Skip to main content
Log in

Complex dynamics of a novel 3D autonomous system without linear terms having line of equilibria: coexisting bifurcations and circuit design

  • Published:
Analog Integrated Circuits and Signal Processing Aims and scope Submit manuscript

Abstract

In recent years, an increasing interest has been devoted to the discovery of new chaotic systems with special properties. In this paper, a novel and singular 3D autonomous system without linear terms is introduced. The singularity of the model is that it is dissipative, possesses rotation symmetry and line of equilibria thus displays complex dynamics. The nonlinear behaviour of the introduced model is studied in terms of bifurcation diagrams, Lyapunov exponent plots, time series, frequency spectra two-parameter diagrams, Lyapunov stability diagrams as well as basins of attraction. Some interesting phenomena are found including, for instance, periodic oscillations, chaotic oscillations, periodic windows, symmetry restoring crises scenario, the coexistence of multiple bifurcations and offset-boosting property while monitoring the system parameters. Coexistence of attractors discovered in this work includes up to six competing attractors with one point attractor. Compared to some few cases previously reported (system without linear terms) (Xu and Wang in Opt Int J Light Electron Opt 125:2526–2530, 2014. https://doi.org/10.1007/s11071-016-3170-x; Kengne in Commun Nonlinear Sci Numer Simul 52:62–76, 2017. https://doi.org/10.1016/j.cnsns.2017.04.017; Mobayen et al. in Int J Syst Sci 49:1–15, 2018. https://doi.org/10.1080/00207721.2017.1410251; Zhang et al. in Int J Non-linear Mech 106:1–12, 2018. https://doi.org/10.1016/j.ijnonlinmec.2018.08.012; Pham et al. in Chaos Solitons Fractals 120:213–221, 2019. https://doi.org/10.1016/j.chaos.2019.02.003), the model considered in this work is the only one in which such type of complex dynamics has already been found. A suitable analog simulator (electronics circuit) is designed and used to support the theoretical analysis.

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

Access this article

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. Kengne, J., Njitacke, T. Z., Kamdoum, T. V., & Nguomkam, N. A. (2015). Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit. Chaos,25, 103126. https://doi.org/10.1063/1.4934653.

    Article  MathSciNet  MATH  Google Scholar 

  2. Zhou, L., Wang, C. H., & Zhou, L. L. (2018). A novel no-equilibrium hyperchaotic multi-wing system via introducing memristor. International Journal of Circuit Theory and Applications,46, 1–15. https://doi.org/10.1002/cta.2339.

    Article  Google Scholar 

  3. Jafari, S., Golpayegania, S. M. R. H., & Sprott, J. C. (2013). Elementary quadratic chaotic flows with no equilibria. Physics Letters A,377, 699–702. https://doi.org/10.1016/j.physleta.2013.01.00.

    Article  MathSciNet  MATH  Google Scholar 

  4. Zhang, S., Zeng, Y. C., Li, Z. J., Wang, M. J., & Xiong, L. (2018). Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability. Chaos,28, 013113. https://doi.org/10.1063/1.5006214.

    Article  MathSciNet  MATH  Google Scholar 

  5. Pham, V. T., Volos, C., Jafari, S., & Kapitaniak, T. (2017). Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dynamics,87, 2001–2010. https://doi.org/10.1007/s11071-016-3170-x.

    Article  Google Scholar 

  6. Folifack Signing, V. R., & Kengne, J. (2019). Reversal of period-doubling and extreme multistability in a novel 4D chaotic system with hyperbolic cosine nonlinearity. International Journal of Dynamics and Control,7, 439. https://doi.org/10.1007/s40435-018-0452-9.

    Article  MathSciNet  Google Scholar 

  7. Folifack Signing, V. R., Kengne, J., & Kana, L. K. (2018). Dynamic analysis and multistability of a novel four-wing chaotic system with smooth piecewise quadratic nonlinearity. Chaos, Solitons and Fractals,113, 263–274. https://doi.org/10.1016/j.chaos.2018.06.008.

    Article  MathSciNet  MATH  Google Scholar 

  8. Folifack Signing, V. R., Kengne, J., & MboupdaPone, J. R. (2019). Antimonotonicity, chaos, quasi-periodicity and coexistence of hidden attractors in a new simple 4-D chaotic system with hyperbolic cosine nonlinearity. Chaos, Solitons and Fractals,118, 187–198. https://doi.org/10.1016/j.chaos.2018.10.018.

    Article  MathSciNet  Google Scholar 

  9. Negou, A. N., & Kengne, J. (2019). A minimal three-term chaotic flow with coexisting routes to chaos, multiple solutions, and its analog circuit realization. Analog Integrated Circuits and Signal Processing. https://doi.org/10.1007/s10470-019-01436-8.

    Article  Google Scholar 

  10. Kutnetsov, N. V., Leonov, G. A., Yuldashev, M. V., & Yuldashev, R. V. (2017). Hidden attractors in dynamical models of phase-locked loop circuits: Limitations of simulation in MATLAB and SPICE. Communications in Nonlinear Science and Numerical Simulation,51, 34–49. https://doi.org/10.1016/j.cnsns.2017.03.010.

    Article  Google Scholar 

  11. Leonov, G. A., Kutnetsov, N. V., & Mokaev, T. N. (2017). Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity. Communications in Nonlinear Science and Numerical Simulation,28(3), 166–174. https://doi.org/10.1016/j.cnsns.2015.04.007.

    Article  MathSciNet  Google Scholar 

  12. Dudkowski, D. S., Jafari, S., Kapitaniak, T., Kutnetsov, N. V., Leonov, G. A., & Prasad, A. (2016). Hidden attractors in dynamical systems. Physics Reports,637, 1–50. https://doi.org/10.1016/j.physrep.2016.05.002.

    Article  MathSciNet  MATH  Google Scholar 

  13. Leonov, G. A., Kuznetsov, N. V., & Vagaitsev, V. I. (2011). Localization of hidden Chua’s attractors. Physics Letters A,375, 2230–2233. https://doi.org/10.1016/j.physleta.2011.04.037.

    Article  MathSciNet  MATH  Google Scholar 

  14. Pham, V. T., Volos, Ch K, Jafari, S., & Wang, X. (2018). Dynamics and circuit of a chaotic system with a curve of equilibrium points. International Journal of Electronics,105(3), 385–397. https://doi.org/10.1080/00207217.2017.1357208.

    Article  Google Scholar 

  15. Wei, Z., & Zhang, W. (2014). Hidden hyperchaotic attractors in a modified Lorenz–Stenflo system with only one stable equilibrium. International Journal of Bifurcation and Chaos, 24(10), Article ID 1450127. https://doi.org/10.1142/S0218127414501272.

  16. Petrzela, J., & Gotthans, T. (2017). New chaotic dynamical system with a conic-shaped equilibrium located on the plane structure. Applied Sciences,7(10), 976–989. https://doi.org/10.3390/app7100976.

    Article  Google Scholar 

  17. Pham, V. T., Jafari, S., Volos, Ch K, Vaidyanathan, S., & Kapitaniak, T. (2016). A chaotic system with infinite equilibria located on a piecewise linear curve. Optik,127(20), 9111–9117. https://doi.org/10.1016/j.ijleo.2016.06.11.

    Article  Google Scholar 

  18. Njitacke, Z. T., Kengne, J., Wafo Tapche, R., & Pelap, F. B. (2018). Uncertain destination dynamics of a novel memristive 4D autonomous system. Solitons and Fractals,91, 177–185. https://doi.org/10.1016/j.chaos.2018.01.004.

    Article  MathSciNet  MATH  Google Scholar 

  19. Zuo, Z. L., & Li, C. (2016). Multiple attractors and dynamic analysis of a no-equilibrium chaotic system. Optik,127(19), 7952–7957. https://doi.org/10.1016/j.ijleo.2016.05.069.

    Article  Google Scholar 

  20. Vaidyanathan, S., Pham, V. T., & Volos, C. K. (2015). A 5-D hyperchaotic Rikitake dynamo system with hidden attractors. European Physical Journal,224(8), 1575–1592. https://doi.org/10.1140/epjst/e2015-02481-0.

    Article  Google Scholar 

  21. Bao, B. C., Hu, F. W., Chen, M., Xu, Q., & Yu, Y. J. (2015). Self-excited and hidden attractors found simultaneously in a modified Chua’s circuit. International Journal of Bifurcation and Chaos,25(5), 1550075. https://doi.org/10.1142/S0218127415500753.

    Article  MATH  Google Scholar 

  22. Chen, M., Yu, J. J., & Bao, B. C. (2015). Finding hidden attractors in an improved memristor based Chua’s circuit. Electronics Letters,51(6), 462–464. https://doi.org/10.1049/el.2014.4341.

    Article  Google Scholar 

  23. Chen, M., Li, M. Y., Yu, Q., Bao, B. C., Xu, Q., & Wang, J. (2015). Dynamics of self-excited attractors and hidden attractors in generalized memristor based Chua’s circuit. Nonlinear Dynamics,81(1–2), 215–226. https://doi.org/10.1007/s11071-015-1983-7.

    Article  MathSciNet  MATH  Google Scholar 

  24. Kengne, J. (2017). On the Dynamics of Chua’s oscillator with a smooth cubic nonlinearity: Occurrence of multiple attractors. Nonlinear Dynamics,87, 363–375. https://doi.org/10.1007/s11071-016-3047-z.

    Article  Google Scholar 

  25. Kengne, J., TagneMogue, R. L., Fozin, T. F., & Kengnou Telem, A. N. (2019). Effects of symmetric and asymmetric nonlinearity on the dynamics of a novel chaotic jerk circuit: Coexisting multiple attractors, period doubling reversals, crisis, and offset boosting. Chaos, Solitons and Fractals,121, 63–84. https://doi.org/10.1016/j.chaos.2019.01.033.

    Article  MathSciNet  Google Scholar 

  26. 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. https://doi.org/10.1007/s11071-015-2364-y.

    Article  MathSciNet  Google Scholar 

  27. 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 and Fractals,113, 275–293. https://doi.org/10.1016/j.chaos.2018.05.017.

    Article  MathSciNet  Google Scholar 

  28. Mogue Tagne, R. L., Kengne, J., & Nguomkam Negou, A. (2018). Multistability and chaotic dynamics of a simple Jerk system with a smoothly tuneable symmetry and nonlinearity. International Journal of Dynamics and Control,7, 476–495. https://doi.org/10.1007/s40435-018-0458-3.

    Article  MathSciNet  Google Scholar 

  29. Negou Nguomkam, A., & Kengne, J. (2018). Dynamic analysis of a unique jerk system with a smoothly adjustable symmetry and nonlinearity: Reversals of period doubling, offset boosting and coexisting bifurcations. International Journal of Electronics and Communications (AEÜ),90, 1–19. https://doi.org/10.1016/j.aeue.2018.04.003.

    Article  Google Scholar 

  30. Kengne, J., Njitacke, Z. T., Fotsin, H. B., Negou, A. N., & Tchiotsop, D. (2016). Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bidge-based Jerk circuit. Chaos, Solitons & Fractals,91, 180–197. https://doi.org/10.1142/S0218127416500814.

    Article  MATH  Google Scholar 

  31. Njitacke, Z. T., Kengne, J., & Fotsin, H. B. (2019). A plethora of behaviors in a memristor based Hopfield neural networks (HNNs). International Journal of Dynamics and Control,7, 36. https://doi.org/10.1007/s40435-018-0435-x.

    Article  MathSciNet  Google Scholar 

  32. Njitacke, Z. T., Kengne, J., Fonzin Fozin, T., Leutcha, B. P., & Fotsin, H. B. (2019). Dynamical analysis of a novel 4-neurons based Hopfield neural network: Emergences of antimonotonicity and coexistence of multiple stable states. International Journal of Dynamics and Control,7, 823–841. https://doi.org/10.1007/s40435-019-00509-w.

    Article  MathSciNet  Google Scholar 

  33. Njitacke, Z. T., & Kengne, J. (2019). Nonlinear dynamics of three-neurons-based Hopfield neural networks (HNNs): Remerging Feigenbaum trees, coexisting bifurcations and multiple attractors. Journal of Circuits, Systems, and Computers,28(7), 1950121. https://doi.org/10.1142/S0218126619501214.

    Article  Google Scholar 

  34. Bao, B. C., Qian, H., Wang, J., Xu, Q., Chen, M., Wu, H. G., et al. (2017). Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network. Nonlinear Dynamics,90, 2359. https://doi.org/10.1007/s11071-017-3808-3.

    Article  MathSciNet  Google Scholar 

  35. Bao, B., Qian, H., Xu, Q., Chen, M., Wang, J., & Yu, Y. (2017). Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network. Frontiers in Computational Neuroscience,81, 1–14. https://doi.org/10.3389/fncom.2017.00081.

    Article  Google Scholar 

  36. Bao, B., Hu, A., Bao, H., Xu, Q., Chen, M., & Wu, H. (2018). Three-dimensional memristive Hindmarsh–Rose neuron model with hidden coexisting asymmetric behaviors. Complexity Volume, Article ID 3872573, 11 pp. https://doi.org/10.1155/2018/3872573.

  37. Bao, H., Liu, W., & Hu, A. (2018). Coexisting multiple firing patterns in two adjacent neurons coupled by memristive electromagnetic induction. Nonlinear Dynamics,95, 43–56. https://doi.org/10.1007/s11071-018-4549-7.

    Article  Google Scholar 

  38. Bao, H., Hu, A., & Liu, W. (2019). Bipolar pulse-induced coexisting firing patterns in two-dimensional Hindmarsh–Rose neuron model. International Journal of Bifurcation and Chaos,29(1), 1950006. https://doi.org/10.1142/S0218127419500068.

    Article  MathSciNet  MATH  Google Scholar 

  39. Xu, Y. H., & Wang, Y. L. (2014). A new chaotic system without linear term and its impulsive synchronization. Optik-International Journal for Light and Electron Optics,125, 2526–2530. https://doi.org/10.1007/s11071-016-3170-x.

    Article  Google Scholar 

  40. Kengne, J., Jafari, S., Njitacke, Z. T., Yousefi Azar Khanian, M., & Cheukem, A. (2017). Dynamic analysis and electronic circuit implementation of a novel 3D autonomous system without linear terms. Communications in Nonlinear Science and Numerical Simulation,52, 62–76. https://doi.org/10.1016/j.cnsns.2017.04.017.

    Article  Google Scholar 

  41. Mobayen, S., Kingni, S. T., Pham, V. T., Nazarimehr, F., & Jafari, S. (2018). Analysis, synchronisation and circuit design of a new highly nonlinear chaotic system. International Journal of Systems Science,49, 1–15. https://doi.org/10.1080/00207721.2017.1410251.

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, S., Zeng, Y., & Li, Z. (2018). Chaos in a novel fractional order system without a linear term. International Journal of Non-linear Mechanics,106, 1–12. https://doi.org/10.1016/j.ijnonlinmec.2018.08.012.

    Article  Google Scholar 

  43. Pham, V. T., Jafari, S., Volos, C., & Fortuna, L. (2019). Simulation and experimental implementation of a line–equilibrium system without linear term. Chaos, Solitons and Fractals,120, 213–221. https://doi.org/10.1016/j.chaos.2019.02.003.

    Article  MathSciNet  Google Scholar 

  44. Hilborn, R. C. (1994). Chaos and nonlinear dynamics—An introduction for scientists and engineers. Oxford: Oxford University Press.

    MATH  Google Scholar 

  45. Wolf, A., Swift, J. B., Swinney, H. L., & Wastano, J. A. (1985). Determining Lyapunov exponents from time series. Physica D,16, 285–317. https://doi.org/10.1016/0167-2789(85)90011-9.

    Article  MathSciNet  MATH  Google Scholar 

  46. Sprott, J. C., Jafari, S., Khalaf, A. J. M., & Kapitaniak, T. (2017). Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping. The European Physical Journal Special Topics,226, 1979–1985. https://doi.org/10.1140/epjst/e2017-70037-1.

    Article  Google Scholar 

  47. Li, C., Lu, T., Chen, G., & Xing, H. (2019). Doubling the coexisting attractors. Chaos,29, 051102. https://doi.org/10.1063/1.5097998.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zeric Tabekoueng Njitacke.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tapche, R.W., Njitacke, Z.T., Kengne, J. et al. Complex dynamics of a novel 3D autonomous system without linear terms having line of equilibria: coexisting bifurcations and circuit design. Analog Integr Circ Sig Process 103, 57–71 (2020). https://doi.org/10.1007/s10470-020-01591-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10470-020-01591-3

Keywords

Navigation