A NEW STABLE INTERNAL STRUCTURE OF THE MANDELBROT SET DURING THE ITERATION PROCESS

Dakuan Yu and Wurui Ta
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Dakuan Yu: Department of Mechanics and Engineering Science, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, P. R. China†School of Oceanography, Shanghai Jiao Tong University, Shanghai, P. R. China
Wurui Ta: Department of Mechanics and Engineering Science, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, P. R. China

FRACTALS (fractals), 2021, vol. 29, issue 01, 1-12

Abstract: In this paper, we focus on the periodic points during the iteration process to understand the convergence characteristics of the Mandelbrot set. A new internal structure of the Mandelbrot set is obtained. In addition, the influences of the number of initial iteration points and computational accuracy on this structure are investigated. The results show that the new internal structure is stable no matter how the computational accuracy and the number of initial iteration points change. However, the boundaries of subsets of the Mandelbrot set are sensitive to the computational accuracy. This finding reveals the true convergence structure of the Mandelbrot set during numerical simulations, which differs from the theoretical convergence point distribution. Thus, it helps us further understand the convergence mechanism of the Mandelbrot set.

Keywords: Stable Internal Structure; Computational Accuracy; Boundary Sensitivity (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X2150002X

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