable equilibrium: analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form. Eur. Phys. J. Plus 129, 76 (2014)CrossRefGoogle Scholar
  79. 79.
    Sprott, J.C.: Generalization of the simplest autonomous chaotic system. Phys. Lett. A 375(12), 1445–1450 (2011)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Laboratoire d’Automatique et Informatique Appliquée (LAIA), Department of Electrical Engineering, IUT-FV BandjounUniversity of DschangDschangCameroon
  2. 2.Laboratory of Electronics and Signal Processing, Department of PhysicsUniversity of DschangDschangCameroon

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