Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Chua, L.O.: Memristor—the missing circuit element. IEEE Trans. Circuits Theory 18(5), 507–519 (1971)

    Article  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  4. Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L.V.: A chaotic circuit based on Hewlett–Packard memristor. Chaos 22(2), 023136 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bao, B., Zhong, L., Xu, J.-P.: Transient chaos in smooth memristor oscillator. Chin. Phys. B 19(3), 030510 (2010)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bao, B., Zou, X., Liu, Z., Hu, F.: Generalized memory element and chaotic memory system. Int. J. Bifurc. Chaos 23(8), 1350135 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Muthuswamy, B., Chua, L.O.: Simplest chaotic circuit. Int. J. Bifurc. Chaos 20(5), 1567–1580 (2010)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  11. Itoh, M., Chua, L.O.: Memristor oscillators. Int. J. Bifurc. Chaos 18, 3183–3206 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Muthuswamy, B.: Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20, 1335–1350 (2010)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  17. Sprott, J.C.: A new chaotic jerk circuit. IEEE Trans. Circuits Syst. II Express Briefs 58, 240–243 (2011)

    Article  Google Scholar 

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

    Article  Google Scholar 

  19. Sprott, J.C.: Some simple Jerk functions. Am. J. Phys. 65, 537–543 (1997)

    Article  Google Scholar 

  20. Sprott, J.C.: Simplest dissipative chaotic flow. Phys. Lett. A 228, 271–274 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sprott, J.C.: Simple chaotic systems and circuits. Am. J. Phys. 68, 758–763 (2000)

    Article  Google Scholar 

  22. Sprott, J.C.: Elegant Chaos: Algebraically Simple Flow. World Scientific Publishing, Singapore (2010)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  28. Letellier, C., Gilmore, R.: Symmetry groups for 3D dynamical systems. J. Phys. A.: Math. Theor. 40, 5597–5620 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  31. Strogatz, S.H.: Nonlinear Dynamics and Chaos. Addison-Wesley, Reading (1994)

    Google Scholar 

  32. Argyris, J., Faust, G., Haase, M.: An Exploration of Chaos. North-Holland, Amsterdam (1994)

    MATH  Google Scholar 

  33. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. Wiley, New York (1995)

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  35. Leonov, G., Kuznetsov, N., Vagaitsev, V.: Localization of hidden Chua’s attractors. Phys. Lett. A 375(23), 2230–2233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Leonov, G., Kuznetsov, N., Vagaitsev, V.: Hidden attractor in smooth Chua systems. Phys. D: Nonlinear Phenom. 241(18), 1482–1486 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  44. Wolf, A., Swift, J.B., Swinney, H.L., Wastano, J.A.: Determining Lyapunov exponents from time series. Phys. D 16, 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  45. Swathy, P.S., Thamilmaran, K.: An experimental study on SC-CNN based canonical Chua’s circuit. Nonlinear Dyn. 71, 505–514 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang, X., Chen, G.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17(3), 1264–1272 (2012)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  48. Molaie, M., Jafari, S.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23(11), 1350188 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  53. Cushing, J.M., Henson, S.M.: Blackburn: multiple mixed attractors in a competition model. J. Biol. Dyn. 1, 347–362 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  55. Li, C., Sprott, J.C.: Coexisting hidden attractors in a 4-D simplified Lorenz system. Int. J. Bifurc. Chaos 24, 1450034 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. Pivka, L., Wu, C.W., Huang, A.: Chua’s oscillator: a compendium of chaotic phenomena. J. Frankl. Inst. 331B(6), 705–741 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  60. Lai, Q., Chen, S.: Generating multiple chaotic attractors from Sprott B system. Int. J. Bifurc. Chaos 26(11), 1650177 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  62. Lai, Q., Chen, S.: Research on a new 3D autonomous chaotic system with coexisting attractors. Optik 127(5), 3000–3004 (2016)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  64. Bao, B.C., Xu, B., Bao, H., Chen, M.: Extreme multistability in a memristive circuit. Electron. Lett. 52(12), 1008–1010 (2016)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  66. Luo, X., Small, M.: On a dynamical system with multiple chaotic attractors. Int. J. Bifurc. Chaos 17(9), 3235–3251 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  67. Pisarchik, A.N., Feudel, U.: Control of multistability. Phys. Rep. 540(4), 167–218 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Manimehan, I., Philominathan, P.: Composite dynamical behaviors in a simple series–parallel LC circuit. Chaos Solitons Fract. 45, 1501–1509 (2012)

    Article  Google Scholar 

  75. Bier, M., Bountis, T.C.: Remerging Feigenbaum trees in dynamical systems. Phys. Lett. A 104, 239–244 (1984)

    Article  MathSciNet  Google Scholar 

  76. Kiers, K., Schmidt, D.: Precision measurement of a simple chaotic circuit. Am. J. Phys. 76(4), 503–509 (2004)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  79. Sprott, J.C.: Generalization of the simplest autonomous chaotic system. Phys. Lett. A 375(12), 1445–1450 (2011)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire d’Automatique et Informatique Appliquée (LAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Dschang, Cameroon

    J. Kengne, A. Nguomkam Negou & D. Tchiotsop

  2. Laboratory of Electronics and Signal Processing, Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon

    A. Nguomkam Negou

Authors
  1. J. Kengne
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Nguomkam Negou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Tchiotsop
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Kengne.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kengne, J., Negou, A.N. & Tchiotsop, D. Antimonotonicity, chaos and multiple attractors in a novel autonomous memristor-based jerk circuit. Nonlinear Dyn 88, 2589–2608 (2017). https://doi.org/10.1007/s11071-017-3397-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3397-1

Keywords

Search

Navigation