Global analytical structure of the Mandelbrot set and its generalization
type: Item
instance of: scholarly article
title: Global analytical structure of the Mandelbrot set and its generalization
zbMATH Open document ID: 0760.58025
published in: Science in China. Series A
publication date: 1993-01-17T00:00:00Z
review text: The author studies the global structure of periodic orbits for a one- dimensional map \(Z_{n+1}=Z^ m_ n+C\) (\(C\) complex, \(m\geq 2\)) using an earlier developed algebraic analytical method. He gives a general formula for the calculation of the orbit number of any orbit with finite period. He verifies that the complex structures of the Mandelbrot set (\(m=2\)) and its generalized form (\(m>2\)) are composed of infinitely many stable regions of different periodic orbits. A simple linear relation between the stable region number of the periodic orbit and its orbit number is derived. The algebraic equations of the boundary of each element and the locations of its cusp and the center are given precisely and the cause for the infinitely nested structures in these complex figures is explained.
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 98040
zbMATH Keywords: fractals
zbMATH Keywords: iteration
zbMATH Keywords: Mandelbrot set
zbMATH Keywords: periodic orbits
author: Yongnian Huang
reviewed by: Klaus Brod
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 98040