scholarly journals Fractal Newton basins

2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
M. L. Sahari ◽  
I. Djellit
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Main Tool ◽  
Chaotic Sequence ◽  
Solutions To Equations ◽  
Main Interest ◽  
Dynamical Plane ◽  
Complex Functions ◽  
Cubic Polynomials

The dynamics of complex cubic polynomials have been studied extensively in the recent years. The main interest in this work is to focus on the Julia sets in the dynamical plane, and then is consecrated to the study of several topics in more detail. Newton's method is considered since it is the main tool for finding solutions to equations, which leads to some fantastic images when it is applied to complex functions and gives rise to a chaotic sequence.

scholarly journals A class of Newton maps with Julia sets of Lebesgue measure zero

2021 ◽  
Author(s):  
Mareike Wolff
Keyword(s):  
Lebesgue Measure ◽  
Newton’S Method ◽  
Newton's Method ◽  
Measure Zero ◽  
Julia Set ◽  
Julia Sets ◽  
Lebesgue Measure Zero ◽  
Almost Everywhere ◽  
General Conditions

AbstractLet $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$ g ( z ) = ∫ 0 z p ( t ) exp ( q ( t ) ) d t + c where p, q are polynomials and $$c\in {\mathbb {C}}$$ c ∈ C , and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of $$g''$$ g ′ ′ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $$f^n(z)$$ f n ( z ) converges to zeros of g almost everywhere in $${\mathbb {C}}$$ C if this is the case for each zero of $$g''$$ g ′ ′ that is not a zero of g or $$g'$$ g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

JULIA SETS OF GENERALIZED NEWTON'S METHOD

Fractals ◽  
2007 ◽  
Vol 15 (04) ◽  
pp. 323-336 ◽  
Author(s):  
XINGYUAN WANG ◽  
TINGTING WANG
Keyword(s):  
Newton’S Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Basins Of Attraction ◽  
Principal Value ◽  
Extraneous Fixed Points ◽  
Sets Theory ◽  
Steffensen Method ◽  
Generalized Newton’S Method

The Julia sets theory of generalized Newton's method is analyzed and the Julia sets of generalized Newton's method are constructed using the iteration method. From the research we find that: (1) the basins of attraction of the Julia sets of generalized Newton's method depend on the roots of the equation and their orders and also the existence of the extraneous fixed points; (2) the Steffensen method is an exception to the law given in (1); and (3) if the order of the root is decimal, then the different choice of the range of the principal value of the phase angle will cause a different evolvement of the Julia sets.

INFINITE BASINS OF JULIA SETS

2008 ◽  
Vol 18 (10) ◽  
pp. 3169-3173
Author(s):  
FİGEN ÇİLİNGİR
Keyword(s):  
Entire Function ◽  
Fixed Points ◽  
Newton’S Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Rational Functions ◽  
Complex Parameter

The goal of this paper is to investigate the iterative behavior of a particular class of rational functions which arise from Newton's method applied to the entire function (z2 + c)eQ(z) where c is a complex parameter and Q is a nonconstant polynomial with deg(Q) ≤ 2. In particular, the basins of attracting fixed points will be described.

MANDELBROT AND JULIA SETS OF ONE-PARAMETER RATIONAL FUNCTION FAMILIES ASSOCIATED WITH NEWTON'S METHOD

Fractals ◽  
2010 ◽  
Vol 18 (02) ◽  
pp. 255-263 ◽  
Author(s):  
XIANG-DONG LIU ◽  
ZHI-JIE LI ◽  
XUE-YE ANG ◽  
JIN-HAI ZHANG
Keyword(s):  
Rational Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Rational Functions ◽  
Periodic Points ◽  
Mandelbrot Sets

In this paper, general Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method were discussed. The bounds of these general Mandelbrot sets and two formulas for calculating the number of different periods periodic points of these rational functions were given. The relations between general Mandelbrot sets and common Mandelbrot sets of zn + c (n ∈ Z, n ≥ 2), along with the relations between general Mandelbrot sets and their corresponding Julia sets were investigated. Consequently, the results were found in the study: there are similarities between the Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method and the Mandelbrot and Julia sets of zn + c (n ∈ Z, n ≥ 2).

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