Fractal Structures of General Mandelbrot Sets and Julia Sets Generated from Complex Non-Analytic Iteration Fm(z)=z¯m+c

2013 ◽  
Vol 347-350 ◽  
pp. 3019-3023
Author(s):  
De Jun Yan ◽  
Xiao Dan Wei ◽  
Hong Peng Zhang ◽  
Nan Jiang ◽  
Xiang Dong Liu
Keyword(s):  
Rotational Symmetry ◽  
Julia Sets ◽  
Boundary Curve ◽  
Mandelbrot Set ◽  
Escape Time ◽  
Fractal Structures ◽  
Mandelbrot Sets ◽  
The Stability ◽  
Definition Of ◽  
Analytic Iteration

In this paper we use the same idea as the complex analytic dynamics to study general Mandelbrot sets and Julia sets generated from the complex non-analytic iteration . The definition of the general critical point is given, which is of vital importance to the complex non-analytic dynamics. The general Mandelbrot set is proved to be bounded, axial symmetry by real axis, and have (m+1)-fold rotational symmetry. The stability condition of periodic orbits and the boundary curve of stability region of one-cycle are given. And the general Mandelbrot sets are constructed by the escape-time method and the periodic scanning algorithm, which present a better understanding of the structure of the Mandelbrot sets. The filled-in Julia sets Km,c have m-fold structures. Similar to the complex analytic dynamics, the general Mandelbrot sets are kinds of mathematical dictionary or atlas that map out the behavior of the filled-in Julia sets for different values of c.

GENERALIZED MANDELBROT SETS FOR MEROMORPHIC COMPLEX AND QUATERNIONIC MAPS

2002 ◽  
Vol 12 (08) ◽  
pp. 1755-1777 ◽  
Author(s):  
WALTER BUCHANAN ◽  
JAGANNATHAN GOMATAM ◽  
BONNIE STEVES
Keyword(s):  
Parameter Space ◽  
Three Dimensional ◽  
Mandelbrot Set ◽  
Rational Maps ◽  
Fractal Structures ◽  
Stability Regions ◽  
Theoretical Support ◽  
The Stability ◽  
Definition Of ◽  
Meromorphic Maps

The concepts of the Mandelbrot set and the definition of the stability regions of cycles for rational maps require careful investigation. The standard definition of the Mandelbrot set for the map f : z → z2+ c (the set of c values for which the iteration of the critical point at 0 remains bounded) is inappropriate for meromorphic maps such as the inverse square map. The notion of cycle sets, introduced by Brooks and Matelski [1978] for the quadratic map and applied to meromorphic maps by Yin [1994], facilitates a precise definition of the Mandelbrot parameter space for these maps. Close scrutiny of the cycle sets of these maps reveals generic fractal structures, echoing many of the features of the Mandelbrot set. Computer representations confirm these features and allow the dynamical comparison with the Mandelbrot set. In the parameter space, a purely algebraic result locates the stability regions of the cycles as the zeros of characteristic polynomials. These maps are generalized to quaternions. The powerful theoretical support that exists for complex maps is not generally available for quaternions. However, it is possible to construct and analyze cycle sets for a class of quaternionic rational maps (QRM). Three-dimensional sections of the cycle sets of QRM are nontrivial extensions of the cycle sets of complex maps, while sharing many of their features.

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.

scholarly journals Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

2019 ◽  
Vol 3 (1) ◽  
pp. 6 ◽  
Author(s):  
Vance Blankers ◽  
Tristan Rendfrey ◽  
Aaron Shukert ◽  
Patrick Shipman
Keyword(s):  
Dynamical Systems ◽  
Natural Number ◽  
Hyperbolic Plane ◽  
Julia Sets ◽  
Number System ◽  
Mandelbrot Set ◽  
Complex Numbers ◽  
Hyperbolic Numbers ◽  
Fractal Sets ◽  
Mandelbrot Sets

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + τ y for x , y ∈ R , and τ 2 = 1 but τ ≠ ± 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

COMPOSED ACCELERATED ESCAPE TIME ALGORITHM TO CONSTRUCT THE GENERAL MANDELBROT SETS

Fractals ◽  
2001 ◽  
Vol 09 (02) ◽  
pp. 149-153 ◽  
Author(s):  
XIANGDONG LIU ◽  
ZHILIANG ZHU ◽  
GUANGXING WANG ◽  
WEIYONG ZHU
Keyword(s):  
Time Algorithm ◽  
Mandelbrot Set ◽  
Escape Time ◽  
Mandelbrot Sets

In this paper, we present a new composed accelerated escape time algorithm by introducing a new composed iterative function. By the new algorithm, the points in the general Mandelbrot set can be decided rapidly with the same precision of the origin escape time algorithm.

Similarity between the Dynamic and Parameter Spaces in Cubic Mappings

Fractals ◽  
1998 ◽  
Vol 06 (03) ◽  
pp. 293-299
Author(s):  
Chia-Chin Cheng ◽  
Sy-Sang Liaw
Keyword(s):  
Julia Sets ◽  
Mandelbrot Set ◽  
Fractal Structures ◽  
Parameter Spaces

We have extended the work of Lei Tan on the similarity between the Mandelbrot set and the Julia sets. We show that the fractal structures of dynamic and parameter spaces are asymtotically similar at Misiurewicz points for the cubic mappings.

DYNAMIC ANALYSIS OF THE CAROTID–KUNDALINI MAP

2008 ◽  
Vol 22 (04) ◽  
pp. 243-262 ◽  
Author(s):  
XINGYUAN WANG ◽  
QINGYONG LIANG ◽  
JUAN MENG
Keyword(s):  
Analytic Function ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Quantitative Criterion ◽  
Boundary Equation ◽  
Computer Aided ◽  
Definition Of ◽  
Mathematics Method ◽  
Periodic Bifurcation

The nature of the fixed points of the Carotid–Kundalini (C–K) map was studied and the boundary equation of the first bifurcation of the C–K map in the parameter plane is presented. Using the quantitative criterion and rule of chaotic system, the paper reveals the general features of the C–K Map transforming from regularity to chaos. The following conclusions are obtained: (i) chaotic patterns of the C–K map may emerge out of double-periodic bifurcation; (ii) the chaotic crisis phenomena are found. At the same time, the authors analyzed the orbit of critical point of the complex C–K Map and put forward the definition of Mandelbrot–Julia set of the complex C–K Map. The authors generalized the Welstead and Cromer's periodic scanning technique and using this technology constructed a series of the Mandelbrot–Julia sets of the complex C–K Map. Based on the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, we investigated the symmetry of the Mandelbrot–Julia set and studied the topological inflexibility of distribution of the periodic region in the Mandelbrot set, and found that the Mandelbrot set contains abundant information of the structure of Julia sets by finding the whole portray of Julia sets based on Mandelbrot set qualitatively.

scholarly journals Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions

2019 ◽  
Vol 3 (3) ◽  
pp. 42 ◽  
Author(s):  
L.K. Mork ◽  
Trenton Vogt ◽  
Keith Sullivan ◽  
Drew Rutherford ◽  
Darin J. Ulness
Keyword(s):  
Fixed Points ◽  
Parameter Space ◽  
Rotational Symmetry ◽  
Julia Sets ◽  
Phase Rotation ◽  
Mandelbrot Sets

Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot sets. The features and characteristics of these related Julia and Mandelbrot sets are discussed and the parameter space, made with a phase rotation and offset shift, is intricately explored. Also studied in this work is the iterative dynamical map, its characteristics and its fixed points.

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.

CHAOS AND FRACTALS IN C–K MAP

2008 ◽  
Vol 19 (09) ◽  
pp. 1389-1409 ◽  
Author(s):  
XING-YUAN WANG ◽  
QING-YONG LIANG ◽  
JUAN MENG
Keyword(s):  
Power Spectrum ◽  
Fixed Points ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Parameter Plane ◽  
Boundary Equation ◽  
Mandelbrot Sets

The characteristic of the fixed points of the Carotid–Kundalini (C–K) map is investigated and the boundary equation of the first bifurcation of the C–K map in the parameter plane is given. Based on the studies of the phase graph, the power spectrum, the correlation dimension and the Lyapunov exponents, the paper reveals the general features of the C–K map transforming from regularity. Meanwhile, using the periodic scanning technology proposed by Welstead and Cromer, a series of Mandelbrot–Julia (M–J) sets of the complex C–K map are constructed. The symmetry of M–J set and the topological inflexibility of distributing of periodic region in the Mandelbrot set are investigated. By founding the whole portray of Julia sets based on Mandelbrot set qualitatively, we find out that Mandelbrot sets contain abundant information of structure of Julia sets.

Sign in / Sign up

Export Citation Format

Share Document