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On fractional difference logistic maps: Dynamic analysis and synchronous control

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Abstract

This paper investigates a logistic map derived from a difference equation in the framework of discrete fractional calculus. Through the Poincaré plots and Julia sets, the map’s chaotic and fractal characteristics are studied comparing with those of a quadratic map to be proposed. The memory effect of fractional difference maps is reflected in these dynamics, and some reasonable explanations are given by combining with quantitative analysis. A coupled controller is designed to realize synchronization between fractional difference logistic map and fractional difference quadratic map.

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Notes

  1. It is calculated by combining Wolf algorithm and bisection method. In the classical case (\(\alpha = 1\)), the relative error from the theoretical value 3.569946 is only 6.99‰.

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

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (Nos. U1806203, 61533011). The authors sincerely thank all anonymous referees for their constructive suggestions and valuable comments that have led to the present improved version of the original manuscript. The first author is also indebted to a referee for providing related works of Prof. Mark Edelman.

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Authors and Affiliations

  1. Institute of Marine Science and Technology, Shandong University, Qingdao, 266237, 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 & Hui Li

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  1. Yupin Wang
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  2. Shutang Liu
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  3. Hui Li
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Correspondence to Shutang Liu.

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We certify that this manuscript is original and has not been published and will not be submitted elsewhere for publication while being considered by Nonlinear Dynamics. And the study is not split up into several parts to increase the quantity of submissions and submitted to various journals or to one journal over time. No data have been fabricated or manipulated (including images) to support our conclusions. No data, text, or theories by others are presented as if they were our own. The submission has been received explicitly from all co-authors. And authors whose names appear on the submission have contributed sufficiently to the scientific work and therefore share collective responsibility and accountability for the results.

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Wang, Y., Liu, S. & Li, H. On fractional difference logistic maps: Dynamic analysis and synchronous control. Nonlinear Dyn 102, 579–588 (2020). https://doi.org/10.1007/s11071-020-05927-6

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