scholarly journals On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

2021 ◽  
Vol 5 (2) ◽  
pp. 55
Author(s):  
Yang Zhao ◽  
Shicun Zhao ◽  
Yi Zhang ◽  
Da Wang
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Initial Point ◽  
Visual Display ◽  
Time Algorithm ◽  
Escape Time ◽  
Polynomial Maps ◽  
Distribution Rule ◽  
Connected Area ◽  
Criterion Method

In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

scholarly journals Fractals Parrondo’s Paradox in Alternated Superior Complex System

2021 ◽  
Vol 5 (2) ◽  
pp. 39
Author(s):  
Yi Zhang ◽  
Da Wang
Keyword(s):  
Complex System ◽  
Julia Set ◽  
Julia Sets ◽  
Time Algorithm ◽  
Mandelbrot Set ◽  
Escape Time ◽  
Parrondo’S Paradox ◽  
The One ◽  
Totally Disconnected ◽  
Superior Mandelbrot Set

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

scholarly journals Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles

2010 ◽  
Vol 30 (6) ◽  
pp. 1869-1902 ◽  
Author(s):  
HIROKI SUMI
Keyword(s):  
Jordan Curve ◽  
Riemann Sphere ◽  
Julia Set ◽  
Julia Sets ◽  
John Domain ◽  
Unbounded Component ◽  
Random Dynamics ◽  
Jordan Curves ◽  
Polynomial Maps

AbstractWe investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere $\CCI $) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups G such that G is generated by a compact family Γ, the planar postcritical set of G is bounded, and G is (semi-) hyperbolic. In one of the classes, we have that, for almost every sequence $\gamma \in \Gamma ^{\NN }$, the Julia set Jγ of γ is a Jordan curve but not a quasicircle, the unbounded component of $\CCI {\setminus } J_{\gamma }$ is a John domain, and the bounded component of $\CC {\setminus } J_{\gamma }$ is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups G such that the planar postcritical set of G is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.

CALCULATION OF JULIA SETS BY EQUIPOTENTIAL POINT ALGORITHM

2013 ◽  
Vol 23 (01) ◽  
pp. 1350015 ◽  
Author(s):  
YUANYUAN SUN ◽  
XUDONG ZHAO ◽  
KAINING HOU
Keyword(s):  
Complex Plane ◽  
Iterative Process ◽  
Single Point ◽  
Julia Sets ◽  
Time Algorithm ◽  
Escape Time ◽  
Classical Algorithm

Escape time algorithm is a classical algorithm to calculate the Julia sets, but it has the disadvantage of dull color and cannot record the iterative process of the points. In this paper, we present the equipotential point algorithm to calculate the Julia sets by recording the strike frequency of the points in the iterative process. We calculate and analyze the Julia sets in the complex plane by using this algorithm. Finally, we discuss the iteration trajectory of a single point.

scholarly journals Problems and solutions by the application of Julia set theory to one-dot and multi-dots numerical methods

2001 ◽  
Vol 28 (9) ◽  
pp. 545-548
Author(s):  
Anna Tomova
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Nonlinear Equations ◽  
Set Theory ◽  
Julia Set ◽  
Time Algorithm ◽  
Escape Time ◽  
Special Cases ◽  
Problems And Solutions ◽  
The One

In 1977 Hubbard developed the ideas of Cayley (1879) and solved in particular the Newton-Fourier imaginary problem. We solve the Newton-Fourier and the Chebyshev-Fourier imaginary problems completely. It is known that the application of Julia set theory is possible to the one-dot numerical method like the Newton's method for computing solution of the nonlinear equations. The secants method is the two-dots numerical method and the application of Julia set theory to it is not demonstrated. Previously we have defined two one-dot combinations: the Newton's-secants and the Chebyshev's-secants methods and have used the escape time algorithm to analyse the application of Julia set theory to these two combinations in some special cases. We consider and solve the Newton's-secants and Tchebicheff's-secants imaginary problems completely.

Pattern Generation Based on Julia Set and its Application in the Design of Seamless Underwear Products

2011 ◽  
Vol 332-334 ◽  
pp. 702-705
Author(s):  
Li Chen ◽  
Rui Zhang
Keyword(s):  
Julia Set ◽  
Time Algorithm ◽  
Pattern Generation ◽  
Escape Time ◽  
Knitted Fabric ◽  
Image Generation ◽  
Pattern Design ◽  
Design Capacity ◽  
Fractal Image

In order to improve the design capacity of knitted fabric pattern and explore new design thought, this article studies the fractal image generation of Julia set and applies it into the pattern design of seamless underwear products. It adopts VB programming to generate Julia set pattern based on escape time algorithm, and then uses part imaging and amplification or changes the function orders to get a lot of colorful patterns. When these generated patterns are imported under the help of Photon software, color part in the patterns will be replaced with pre-designed fabric patterns and finally comes with patterns identifiable to seamless underwear knitting machine. Such method greatly expands the design thoughts of seamless underwear products.

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.

Fractal analysis and control of the competition model

2016 ◽  
Vol 09 (03) ◽  
pp. 1650045 ◽  
Author(s):  
Mianmian Zhang ◽  
Yongping Zhang
Keyword(s):  
Feedback Control ◽  
Mathematical Models ◽  
Fractal Analysis ◽  
Fractal Geometry ◽  
Julia Set ◽  
Julia Sets ◽  
Competition Model ◽  
Simulation Results ◽  
Population Competition ◽  
And Control

Lotka–Volterra population competition model plays an important role in mathematical models. In this paper, Julia set of the competition model is introduced by use of the ideas and methods of Julia set in fractal geometry. Then feedback control is taken on the Julia set of the model. And synchronization of two different Julia sets of the model with different parameters is discussed, which makes one Julia set change to be another. The simulation results show the efficacy of these methods.

Rendering and Comparing Research of 3D Julia Sets Based on CPU/GPU

2012 ◽  
Vol 542-543 ◽  
pp. 1434-1437
Author(s):  
Xiao Ping Xiao ◽  
Zi Sheng Li ◽  
Wei Gong
Keyword(s):  
Ray Tracing ◽  
Volume Rendering ◽  
Julia Sets ◽  
Time Algorithm ◽  
High Quality

Aiming at the problem that rendering 3D Julia sets on CPU is slowly, a method of rendering 3D Julia sets on GPU is presented in this paper. After introducing the advantages of GPU and the operations of quaternion, the generating process of 3D Julia sets is discussed in detail. Ray tracing volume rendering algorithm is applied to obtain high quality 3D Julia sets, and escaping time algorithm is used to generate the discreet data of Julia sets, of which normal is estimated according to the original of ray and accelerated by using unbounding sphere algorithm, and the graphics examples are given to illustrate this algorithm. Finally, the factors of affecting rendering speed and refined effect are summarized. The results show that the speed of 3D Julia sets rendering on GPU is much faster than CPU, and the interactivity of rendering process is also enhanced.

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