Skip to main content
SpringerLink
Log in

Desired number of coexisting chaotic attractors using quaternionic fractal

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

Abstract

Multistability and multiscroll are two interesting phenomena in chaotic systems. When multiscroll attractors can be generated easily by some approaches, there is few generic method to construct desired number of coexisting chaotic attractors. To address this issue, the quaternionic fractal process inspired by quaternionic Julia sets iterative mapping is presented in this paper. The proposed methodology can not only apply to arbitrary existing continuous chaotic systems, but also generate any number of coexisting attractors based on the seed chaotic systems. Moreover, the design can be completed with simple variable substitution. In order to evaluate the performance of the quaternionic fractal process, we employ it to different types of 4D chaotic systems in a specific set of parameter regions. Moreover, the general form of quaternionic fractal process is presented so that other dimensional chaotic systems can be transformed. The dynamic behavior of fractal-processed system is investigated by means of phase portraits, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The result indicates that the quaternionic fractal process is applicable in different cases. Furthermore, a microcontroller-based hardware platform is developed to show its feasibility in industrial field. The experimental results imply that desired number of coexisting attractors are generated with preloading corresponding initial values.

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

Similar content being viewed by others

Data availability

The data that support the findings of this study are available on request from the corresponding author.

References

  1. Li, H., Hua, Y., Bao, H., Zhu, L., Chen, M., Bao, B.: Two-dimensional memristive hyperchaotic maps and application in secure communication. IEEE Trans. Ind. Electron. 68(10), 9931–9940 (2021)

    Article  Google Scholar 

  2. Shi, X., Duan, S., Wang, L., Huang, T., Li, C.: A novel memristive electronic synapse-based Hermite chaotic neural network with application in cryptography. Neurocomputing 166, 487–495 (2015)

    Article  Google Scholar 

  3. Shi, H., Yan, D., Wang, L., Duan, S.: A novel memristor-based chaotic image encryption algorithm with Hash process and S-box. Eur. Phys. J. Spec. Top. 231, 465–480 (2022)

    Article  Google Scholar 

  4. Chen, B., Yu, S., Chen, P., Xiao, L., Lu, J.: Design and virtex-7-based implementation of video chaotic secure communications. Int. J. Bifurc. Chaos 30(5), 24 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bao, H., Hua, Z., Wang, N., Zhu, L., Chen, M., Bao, B.: Initials-boosted coexisting chaos in a 2-D sine map and its hardware implementation. IEEE Trans. Ind. Inform. 17(2), 1132–1140 (2021)

    Article  Google Scholar 

  6. Morfu, S., Nofiele, B., Marquie, P.: On the use of multistability for image processing. Phys. Lett. A 367(3), 192–198 (2007)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, C., Sprott, J.: Multistability in the lorenz system: a broken butterfly. Int. J. Bifurc. Chaos 24(10), 7 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bao, B., Jiang, T., Wang, G., Jin, P., Bao, H., Chen, M.: Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonlinear Dyn. 89(2), 1157–1171 (2017)

    Article  Google Scholar 

  10. Xu, Q., Lin, Y., Bao, B., Chen, M.: Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit. Chaos Soliton Fract. 83, 186–200 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lai, Q., Kuate, P.D.K., Liu, F., Iu, H.H.C.: An extremely simple chaotic system with infinitely many coexisting attractors. IEEE Trans. Circuits Syst. II 67(6), 1129–1133 (2020)

    Article  Google Scholar 

  12. Wang, M., Li, J., Zhang, X., Iu, H.H.C., Fernando, T., Li, Z., Zeng, Y.: A novel non-autonomous chaotic system with infinite 2-D lattice of attractors and bursting oscillations. IEEE Trans. Circuits Syst. II 68(3), 1023–1027 (2021)

    Article  Google Scholar 

  13. Ye, X., Wang, X.: Characteristic analysis of a simple fractional-order chaotic system with infinitely many coexisting attractors and Its DSP implementation. Phys. Scr. 95(7), 9 (2020)

    Article  Google Scholar 

  14. Li, C., Wang, R., Ma, X., Jiang, Y., Liu, Z.: Embedding any desired number of coexisting attractors in memristive system. Chin. Phys. B 30(12), 10 (2021)

    Article  Google Scholar 

  15. Zhang, S., Li, C., Zheng, J., Wang, X., Zeng, Z., Peng, X.: Generating any number of initial offset-boosted coexisting chua’s double-scroll attractors via piecewise-nonlinear memristor. IEEE Trans. Ind. Electron. 69(7), 7202–7212 (2022)

    Article  Google Scholar 

  16. Mandelbrot, B.: How long is coast of britain-statistical self-similarity and fractional dimension. Science 156(3775), 636 (1967)

    Article  Google Scholar 

  17. Lin, H., Wang, C., Yu, F., Xu, C., Hong, Q., Yao, W., Sun, Y.: An extremely simple multiwing chaotic system: dynamics analysis, encryption application, and hardware implementation. IEEE Trans. Ind. Electron. 68(12), 12708–12719 (2021)

    Article  Google Scholar 

  18. Bouallegue, K., Chaari, A., Toumi, A.: Multi-scroll and multi-wing chaotic attractor generated with Julia process fractal. Chaos Soliton Fract. 44(1–3), 79–85 (2011)

    Article  MathSciNet  Google Scholar 

  19. Bouallegue, K.: Chaotic attractors with separated scrolls. Chaos 25(7), 11 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Atangana Bouallegue, G., Bouallegue, K.: New multi-scroll attractors obtained via Julia set mapping. Chaos Soliton Fract. 134, 11 (2020)

    MathSciNet  MATH  Google Scholar 

  21. Yan, D., Ji’e, M., Wang, L., Duan, S., Du, X.: Generating novel multi-scroll chaotic attractors via fractal transformation. Nonlinear Dyn. 107(4), 3919–3944 (2022)

    Article  Google Scholar 

  22. Guo, Y., Qi, G., Hamam, Y.: A multi-wing spherical chaotic system using fractal process. Nonlinear Dyn. 85(4), 2765–2775 (2016)

    Article  MATH  Google Scholar 

  23. Du, X., Wang, L., Yan, D., Duan, S.: A multiring julia fractal chaotic system with separated-scroll attractors. IEEE Trans. VLSI Syst. 29(12), 2210–2219 (2021)

    Article  Google Scholar 

  24. Norton, A.: Julia sets in the quaternions. Comput. Graph. 13(2), 267–278 (1989)

    Article  Google Scholar 

  25. Nakane, S.: Dynamics of a family of quadratic maps in the quaternion space. Int. J. Bifurc Chaos 15(8), 2535–2543 (2005)

    Article  MATH  Google Scholar 

  26. Wang, X., Sun, Y.: The general quaternionic M-J sets on the mapping z <- z(alpha)+c (alpha is an element of N). Comput. Math. Appl. 53(11), 1718–1732 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sun, Y., Li, P., Lu, Z.: Generalized quaternion M sets and Julia sets perturbed by dynamical noises. Nonlinear Dyn. 82(1–2), 143–156 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lan, R., He, J., Wang, S., Liu, Y., Luo, X.: A parameter-selection-based chaotic system. IEEE Trans. Circuits Syst. II 66(3), 492–496 (2019)

    Article  Google Scholar 

  29. Dadras, S., Momeni, H., Qi, G., Wang, Z.: Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form. Nonlinear Dyn. 67(2), 1161–1173 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining lyapunov exponents from a time-series. Physica D 16(3), 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lai, Q., Nestor, T., Kengne, J., Zhao, X.: Coexisting attractors and circuit implementation of a new 4D chaotic system with two equilibria. Chaos Soliton Fract. 107, 92–102 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu, X., Li, G., Dai, W., Song, X.: Multi-direction chain and grid chaotic system based on julia fractal. Fractals 29(08), 20 (2021)

    Article  MATH  Google Scholar 

  33. Ji’e, M., Yan, D., Du, X., Duan, S., Wang, L.: A novel conservative system with hidden flows evolved from the simplest memristive circuit. Chaos 32(3), 15 (2022)

    Article  MathSciNet  Google Scholar 

  34. Yan, D., Ji’e, M., Wang, L., Duan, S.: Memristor-based chaotic system with abundant dynamical behaviors and its application. Eur. Phys. J. Plus 136(10), 27 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

Project supported by the National Key Research and Development Program of China (Grant No. 2018YFB1306600), the National Natural Science Foundation of China (Grant Nos. 62076207, 62076208, and U20A20227), and the Science and Technology Plan Program of Yubei District of Chongqing (Grant No. 2021-17).

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. College of Artificial Intelligence, Southwest University, Chongqing, 400715, China

    Xinyu Du, Dengwei Yan, Hang Shi, Shukai Duan & Lidan Wang

Authors
  1. Xinyu Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dengwei Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hang Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shukai Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Lidan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lidan Wang.

Ethics declarations

Conflict of interest

We declare that we have no conflict of interest. All authors contribute equally to this work.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Du, X., Yan, D., Shi, H. et al. Desired number of coexisting chaotic attractors using quaternionic fractal. Nonlinear Dyn 111, 831–845 (2023). https://doi.org/10.1007/s11071-022-07825-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07825-5

Keywords

Search

Navigation