The Mandelbrot Set

1986 ◽  
pp. 56-62 ◽  
Author(s):  
Heinz-Otto Peitgen ◽  
Peter H. Richter
Keyword(s):  
Mandelbrot Set

A Generalized Mandelbrot Set of Polynomials of Type 𝐸𝑑 for Bicomplex Numbers

2008 ◽  
Vol 15 (1) ◽  
pp. 189-194
Author(s):  
Ahmad Zireh
Keyword(s):  
Mandelbrot Set ◽  
Complex Numbers ◽  
Quadratic Polynomials ◽  
Bicomplex Numbers

Abstract We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑. Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤2 + 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑, is connected.

scholarly journals On the Fibonacci–Mandelbrot set

2015 ◽  
Vol 26 (1) ◽  
pp. 174-190 ◽  
Author(s):  
Víctor F. Sirvent ◽  
Jörg M. Thuswaldner
Keyword(s):  
Mandelbrot Set

Drawing and computing external rays in the multiple-spiral medallions of the Mandelbrot set

2008 ◽  
Vol 32 (5) ◽  
pp. 597-610 ◽  
Author(s):  
M. Romera ◽  
G. Alvarez ◽  
D. Arroyo ◽  
A.B. Orue ◽  
V. Fernandez ◽  
...  
Keyword(s):  
Mandelbrot Set ◽  
External Rays

A Study on Mandelbrot Sets to Generate Visual Aesthetic Fractal Patterns

2013 ◽  
Vol 311 ◽  
pp. 111-116 ◽  
Author(s):  
Zong Wen Cai ◽  
Artde D. Kin Tak Lam
Keyword(s):  
Information System ◽  
Program Development ◽  
Development Program ◽  
Visual Basic ◽  
Time Algorithm ◽  
Mandelbrot Set ◽  
Escape Time ◽  
Mandelbrot Sets ◽  
Fractal Patterns ◽  
Power Orders

The fractal pattern is a highly visual aesthetic image. This article describes the generation method of Mandelbrot set to generate fractal art patterns. Based on the escape time algorithm on complex plane, the visual aesthetic fractal patterns are generated from Mandelbrot sets. The generated program development, a pictorial information system, is integrated through the application of Visual Basic programming language and development integration environment. Application of the development program, this article analyzes the shape of the fractal patterns generated by the different power orders of the Mandelbrot sets. Finally, the escape time algorithm has been proposed as the generation tools of highly visual aesthetic fractal patterns.

The Mandelbrot Set: Ordering the Julia Sets

1992 ◽  
pp. 415-473
Author(s):  
Heinz-Otto Peitgen ◽  
Hartmut Jürgens ◽  
Dietmar Saupe
Keyword(s):  
Julia Sets ◽  
Mandelbrot Set

The Mandelbrot Set

2018 ◽  
pp. 246-262
Author(s):  
Robert L. Devaney
Keyword(s):  
Mandelbrot Set

scholarly journals Controlling Robots with Fractal Gene Regulatory Networks

2005 ◽  
pp. 320-339 ◽  
Author(s):  
Peter J. Bentley
Keyword(s):  
Gene Regulatory Networks ◽  
Protein Interactions ◽  
Regulatory Networks ◽  
Developmental Process ◽  
Mandelbrot Set ◽  
Gene Regulatory ◽  
Robot Path

Fractal proteins are a new evolvable method of mapping genotype to phenotype through a developmental process, where genes are expressed into proteins comprised of subsets of the Mandelbrot set. The resulting network of gene and protein interactions can be designed by evolution to produce specific patterns, which in turn can be used to solve problems. This chapter introduces the fractal development algorithm in detail and describes the use of fractal gene regulatory networks for learning a robot path through a series of obstacles. The results indicate the ability of this system to learn regularities in solutions and automatically create and use modules.

Warped midgets in the Mandelbrot set

1994 ◽  
Vol 18 (2) ◽  
pp. 239-248 ◽  
Author(s):  
A.G.Davis Philip ◽  
Michael Frame ◽  
Adam Robucci
Keyword(s):  
Mandelbrot Set

LIMITING DYNAMICS FOR THE COMPLEX STANDARD FAMILY

1995 ◽  
Vol 05 (03) ◽  
pp. 673-699 ◽  
Author(s):  
NÚRIA FAGELLA
Keyword(s):  
Cross Sections ◽  
Julia Set ◽  
Three Dimensional ◽  
Mandelbrot Set ◽  
Dimensional Parameter ◽  
Quadratic Polynomials ◽  
New Family ◽  
Standard Family ◽  
The Family ◽  
Characteristic Features

The complexification of the standard family of circle maps Fαβ(θ)=θ+α+β+β sin(θ) mod (2π) is given by Fαβ(ω)=ωeiαe(β/2)(ω−1/ω) and its lift fαβ(z)=z+a+β sin(z). We investigate the three-dimensional parameter space for Fαβ that results from considering a complex and β real. In particular, we study the two-dimensional cross-sections β=constant as β tends to zero. As the functions tend to the rigid rotation Fα,0, their dynamics tend to the dynamics of the family Gλ(z)=λzez where λ=e−iα. This new family exhibits behavior typical of the exponential family together with characteristic features of quadratic polynomials. For example, we show that the λ-plane contains infinitely many curves for which the Julia set of the corresponding maps is the whole plane. We also prove the existence of infinitely many sets of λ values homeomorphic to the Mandelbrot set.

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