FRACTAL PROPERTIES OF THE GENERALIZED MANDELBROT SET WITH COMPLEX EXPONENT

Shuai Liu, Xiyu Xu, Gautam Srivastava and Hari M. Srivastava
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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

FRACTALS (fractals), 2024, vol. 32, issue 04, 1-8

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.

Keywords: Fractal; Mandelbrot Set; Generalized Mandelbrot Set; Complex Exponent; Fractal Property (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1142/S0218348X23401217

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