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A brief note on fractal dynamics of fractional Mandelbrot sets

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  • Wang, Yupin
  • Li, Xiaodi
  • Wang, Da
  • Liu, Shutang

Abstract

This paper preliminary examines a kind of Mandelbrot set generated by a fractional difference quadratic map involving Caputo-like fractional h-difference operators. A connectivity index is proposed based on numerical methods, which avoids difficulties in discussion at the topological level. The dynamics of those sets in two kinds of noise environments are considered involving connectivity, symmetry and dimension. Several typical cases are visualized to illustrate the main conclusions.

Suggested Citation

  • Wang, Yupin & Li, Xiaodi & Wang, Da & Liu, Shutang, 2022. "A brief note on fractal dynamics of fractional Mandelbrot sets," Applied Mathematics and Computation, Elsevier, vol. 432(C).
    • Handle: RePEc:eee:apmaco:v:432:y:2022:i:c:s0096300322004271
    DOI: 10.1016/j.amc.2022.127353
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    1. Chu, Yu-Ming & Bekiros, Stelios & Zambrano-Serrano, Ernesto & Orozco-López, Onofre & Lahmiri, Salim & Jahanshahi, Hadi & Aly, Ayman A., 2021. "Artificial macro-economics: A chaotic discrete-time fractional-order laboratory model," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    2. Negi, Ashish & Rani, Mamta, 2008. "Midgets of superior Mandelbrot set," Chaos, Solitons & Fractals, Elsevier, vol. 36(2), pages 237-245.
    3. Yupin Wang & Shutang Liu & Hui Li, 2021. "New Fractal Sets Coined From Fractional Maps," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(08), pages 1-12, December.
    4. Wu, Guo-Cheng & Baleanu, Dumitru & Luo, Wei-Hua, 2017. "Lyapunov functions for Riemann–Liouville-like fractional difference equations," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 228-236.
    5. Xingyuan Wang & Ruihong Jia & Yuanyuan Sun, 2009. "The Generalized Julia Set Perturbed by Composing Additive and Multiplicative Noises," Discrete Dynamics in Nature and Society, Hindawi, vol. 2009, pages 1-18, March.
    6. Ran, Jie & Li, Yu-Qin & Xiong, Yi-Bin, 2022. "On the dynamics of fractional q-deformation chaotic map," Applied Mathematics and Computation, Elsevier, vol. 424(C).
    7. Luo, Cheng & Liu, Bao-Qing & Hou, Hu-Shuang, 2021. "Fractional chaotic maps with q–deformation," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    8. Ze-Yu Liu & Tie-Cheng Xia & Hua-Rong Feng & Chang-You Ma, 2021. "Image Encryption Based On Two-Dimensional Fractional Quadric Polynomial Map," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(08), pages 1-16, December.
    9. Wang, Yupin & Liu, Shutang & Li, Hui & Wang, Da, 2019. "On the spatial Julia set generated by fractional Lotka-Volterra system with noise," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 129-138.
    10. Wu, Junru, 2020. "On a linearity between fractal dimension and order of fractional calculus in Hölder space," Applied Mathematics and Computation, Elsevier, vol. 385(C).
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    5. Wang, Yupin, 2023. "Fractional quantum Julia set," Applied Mathematics and Computation, Elsevier, vol. 453(C).

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