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On fractional coupled logistic maps: chaos analysis and fractal control

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Abstract

This paper investigates chaotic and fractal dynamics of fractional coupled logistic maps constructed based on the Caputo fractional h-difference. The chaos of this map, affected by the memory and scale derived from the fractional operator, is examined through phase portrait, “0–1” test and Lyapunov exponent. Fractal synchronization is achieved by designing a coupled controller between Julia sets generated from two fractional coupled maps with different structures. Numerical simulations are presented to validate the main findings.

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Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. Such numerical calculation processes in this work takes advantage of Frédéric Moisy’s MATLAB® function shared on https://ww2.mathworks.cn/matlabcentral/fileexchange/13063-boxcount

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Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. The first author is also indebted to Dr. Hui Li for numerous helpful discussions.

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 62203283, U1806203 and 61533011), Shandong Provincial Natural Science Foundation (Grant No. ZR2022QF009) and the China Postdoctoral Science Foundation (Grant No. 2022M711981).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250358, Shandong, People’s Republic of China

    Yupin Wang

  2. School of Control Science and Engineering, Shandong University, Jinan, 250061, Shandong, People’s Republic of China

    Shutang Liu

  3. Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia

    Aziz Khan

Authors
  1. Yupin Wang
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  2. Shutang Liu
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  3. Aziz Khan
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Yupin Wang, Shutang Liu and Aziz Khan. The first draft of the manuscript was written by Yupin Wang and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Yupin Wang.

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Appendices

Appendix A: Pseudo-code

figure a
figure b

Appendix B: Program script

The MATLAB® code for phase portraits of FCLM (2).

figure c

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Wang, Y., Liu, S. & Khan, A. On fractional coupled logistic maps: chaos analysis and fractal control. Nonlinear Dyn 111, 5889–5904 (2023). https://doi.org/10.1007/s11071-022-08141-8

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