Accurate computation of periodic regions' centers in the general \(M\)-set with integer index number
type: Item
instance of: scholarly article
title: Accurate computation of periodic regions' centers in the general \(M\)-set with integer index number
zbMATH Open document ID: 1196.28020
DOI: 10.1155/2010/653816
author: He Yijie
author: Sun Yuanyuan
author: Xingyuan Wang
published in: Discrete Dynamics in Nature and Society
publication date: 2010-09-29T00:00:00Z
full work available at URL: https://eudml.org/doc/224593
review text: Summary: This paper presents two methods for accurately computing the periodic regions' centers. One method fits for the general M-sets with integer index number, the other fits for the general M-sets with negative integer index number. Both methods improve the precision of computation by transforming the polynomial equations which determine the periodic regions' centers. We primarily discuss the general M-sets with negative integer index, and analyze the relationship between the number of periodic regions' centers on the principal symmetric axis and in the principal symmetric interior. We can get the centers' coordinates with at least 48 significant digits after the decimal point in both real and imaginary parts by applying the Newton's method to the transformed polynomial equation which determine the periodic regions' centers. In this paper, we list some centers' coordinates of general M-sets' \(k\)-periodic regions \((k=3,4,5,6)\) for the index numbers \(\alpha = - 25, - 24,\cdots , - 1\) , all of which have highly numerical accuracy.
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 5793503
Wikidata QID: Q58650858
MaRDI profile type: Publication
OpenAlex ID: W1996022279
cites work: Q4740120
cites work: Mixing Markov Chains and Their Images
cites work: Predicting continuum breakdown in hypersonic viscous flows
cites work: An approach to the ordering of one-dimensional quadratic maps.
cites work: Determination of Mandelbrot set's hyperbolic component centres
cites work: ANALYSIS OF C-PLANE FRACTAL IMAGES FROM z ← zα + c FOR (α < 0)
cites work: GENERALIZED MANDELBROT SETS FOR MEROMORPHIC COMPLEX AND QUATERNIONIC MAPS
cites work: Equivalence between subshrubs and chaotic bands in the Mandelbrot set
cites work: Accurate computation of component centers in the degree-\(n\) bifurcation set
cites work: Research on fractal structure of generalized M--J sets utilized Lyapunov exponents and periodic scanning techniques
cites work: On the dynamics of polynomial-like mappings
cites work: RESEARCH ON BROWNIAN MOVEMENT BASED ON GENERALIZED MANDELBROT–JULIA SETS FROM A CLASS COMPLEX MAPPING SYSTEM
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description: scientific article; zbMATH DE number 5793503