Fractal Structures of General Mandelbrot Sets and Julia Sets Generated from Complex Non-Analytic Iteration Fm(z)=z¯m+c

• Published in: Applied Mechanics and Materials Vol. 347-350; pp. 3019 - 3023
• Main Authors: Jiang, Nan, Wei, Xiao Dan, Zhang, Hong Peng, Yan, De Jun, Liu, Xiang Dong
• Format: Journal Article
• Language: English
• Published: Zurich Trans Tech Publications Ltd 01.08.2013
• Subjects: Complex Non-Analytic Iteration; General Mandelbrot Set; Julia Set; Critical Point
• ISBN: 3037857846; 9783037857847
• ISSN: 1660-9336; 1662-7482; 1662-7482
• DOI: 10.4028/www.scientific.net/AMM.347-350.3019

In this paper we use the same idea as the complex analytic dynamics to study general Mandelbrot sets and Julia sets generated from the complex non-analytic iteration . The definition of the general critical point is given, which is of vital importance to the complex non-analytic dynamics. The general Mandelbrot set is proved to be bounded, axial symmetry by real axis, and have (m+1)-fold rotational symmetry. The stability condition of periodic orbits and the boundary curve of stability region of one-cycle are given. And the general Mandelbrot sets are constructed by the escape-time method and the periodic scanning algorithm, which present a better understanding of the structure of the Mandelbrot sets. The filled-in Julia sets Km,c have m-fold structures. Similar to the complex analytic dynamics, the general Mandelbrot sets are kinds of mathematical dictionary or atlas that map out the behavior of the filled-in Julia sets for different values of c.