A broken symmetry approach for the modeling and analysis of antiparallel diodes-based chaotic circuits: a case study

Abstract

This paper focuses on the modeling and symmetry breaking analysis of the large class of chaotic electronic circuits utilizing an antiparallel diodes pair as nonlinear device necessary for chaotic oscillations. A new and relatively simple autonomous jerk circuit is used as a paradigm. Unlike current approaches assuming identical diodes (and thus a perfect symmetric circuit), we consider the more realistic situation where both antiparallel diodes present different electrical properties in spite of unavoidable scattering of parameters. Hence, the nonlinear component synthesized by the diodes pair exhibits an asymmetric current–voltage characteristic which engenders the explicit symmetry break of the whole electronic circuit. The mathematical model of the new circuit consists of a continuous time 3D autonomous system with exponential nonlinear terms. We investigate the chaos mechanism with respect to model parameters and initial conditions as well, both in the symmetric and asymmetric modes of operation by using bifurcation diagrams and phase space trajectories plots as main indicators. We report period doubling route to chaos, spontaneous symmetry breaking, merging crisis and the coexistence of two, four, or six mutually symmetric attractors in the symmetric regime of operation. More intriguing and complex nonlinear behaviors are revealed in the asymmetric system such as coexisting asymmetric bubbles of bifurcation (a new kind of phenomenon discovered in this work), hysteretic dynamics, critical phenomena, and coexisting multiple (i.e. two, three, four, or five) asymmetric attractors for some appropriately selected values of parameters. Laboratory experimental tests are conducted to support the theoretical analysis. The results obtained in this work clearly indicate that chaotic circuits with back to back diodes can demonstrate much more complex dynamics than what is reported in the relevant literature and thus should be reconsidered accordingly following the method described in this paper.

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

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
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25

References

  1. 1.

    Chen, X., Lin, Y., & Bao, B. (2017). Multistability induced by two symmetric stable node-foci in modified canonical Chua’s circuit. Nonlinear Dynamics,87, 789–802.

    Google Scholar 

  2. 2.

    Maggio, G. M., De Feo, O., & Kennedy, M. P. (1999). Nonlinear analysis of the Colpitts oscillator and application to design. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications,46, 1118–1130.

    MATH  Google Scholar 

  3. 3.

    Freire, E., Franquelo, L. G., & Aracil, J. (1994). Periodicity and chaos in an autonomous electrical system. IEEE Transactions on Circuits and Systems,31(3), 237–247.

    Google Scholar 

  4. 4.

    Kengne, J., Njitacke, Z. T., Nguomkam, N. A., Fouodji, T. M., & Fotsin, H. B. (2015). Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit. International Journal of Bifurcation and Chaos,25(4), 1550052.

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Fozin, F. T., Kengne, J., & Pelap, F. B. (2018). Dynamical analysis and multistability in autonomous hyperchaotic oscillator with experimental verification. Nonlinear Dynamics,93(2), 653–669.

    Google Scholar 

  6. 6.

    Jeevarekha, A., Sabarathinam, S., Thamilmaran, K., & Philomenathan, P. (2016). Analysis of a 4D autonomous system with volume-expanding phase space. Nonlinear Dynamics,84(4), 2273–2284.

    Google Scholar 

  7. 7.

    Kengne, J., Tabekoueng, Z. N., & Fotsin, H. B. (2016). Coexistence of multiple attractors and crisis route to chaos in autonomous third order Duffing–Holmes type chaotic oscillators. Communications in Nonlinear Science and Numerical Simulation,36, 29–44.

    MathSciNet  Google Scholar 

  8. 8.

    Leutcho, G. D., Kengne, J., & Kamdjeu, K. L. (2018). Dynamical analysis of a novel autonomous 4-D hyperjerk circuit with hyperbolic sine nonlinearity: Chaos, antimonotonicity and a plethora of coexisting attractors Chaos. Solitons and Fractals,107, 67–87.

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Kahllert, C. (1993). The effects of symmetry breaking in Chua’s circuit and related piecewise-linear dynamical system. International Journal of Bifurcation and Chaos,3(4), 963–979.

    MathSciNet  Google Scholar 

  10. 10.

    Dana, S. K., Chakraborty, S., & Ananthakrishna, G. (2005). Homoclinic bifurcation in Chua’s circuit. Pramana Journal of Physics,64(3), 44344.

    Google Scholar 

  11. 11.

    Cao, H., Seoane, J. M., & Sanjuan, M. A. F. (2007). Symmetry-breaking analysis for the general Helmholz–Duffing oscillator. Chaos, Solitons & Fractals,34, 197–212.

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Sofroniou, A., & Bishop, S. R. (2006). Breaking the symmetry of the parametrically excited pendulum. Chaos, Solitons & Fractals,28, 673–681.

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Bishop, S. R., Sofroniou, A., & Shi, P. (2005). Symmetry-breaking in the response of the parameterically excited pendulum model. Chaos, Solitons & Fractals,25(2), 27–264.

    MATH  Google Scholar 

  14. 14.

    Rynio, R., & Okninski, A. (1998). Symmetry breaking and fractal dependence on initial conditions in dynamical systems: Ordinary differential equations of thermal convection. Chaos, Solitons & Fractals,9(10), 1723–1732.

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Henrich, M., Dahms, T., Flunkert, V., Teitsworth, S. W., & Scholl, E. (2010). Symmetry breaking transitions in networks of nonlinear circuits elements. New Journal of Physics,12, 113030.

    Google Scholar 

  16. 16.

    Cao, H., & Jing, Z. (2001). Chaotic dynamics of Josephson equation driven by constant and ac forcings. Chaos, Solitons & Fractals,12, 1887–1895.

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Kengne, J., Njitacke, Z. T., & Fotsin, H. B. (2016). Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dynamics,83, 751–765.

    MathSciNet  Google Scholar 

  18. 18.

    Kengne, J., Folifack Signing, V. R., Chedjou, J. C., & Leutcho, G. D. (2017). Nonlinear behavior of a novel chaotic jerk system: Antimonotonicity, crises, and multiple coexisting attractors. International Journal of Dynamics and Control,6, 468–485. https://doi.org/10.1007/s40435-017-0318-6.

    MathSciNet  Article  Google Scholar 

  19. 19.

    Kengne, J., Njikam, S. M., & Folifack, V. R. (2018). A plethora of coexisting strange attractors in a simple jerk system with hyperbolic tangent nonlinearity. Chaos, Solitons & Fractals,106, 201–213.

    MathSciNet  MATH  Google Scholar 

  20. 20.

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

    MATH  Google Scholar 

  21. 21.

    Kengne, J., & Mogue, R. L. T. (2018). Dynamic analysis of a novel jerk system with composite tanh-cubic nonlinearity: Chaos, multi-scroll, and multiple coexisting attractors. International Journal of Dynamics and Control. https://doi.org/10.1007/s40435-018-0444-9.

    Article  Google Scholar 

  22. 22.

    Kingni, S. T., Pone, J. R. M., Kuiate, G. F., & Pham, V. T. (2019). Coexistence of attractors in integer-and fractional-order three-dimensional autonomous systems with hyperbolic sine nonlinearity: Analysis, circuit design and combination synchronization. Pramana,93(1), 12.

    Google Scholar 

  23. 23.

    Joshi, M., & Ranjan, A. (2019). An autonomous chaotic and hyperchaotic oscillator using OTRA. Analog Integrated Circuits and Signal Processing,101(3), 401–413.

    Google Scholar 

  24. 24.

    Tamaševičius, A., Bumelienė, S., Kirvaitis, R., Mykolaitis, G., Tamaševičiūtė, E., & Lindberg, E. (2009). Autonomous Duffing–Holmes type chaotic oscillator. Elektronika ir Elektrotechnika,3(5), 43–46.

    MATH  Google Scholar 

  25. 25.

    Joshi, M., & Ranjan, A. (2019). New simple chaotic and hyperchaotic system with an unstable node. AEU-International Journal of Electronics and Communications,108, 1–9.

    Google Scholar 

  26. 26.

    Pone, J. R. M., Çiçek, S., Kingni, S. T., Tiedeu, A., & Kom, M. (2020). Passive–active integrators chaotic oscillator with anti-parallel diodes: Analysis and its chaos-based encryption application to protect electrocardiogram signals. Analog Integrated Circuits and Signal Processing,103, 1–15.

    Google Scholar 

  27. 27.

    Sprott, J. C. (2011). A new chaotic jerk circuit. IEEE Transactions on Circuits and Systems II: Express Briefs,58, 240–243.

    Google Scholar 

  28. 28.

    Louodop, P., Kountchou, M., Fotsin, H., & Bowong, S. (2014). Practical finite-time synchronization of jerk systems: Theory and experiment. Nonlinear Dynamics,78, 597–607.

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Hanias, M. P., Giannaris, G., & Spyridakis, R. A. (2006). Time series analysis in chaotic diode resonator circuit. Chaos, Solitons & Fractals,27, 569–573.

    MATH  Google Scholar 

  30. 30.

    Sukov, D. W., Bleich, M. E., Gauthier, J., & Socolar, J. E. S. (1997). Controlling chaos in a fast diode resonator using extended time-delay autosynchronization: Experimental observations and theoretical analysis. Chaos,7, 560–576.

    Google Scholar 

  31. 31.

    Sprott, J. C. (2010). Elegant chaos: Algebraically simple flow. Singapore: World Scientific Publishing.

    Google Scholar 

  32. 32.

    Li, C., & Sprott, J. C. (2013). Amplitude control approach for chaotic signals. Nonlinear Dynamics,73, 1335–1341.

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Strogatz, S. H. (1994). Nonlinear dynamics and chaos. Reading: Addison-Wesley.

    Google Scholar 

  34. 34.

    Nayfeh, A. H., & Balachandran, B. (1995). Applied nonlinear dynamics: Analytical, computational and experimental methods. New York: Wiley.

    Google Scholar 

  35. 35.

    Kuznetsov, Y. A. (1995). Elements of applied bifurcation theory. New York: Springer.

    Google Scholar 

  36. 36.

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

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Leonov, G. A., Kuznetsov, N. V., & Mokaev, T. N. (2015). Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. European Physical Journal Special Topics,224, 1421–1458.

    Google Scholar 

  38. 38.

    Pham, V. T., Jafari, S., Volos, C., Giakoumis, A., Vaidyanathan, S., & Kapitaniak, T. (2016). A chaotic system with equilibria located on the rounded square loop and its circuit implementation. IEEE Transactions on Circuits and Systems II: Express Briefs,6(9), 878–882.

    Google Scholar 

  39. 39.

    Jafari, S., Sprott, J. C., & Golpayegani, S. M. R. H. (2013). Elementary quadratic chaotic flows with no equilibria. Physics Letters A,377(9), 699–702.

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Jafari, S., Pham, V. T., & Kapitaniak, T. (2016). Multiscroll chaotic sea obtained from a simple 3D system without equilibrium. International Journal of Bifurcation and Chaos,26(02), 1650031.

    MathSciNet  MATH  Google Scholar 

  41. 41.

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

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Bier, M., & Bountis, T. C. (1994). Remerging Feigenbaum trees in dynamical systems. Physics Letters A,104, 239–244.

    MathSciNet  Google Scholar 

  43. 43.

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

    MathSciNet  Google Scholar 

  44. 44.

    Kyprianidis, I., Stouboulos, I., Haralabidis, P., & Bountis, T. (2000). Antimonotonicity and chaotic dynamics in a fourth-order autonomous nonlinear electric circuit. International Journal of Bifurcation and Chaos,10, 1903–1915.

    Google Scholar 

  45. 45.

    Kocarev, L., Halle, K. S., Eckert, K., & Chua, L. O. (1993). Experimental observation of antimonotonicity in Chua’s circuit. International Journal of Bifurcation and Chaos,3(4), 1051–1055.

    MATH  Google Scholar 

  46. 46.

    Kengne, J. (2017). On the dynamics of Chua’s oscillator with a smooth cubic nonlinearity: Occurrence of multiple attractors. Nonlinear Dynamics,87(1), 363–375.

    Google Scholar 

  47. 47.

    Lai, Q., & Chen, S. (2016). Generating multiple chaotic attractors from Sprott B system. International Journal of Bifurcation and Chaos,26(11), 1650177.

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Li, C., & Sprott, J. C. (2014). Coexisting hidden attractors in a 4-D simplified Lorenz system. International Journal of Bifurcation and Chaos,24, 1450034.

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Li, C., Hu, W., Sprott, J. C., & Wang, X. (2015). Multistability in symmetric chaotic systems. The European Physical Journal Special Topics,224, 1493–1506.

    Google Scholar 

  50. 50.

    Leipnik, R. B., & Newton, T. A. (1981). Double strange attractors in rigid body motion with linear feedback control. Physics Letters A,86, 63–87.

    MathSciNet  Google Scholar 

  51. 51.

    Leutcho, G. D., & Kengne, J. (2018). A unique chaotic snap system with a smoothly adjustable symmetry and nonlinearity: Chaos, offset-boosting, antimonotonicity, and coexisting multiple attractors. Chaos, Solitons & Fractals,113, 275–293.

    MathSciNet  Google Scholar 

  52. 52.

    Luo, X., & Small, M. (2007). On a dynamical system with multiple chaotic attractors. International Journal of Bifurcation and Chaos,17(9), 3235–3251.

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Pisarchik, A. N., & Feudel, U. (2014). Control of multistability. Physics Reports,540(4), 167–218.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jacques Kengne.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kamdjeu Kengne, L., Kamdem Tagne, H.T., Kengnou Telem, A.N. et al. A broken symmetry approach for the modeling and analysis of antiparallel diodes-based chaotic circuits: a case study. Analog Integr Circ Sig Process 104, 205–227 (2020). https://doi.org/10.1007/s10470-020-01664-3

Download citation

Keywords

  • Chaotic circuit with a diodes pair
  • Coexisting bubbles
  • Critical transitions
  • Coexisting multiple stable states
  • Experimental study