Numeric characteristics of generalized M-set with its asymptote

Liu, Shuai, Cheng, Xiaochun ORCID: https://orcid.org/0000-0003-0371-9646, Fu, Weina, Zhou, Yunpeng and Li, Qianzhong (2014) Numeric characteristics of generalized M-set with its asymptote. Applied Mathematics and Computation, 243 . pp. 767-774. ISSN 0096-3003 [Article] (doi:10.1016/j.amc.2014.06.016)

Abstract

The Mandelbrot set (M-set) formulated by fc(z) = z2 + c is important in chaos theory. It has been extended into generalized M set (k-M set) fc(z) = zk + c by using different exponent k. The escape time algorithm is used to draw fractals, which need preset thresholds and a maximum iteration times as input. This paper presents and proves the upper and lower bounds of generalized M set thresholds. A new method is also introduced to draw k-M set when k is a rational number. This method uses asymptotes of k-M set to approach k-M set from the outside. We prove that k-M set is the limit of these asymptotes. Finally, some experiments are presented to validate our methods.

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