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Dynamical analysis of the improper fractional-order 2D-SCLMM and its DSP implementation

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Abstract

In this paper, the definition of fractional calculus introduces a newly constructed discrete chaotic map. Interestingly, when the order is extended to the improper fractional order, the map still shows a chaotic state with appropriate parameters. The related dynamical behavior is analyzed by a phase diagram, bifurcation model, maximum Lyapunov exponent diagram, and approximate entropy algorithm method. The research results show that both the chaotic range of the fractional-order and the improper fractional-order are larger than the chaotic range of the integer-order, and they have richer dynamical behaviors. Besides, we found that multi-stability phenomena also exist in discrete chaotic systems, and the multiple stability of fractional order is more complicated than that of integer order, and the types of coexisting attractors are more abundant. Finally, the digital circuit and pseudo-random sequence generator of the discrete system are designed. This research guides the application and teaching of discrete fractional-order systems.

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Acknowledgements

This work was supported by Provincial Natural Science Foundation of Liaoning (Grant Nos. 2020-MS-274); National Natural Science Foundation of China (Grant Nos. 62061014).

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Authors and Affiliations

  1. School of Information Science and Engineering, Dalian polytechnic University, Dalian, 116034, China

    Tianming Liu, Huizhen Yan & Jun Mou

  2. Department of Mathematical Sciences, Giuseppe Luigi Lagrange, Politecnico di Torino, Corso Duca degli Abruzzi 24, Turin, Italy

    Santo Banerjee

Authors
  1. Tianming Liu
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  2. Santo Banerjee
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  3. Huizhen Yan
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  4. Jun Mou
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Contributions

Tianming Liu designed and carried out experiments, analyzed data and wrote manuscript . Jun Mou and Santo Banerjee made the theoretical guidance for this paper. Jun Mou carried out experiment. Tianming Liu improved the algorithm. All authors reviewed the manuscript.

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Correspondence to Santo Banerjee or Jun Mou.

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Liu, T., Banerjee, S., Yan, H. et al. Dynamical analysis of the improper fractional-order 2D-SCLMM and its DSP implementation. Eur. Phys. J. Plus 136, 506 (2021). https://doi.org/10.1140/epjp/s13360-021-01503-y

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