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Fractal Properties Of The Generalized Mandelbrot Set With Complex Exponent

Author

Listed:
  • SHUAI LIU

    (School of Educational Science, Hunan Normal University, No. 36, Lushan Road, Changsha 410081, P. R. China†College of Information Science and Engineering, Hunan Normal University, Changsha 410081, P. R. China‡Institute of Interdisciplinary Studies, Hunan Normal University, No. 36, Lushan Road, Changsha 410081, P. R. China)

  • XIYU XU

    (School of Educational Science, Hunan Normal University, No. 36, Lushan Road, Changsha 410081, P. R. China†College of Information Science and Engineering, Hunan Normal University, Changsha 410081, P. R. China‡Institute of Interdisciplinary Studies, Hunan Normal University, No. 36, Lushan Road, Changsha 410081, P. R. China)

  • GAUTAM SRIVASTAVA

    (�Department of Mathematics & Computer Science, Brandon University, Brandon R7A6A9, Canada∥Research Center for Interneural Computing, China Medical University, Taichung 40402, Taiwan)

  • HARI M. SRIVASTAVA

    (�Department of Mathematics and Statistics, University of Victoria, Victoria (British Columbia), V8W3R4, Canada**Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan††Center for Converging Humanities, Kyung Hee University, Seoul 02447, Republic of Korea)

Abstract

Mandelbrot set, which was provided as a highlight in fractal and chaos, is studied by many researchers. With the extension of Mandelbrot set to generalized M set with different kinds of exponent k (k − M set), properties are hard to understand when k is a complex number. In this paper, fractal property of generalized M set with complex exponent z is studied. First, a relation is constructed between generalized M set with complex and real exponent. Then, distribution of z − M set on complex plane is researched. Meanwhile, symmetry of generalized M set is proved. Finally, graphics, generated by escape time algorithm, are the validated results of this paper.

Suggested Citation

  • Shuai Liu & Xiyu Xu & Gautam Srivastava & Hari M. Srivastava, 2024. "Fractal Properties Of The Generalized Mandelbrot Set With Complex Exponent," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 32(04), pages 1-8.
    • Handle: RePEc:wsi:fracta:v:32:y:2024:i:04:n:s0218348x23401217
    DOI: 10.1142/S0218348X23401217
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