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Статті в журналах з теми "Mandelbrot sets"

Автор: Grafiati

Опубліковано: 4 червня 2021

Оновлено: 30 липня 2024

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1

LIU, XIANG-DONG, ZHI-JIE LI, XUE-YE ANG, and JIN-HAI ZHANG. "MANDELBROT AND JULIA SETS OF ONE-PARAMETER RATIONAL FUNCTION FAMILIES ASSOCIATED WITH NEWTON'S METHOD." Fractals 18, no. 02 (June 2010): 255–63. http://dx.doi.org/10.1142/s0218348x10004841.

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Анотація:

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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2

Mu, Beining. "Fuzzy Julia Sets and Fuzzy Superior Julia Sets." Highlights in Science, Engineering and Technology 72 (December 15, 2023): 375–80. http://dx.doi.org/10.54097/5c5hp748.

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This article examine the past study of fuzzy Mandelbrot set and fuzzy superior Mandelbrot set, then give the definition of fuzzy Julia sets and fuzzy superior Julia sets with the idea of utilizing membership functions to represent the escape velocity of Julia sets or superior Julia sets of each complex number in the definition of fuzzy Mandelbrot set and fuzzy superior Mandelbrot set inherited while the membership functions are selected to distinguish complex numbers with different escape velocity and same orbit. Then some examples of fuzzy Julia sets and fuzzy superior Julia sets are presented. With some observation of the examples, some analytical and topological properties of fuzzy Julia sets are demonstrated and sketches of the proofs are also presented. In the part of fuzzy superior Julia sets, some trivial conclusions as well as an example is presented where more complicated properties which involves the study into very chaotic behaviors are left open.

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3

Jha, Ketan, and Mamta Rani. "Control of Dynamic Noise in Transcendental Julia and Mandelbrot Sets by Superior Iteration Method." International Journal of Natural Computing Research 7, no. 2 (April 2018): 48–59. http://dx.doi.org/10.4018/ijncr.2018040104.

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Researchers and scientists are attracted towards Julia and Mandelbrot sets constantly. They analyzed these sets intensively. Researchers have studied the perturbation in Julia and Mandelbrot sets which is due to different types of noises, but transcendental Julia and Mandelbrot sets remained ignored. The purpose of this article is to study the perturbation in transcendental Julia and Mandelbrot sets. Also, we made an attempt to control the perturbation in transcendental sets by using superior iteration method.

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4

Danca, Marius-F. "Mandelbrot Set as a Particular Julia Set of Fractional Order, Equipotential Lines and External Rays of Mandelbrot and Julia Sets of Fractional Order." Fractal and Fractional 8, no. 1 (January 19, 2024): 69. http://dx.doi.org/10.3390/fractalfract8010069.

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This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the differences between the fractional-order Mandelbrot set and Julia sets from their integer-order variants are revealed. Equipotential lines and external rays of a Mandelbrot set and Julia sets of fractional order are determined.

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5

Tassaddiq, Asifa, Muhammad Tanveer, Muhammad Azhar, Waqas Nazeer, and Sania Qureshi. "A Four Step Feedback Iteration and Its Applications in Fractals." Fractal and Fractional 6, no. 11 (November 9, 2022): 662. http://dx.doi.org/10.3390/fractalfract6110662.

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Fractals play a vital role in modeling the natural environment. The present aim is to investigate the escape criterion to generate specific fractals such as Julia sets, Mandelbrot sets and Multi-corns via F-iteration using complex functions h(z)=zn+c, h(z)=sin(zn)+c and h(z)=ezn+c, n≥2,c∈C. We observed some beautiful Julia sets, Mandelbrot sets and Multi-corns for n = 2, 3 and 4. We generalize the algorithms of the Julia set and Mandelbrot set to visualize some Julia sets, Mandelbrot sets and Multi-corns. Moreover, we calculate image generation time in seconds at different values of input parameters.

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6

Yan, De Jun, Xiao Dan Wei, Hong Peng Zhang, Nan Jiang, and Xiang Dong Liu. "Fractal Structures of General Mandelbrot Sets and Julia Sets Generated from Complex Non-Analytic Iteration F_m(z)=z¯^m+c." Applied Mechanics and Materials 347-350 (August 2013): 3019–23. http://dx.doi.org/10.4028/www.scientific.net/amm.347-350.3019.

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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.

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7

KOZMA, ROBERT T., and ROBERT L. DEVANEY. "Julia sets converging to filled quadratic Julia sets." Ergodic Theory and Dynamical Systems 34, no. 1 (August 21, 2012): 171–84. http://dx.doi.org/10.1017/etds.2012.115.

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AbstractIn this paper we consider singular perturbations of the quadratic polynomial $F(z) = z^2 + c$ where $c$ is the center of a hyperbolic component of the Mandelbrot set, i.e., rational maps of the form $z^2 + c + \lambda /z^2$. We show that, as $\lambda \rightarrow 0$, the Julia sets of these maps converge in the Hausdorff topology to the filled Julia set of the quadratic map $z^2 + c$. When $c$ lies in a hyperbolic component of the Mandelbrot set but not at its center, the situation is much simpler and the Julia sets do not converge to the filled Julia set of $z^2 + c$.

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8

Al-Salami, Hassanein Q. "Some Properties of the Mandelbrot Sets M(Q_α)". JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences 31, № 2 (29 червня 2023): 263–69. http://dx.doi.org/10.29196/jubpas.v31i2.4683.

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The aim of this work is to study the concept of the parameter plane, so we are interested in introducing the Mandebrot set,as well as the Mandelbrot set is a fractals ,whereas, if we zoom in on any small piece in the border regions of the shape, we notice that it is similar to the original shape, that is the Mandelbrot set, with special topological specifications, and we introduce of some properties of the Mandelbrot sets for the map in the form , where is a complex constant, and we prove that the conjecture is the Duadi-Hubbard theorem . By studying of the parameter plane that the Mandelbrot set is a component form in one of its details in the dynamical plane it is the Julia set , in other words every small detail of the Mandelbrot set represents the Julia set, so there is more than one characteristic of the Julia set, we will limit ourselves only to presenting the connection of the Julia set for .

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9

Sekovanov, Valeriy S., Larisa B. Rybina, and Kseniya Yu Strunkina. "The study of the frames of Mandelbrot sets of polynomials of the second degree as a means of developing the originality of students' thinking." Vestnik Kostroma State University. Series: Pedagogy. Psychology. Sociokinetics, no. 4 (2019): 193–99. http://dx.doi.org/10.34216/2073-1426-2019-25-4-193-199.

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The article presents a methodology for studying the frames of Mandelbrot sets of polynomials of the second degree of a complex variable, based on the integration of analytical methods, mathematical programming and the use of computer graphics. A connection is established between the frames of the first and second orders of Mandelbrot sets of functions and with the curves – cardioid, lemniscate and circle. Algorithms for constructing the frames of the Mandelbrot sets of the functions under consideration in the MathCad mathematical package are presented. The task is to describe 3-order frames (where) of the Mandelbrot sets of functions and, which correspond to the existence of attracting fixed points of period 3. It is shown that the establishment of associative relations between classes of various mathematical objects (polynomials of a complex variable, curves, Mandelbrot sets) contributes to the development of original thinking and creative potential of students.

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10

Wang, Feng Ying, Li Ming Du, and Zi Yang Han. "The Construction for Generalized Mandelbrot Sets of the Frieze Group." Advanced Materials Research 756-759 (September 2013): 2562–66. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.2562.

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By an analysis of symmetric features of equivalent mappings of the frieze group, a definition of their generalized Mandelbrot sets is given and a novel method for constructing generalized Mandelbrot sets of equivalent mappings of frieze group is presented via utilizing the Ljapunov exponent as the judgment standard. Based on generating parameter space of dynamical system, lots of patterns of generalized Mandelbrot sets are produced.

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11

Kauko, Virpi. "Shadow trees of Mandelbrot sets." Fundamenta Mathematicae 180, no. 1 (2003): 35–87. http://dx.doi.org/10.4064/fm180-1-4.

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12

Sun, Y. Y., and X. Y. Wang. "Noise-perturbed quaternionic Mandelbrot sets." International Journal of Computer Mathematics 86, no. 12 (December 2009): 2008–28. http://dx.doi.org/10.1080/00207160903131228.

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13

Wang, Xingyuan, Zhen Wang, Yahui Lang, and Zhenfeng Zhang. "Noise perturbed generalized Mandelbrot sets." Journal of Mathematical Analysis and Applications 347, no. 1 (November 2008): 179–87. http://dx.doi.org/10.1016/j.jmaa.2008.04.032.

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14

Jha, Ketan, and Mamta Rani. "Estimation of Dynamic Noise in Mandelbrot Map." International Journal of Artificial Life Research 7, no. 2 (July 2017): 1–20. http://dx.doi.org/10.4018/ijalr.2017070101.

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Julia and Mandelbrot sets have been studied continuously attracting fractal scientists since their creation. As a result, Julia and Mandelbrot sets have been analyzed intensively. In this article, researchers have studied the effect of noise on these sets and analyzed perturbation. Continuing the trend in this article, they analyze perturbation and find the corresponding amount of dynamic noise in the Mandelbrot map. Further, in order to recover a distorted fractal image, a restoration algorithm is presented.

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15

CHEN, YI-CHIUAN, TOMOKI KAWAHIRA, HUA-LUN LI, and JUAN-MING YUAN. "FAMILY OF INVARIANT CANTOR SETS AS ORBITS OF DIFFERENTIAL EQUATIONS II: JULIA SETS." International Journal of Bifurcation and Chaos 21, no. 01 (January 2011): 77–99. http://dx.doi.org/10.1142/s0218127411028295.

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The Julia set of the quadratic map fμ(z) = μz(1 - z) for μ not belonging to the Mandelbrot set is hyperbolic, thus varies continuously. It follows that a continuous curve in the exterior of the Mandelbrot set induces a continuous family of Julia sets. The focus of this article is to show that this family can be obtained explicitly by solving the initial value problem of a system of infinitely coupled differential equations. A key point is that the required initial values can be obtained from the anti-integrable limit μ → ∞. The system of infinitely coupled differential equations reduces to a finitely coupled one if we are only concerned with some invariant finite subset of the Julia set. Therefore, it can be employed to find periodic orbits as well. We conduct numerical approximations to the Julia sets when parameter μ is located at the Misiurewicz points with external angle 1/2, 1/6, or 5/12. We approximate these Julia sets by their invariant finite subsets that are integrated along the reciprocal of corresponding external rays of the Mandelbrot set starting from the anti-integrable limit μ = ∞. When μ is at the Misiurewicz point of angle 1/128, a 98-period orbit of prescribed itinerary obtained by this method is presented, without having to find a root of a 298-degree polynomial. The Julia sets (or their subsets) obtained are independent of integral curves, but in order to make sure that the integral curves are contained in the exterior of the Mandelbrot set, we use the external rays of the Mandelbrot set as integral curves. Two ways of obtaining the external rays are discussed, one based on the series expansion (the Jungreis–Ewing–Schober algorithm), the other based on Newton's method (the OTIS algorithm). We establish tables comparing the values of some Misiurewicz points of small denominators obtained by these two algorithms with the theoretical values.

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16

Ashish, Mamta Rani, and Renu Chugh. "Julia sets and Mandelbrot sets in Noor orbit." Applied Mathematics and Computation 228 (February 2014): 615–31. http://dx.doi.org/10.1016/j.amc.2013.11.077.

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17

DOLOTIN, V., and A. MOROZOV. "ON THE SHAPES OF ELEMENTARY DOMAINS OR WHY MANDELBROT SET IS MADE FROM ALMOST IDEAL CIRCLES?" International Journal of Modern Physics A 23, no. 22 (September 10, 2008): 3613–84. http://dx.doi.org/10.1142/s0217751x08040330.

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Анотація:

Direct look at the celebrated "chaotic" Mandelbrot Set (in Fig. 1) immediately reveals that it is a collection of almost ideal circles and cardioids, unified in a specific forest structure. In the paper arXiv:hep-th/0501235, a systematic algebro-geometric approach was developed to the study of generic Mandelbrot sets, but emergency of nearly ideal circles in the special case of the family x2 + c was not fully explained. In the present, paper, the shape of the elementary constituents of Mandelbrot Set is explicitly calculated, and difference between the shapes of root and descendant domains (cardioids and circles respectively) is explained. Such qualitative difference persists for all other Mandelbrot sets: descendant domains always have one less cusp than the root ones. Details of the phase transition between different Mandelbrot sets are explicitly demonstrated, including overlaps between elementary domains and dynamics of attraction/repulsion regions. Explicit examples of three-dimensional sections of Universal Mandelbrot Set are given. Also a systematic small-size approximation is developed for evaluation of various Feigenbaum indices.

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18

Mork, Leah K., and Darin J. Ulness. "Visualization of Mandelbrot and Julia Sets of Möbius Transformations." Fractal and Fractional 5, no. 3 (July 17, 2021): 73. http://dx.doi.org/10.3390/fractalfract5030073.

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This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

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19

WANG, XING-YUAN, QING-YONG LIANG, and JUAN MENG. "CHAOS AND FRACTALS IN C–K MAP." International Journal of Modern Physics C 19, no. 09 (September 2008): 1389–409. http://dx.doi.org/10.1142/s0129183108012935.

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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.

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20

LIAW, SY-SANG. "FIND THE MANDELBROT-LIKE SETS IN ANY MAPPING." Fractals 10, no. 02 (June 2002): 137–46. http://dx.doi.org/10.1142/s0218348x02001282.

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The Mandelbrot-like sets appear in the parameter spaces of many one-parameter complex mappings. We find that the type of these sets depends on the multiplicity of the critical points of the mappings. We give the effective map for parameters in these sets, and accordingly calculate the positions, the sizes, and the orientations of these Mandelbrot-like sets for any one-parameter complex mappings.

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21

YAN, DEJUN, XIANGDONG LIU, and WEIYONG ZHU. "A STUDY OF MANDELBROT AND JULIA SETS GENERATED FROM A GENERAL COMPLEX CUBIC ITERATION." Fractals 07, no. 04 (December 1999): 433–37. http://dx.doi.org/10.1142/s0218348x99000438.

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The structures of Mandelbrot and Julia sets generated from a general complex cubic iteration are studied, and the relation of the critical points and Julia sets is analyzed. In this paper, we put forward the theorems on the range of the Mandelbrot and Julia sets generated from a general complex cubic iteration. These theorems are important for plotting of the sets.

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22

BUCHANAN, WALTER, JAGANNATHAN GOMATAM, and BONNIE STEVES. "GENERALIZED MANDELBROT SETS FOR MEROMORPHIC COMPLEX AND QUATERNIONIC MAPS." International Journal of Bifurcation and Chaos 12, no. 08 (August 2002): 1755–77. http://dx.doi.org/10.1142/s0218127402005443.

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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.

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23

Abbas, Mujahid, Hira Iqbal, and Manuel De la Sen. "Generation of Julia and Mandelbrot Sets via Fixed Points." Symmetry 12, no. 1 (January 2, 2020): 86. http://dx.doi.org/10.3390/sym12010086.

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The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form T ( x ) = x n + m x + r where m , r ∈ C and n ≥ 2 . Fractals represent the phenomena of expanding or unfolding symmetries which exhibit similar patterns displayed at every scale. We prove some escape time results for the generation of Julia and Mandelbrot sets using a Picard Ishikawa type iterative process. A visualization of the Julia and Mandelbrot sets for certain complex polynomials is presented and their graphical behaviour is examined. We also discuss the effects of parameters on the color variation and shape of fractals.

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24

Bandt, Christoph, and Nguyen Viet Hung. "Fractaln-gons and their Mandelbrot sets." Nonlinearity 21, no. 11 (October 10, 2008): 2653–70. http://dx.doi.org/10.1088/0951-7715/21/11/009.

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25

SHIAH, AICHYUN, KIM-KHOON ONG, and ZDZISLAW E. MUSIELAK. "FRACTAL IMAGES OF GENERALIZED MANDELBROT SETS." Fractals 02, no. 01 (March 1994): 111–21. http://dx.doi.org/10.1142/s0218348x94000107.

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Анотація:

The transformation function z ← zα+βi+c, with both α and β being either positive or negative integers or real numbers, is used to generate families of mostly new fractal images in the complex plane [Formula: see text]. The calculations are restricted to the principal value of zα+βi and the obtained fractal images are called the generalized Mandelbrot sets, ℳ (α, β). Three general classes of ℳ (α, β) are considered: (1) α ≠ 0 and β = 0; (2) α ≠ 0 and β ≠ 0; and (3) α = 0 and β ≠ 0. Our results demonstrate that the shapes of fractal images representing ℳ (α, 0) are usually significantly deformed when β ≠ 0, and that the size of either stable (α > 0) or unstable (α < 0) regions in the complex plane may increase as a result of non-zero β. It is also shown that fractal images of the generalized Mandelbrot sets ℳ (0, β) are significantly different than those obtained with a non-zero α.

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26

Pickover, Clifford A. "A note on inverted mandelbrot sets." Visual Computer 6, no. 4 (July 1990): 227–29. http://dx.doi.org/10.1007/bf02341047.

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27

Zhang, Yongping, and Weihua Sun. "Synchronization and coupling of Mandelbrot sets." Nonlinear Dynamics 64, no. 1-2 (October 9, 2010): 59–63. http://dx.doi.org/10.1007/s11071-010-9845-9.

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28

Wang, Xing-yuan, Pei-jun Chang, and Ni-ni Gu. "Additive perturbed generalized Mandelbrot–Julia sets." Applied Mathematics and Computation 189, no. 1 (June 2007): 754–65. http://dx.doi.org/10.1016/j.amc.2006.11.137.

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29

Smirnova, Elena Sa, Valery S. Sekovanov, Larisa B. Rybina, and Roman Al Shchepin. "Performing a multi-stage mathematical information task "Framing the Mandelbrot set of families of polynomials of the third degree and remarkable curves"." Vestnik of Kostroma State University. Series: Pedagogy. Psychology. Sociokinetics 30, no. 1 (June 28, 2024): 63–72. http://dx.doi.org/10.34216/2073-1426-2024-30-1-63-72.

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Анотація:

In this article, within the framework of a multi-stage mathematical information task, the methodology for students to study Mandelbrot sets and frames of Mandelbrot sets of a family of polynomials of the third degree is indicated. Algorithms for constructing these sets in various environments are described. The connections of the frames of the Mandelbrot set with remarkable curves are studied: lemniscate, epicycloid and others. This work is aimed at developing students' creativity and research competencies. When performing a multi-stage mathematical and informational task, the student acts as a mathematician, programmer and computer artist, which is aimed at developing his creativity and increasing motivation for mathematics and programming.

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30

Zou, Cui, Abdul Aziz Shahid, Asifa Tassaddiq, Arshad Khan, and Maqbool Ahmad. "Mandelbrot Sets and Julia Sets in Picard-Mann Orbit." IEEE Access 8 (2020): 64411–21. http://dx.doi.org/10.1109/access.2020.2984689.

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31

Farris, Salma M. "Generalized Mandelbrot Sets of a Family of Polynomials P n z = z n + z + c ; n ≥ 2." International Journal of Mathematics and Mathematical Sciences 2022 (February 22, 2022): 1–9. http://dx.doi.org/10.1155/2022/4510088.

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Анотація:

In this paper, we study the general Mandelbrot set of the family of polynomials P n z = z n + z + c ; n ≥ 2 , denoted by GM( P n ). We construct the general Mandelbrot set for these polynomials by the escaping method. We determine the boundaries, areas, fractals, and symmetry of the previous polynomials. On the other hand, we study some topological properties of GM P n . We prove that GM P n is bounded and closed; hence, it is compact. Also, we characterize the general Mandelbrot set as a union of basins of attraction. Finally, we make a comparison between the properties of famous Mandelbrot set M z 2 + c and our general Mandelbrot sets.

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32

Wang, Feng Ying, Li Ming Du, and Zi Yang Han. "Two Partitioning Algorithms for Generating of M Sets of the Frieze Group." Applied Mechanics and Materials 336-338 (July 2013): 2238–41. http://dx.doi.org/10.4028/www.scientific.net/amm.336-338.2238.

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Анотація:

Symmetric features of the frieze group equivalent mappings were analysed, and two partitioning algorithms are given for constructing generalized Mandelbrot sets of frieze group equivalent mappings in order to study the characteristics of generalized Msets. Based on generating parameter space of dynamical system, lots of patterns of generalized Mandelbrot sets are produced.

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33

Cai, Zong Wen, and Artde D. Kin Tak Lam. "A Study on Mandelbrot Sets to Generate Visual Aesthetic Fractal Patterns." Applied Mechanics and Materials 311 (February 2013): 111–16. http://dx.doi.org/10.4028/www.scientific.net/amm.311.111.

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Анотація:

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.

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34

WANG, XINGYUAN, QINGYONG LIANG, and JUAN MENG. "DYNAMIC ANALYSIS OF THE CAROTID–KUNDALINI MAP." Modern Physics Letters B 22, no. 04 (February 10, 2008): 243–62. http://dx.doi.org/10.1142/s0217984908014717.

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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.

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35

Kang, Shinmin, Arif Rafiq, Abdul Latif, Abdul Shahid, and Faisal Alif. "Fractals through modified iteration scheme." Filomat 30, no. 11 (2016): 3033–46. http://dx.doi.org/10.2298/fil1611033k.

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Анотація:

In this paper we study the geometry of relative superior Mandelbrot sets through S-iteration scheme. Our results are quit significant from other Mandelbrot sets existing in the literature. Besides this, we also observe that S-iteration scheme converges faster than Ishikawa iteration scheme. We believe that the results of this paper can be inspired thosewho are interested in creating automatically aesthetic patterns.

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36

PEHERSTORFER, FRANZ, and CHRISTOPH STROH. "JULIA AND MANDELBROT SETS OF CHEBYSHEV FAMILIES." International Journal of Bifurcation and Chaos 11, no. 09 (September 2001): 2463–81. http://dx.doi.org/10.1142/s0218127401003577.

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We present one-parameter families of rational functions of arbitrary degree d which are globally generalized polynomial-like of degree d and roughly speaking locally quadratic-like everywhere, where the parameter appears not only as a purely multiplicative factor but also in a more complicated nonlinear way. The connectedness locus of these families contains homeomorphic copies of the Mandelbrot set. Main emphasis is put on the explicit construction (and not as usual on the existence only) of the sets on which generalized polynomial-likeness and quadratic-likeness are given as well as on the explicit description of the regions where the homeomorphic copies of the Mandelbrot set are located.

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37

Murali, Arunachalam, and Krishnan Muthunagai. "Generation of Julia and Mandelbrot fractals for a generalized rational type mapping via viscosity approximation type iterative method extended with $ s $-convexity." AIMS Mathematics 9, no. 8 (2024): 20221–44. http://dx.doi.org/10.3934/math.2024985.

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<abstract><p>A dynamic visualization of Julia and Mandelbrot fractals involves creating animated representations of these fractals that change over time or in response to user interaction which allows users to gain deeper insights into the intricate structures and properties of these fractals. This paper explored the dynamic visualization of fractals within Julia and Mandelbrot sets, focusing on a generalized rational type complex polynomial of the form $ S_{c}(z) = a z^{n}+\frac{b}{z^{m}}+c $, where $ a, b, c \in \mathbb{C} $ with $ |a| > 1 $ and $ n, m \in \mathbb{N} $ with $ n > 1 $. By applying viscosity approximation-type iteration processes extended with $ s $-convexity, we unveiled the intricate dynamics inherent in these fractals. Novel escape criteria was derived to facilitate the generation of Julia and Mandelbrot sets via the proposed iteration process. We also presented graphical illustrations of Mandelbrot and Julia fractals, highlighting the change in the structure of the generated sets with respect to the variations in parameters.</p></abstract>

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38

Blankers, Vance, Tristan Rendfrey, Aaron Shukert, and Patrick Shipman. "Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers." Fractal and Fractional 3, no. 1 (February 20, 2019): 6. http://dx.doi.org/10.3390/fractalfract3010006.

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Анотація:

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.

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39

Cheng, Jin, and Jian-rong Tan. "Generalization of 3D Mandelbrot and Julia sets." Journal of Zhejiang University-SCIENCE A 8, no. 1 (January 2007): 134–41. http://dx.doi.org/10.1631/jzus.2007.a0134.

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40

Qi, Hengxiao, Muhammad Tanveer, Muhammad Shoaib Saleem, and Yuming Chu. "Anti Mandelbrot Sets via Jungck-M Iteration." IEEE Access 8 (2020): 194663–75. http://dx.doi.org/10.1109/access.2020.3033733.

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41

Álvarez, G., M. Romera, G. Pastor, and F. Montoya. "Determination of Mandelbrot Sets Hyperbolic Component Centres." Chaos, Solitons & Fractals 9, no. 12 (December 1998): 1997–2005. http://dx.doi.org/10.1016/s0960-0779(98)00046-0.

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42

Beck, Christian. "Physical meaning for Mandelbrot and Julia sets." Physica D: Nonlinear Phenomena 125, no. 3-4 (January 1999): 171–82. http://dx.doi.org/10.1016/s0167-2789(98)00243-7.

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43

Agarwal, Rashi, and Vishal Agarwal. "Dynamic noise perturbed generalized superior Mandelbrot sets." Nonlinear Dynamics 67, no. 3 (July 13, 2011): 1883–91. http://dx.doi.org/10.1007/s11071-011-0115-2.

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44

Zhang, Yong-Ping. "Feedback control and synchronization of Mandelbrot sets." Chinese Physics B 22, no. 1 (January 2013): 010502. http://dx.doi.org/10.1088/1674-1056/22/1/010502.

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45

Endler, Antonio, and Paulo C. Rech. "From Mandelbrot-like sets to Arnold tongues." Applied Mathematics and Computation 222 (October 2013): 559–63. http://dx.doi.org/10.1016/j.amc.2013.08.001.

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46

Romera, M., G. Pastor, A. B. Orue, D. Arroyo, and F. Montoya. "Coupling Patterns of External Arguments in the Multiple-Spiral Medallions of the Mandelbrot Set." Discrete Dynamics in Nature and Society 2009 (2009): 1–14. http://dx.doi.org/10.1155/2009/135637.

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The multiple-spiral medallions are beautiful decorations situated in the proximity of the small copies of the Mandelbrot set. They are composed by an infinity of babies Mandelbrot sets that have external arguments with known structure. In this paper we study the coupling patterns of the external arguments of the baby Mandelbrot sets in multiple-spiral medallions, that is, how these external arguments are grouped in pairs. Based on our experimental data, we obtain that the canonical nonspiral medallions have a nested pairs pattern, the canonical single-spiral medallions have an adjacent pairs pattern, and we conjecture that the canonical double, triple, quadruple-spiral medallions have a 1-nested/adjacent pairs pattern.

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47

Tassaddiq, Asifa, Amna Kalsoom, Maliha Rashid, Kainat Sehr, and Dalal Khalid Almutairi. "Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes." Axioms 13, no. 3 (March 18, 2024): 204. http://dx.doi.org/10.3390/axioms13030204.

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Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form QC(p)=apn+mp+c, where n≥2. It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals.

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48

WANG, XING-YUAN, and LI-NA GU. "RESEARCH FRACTAL STRUCTURES OF GENERALIZED M-J SETS USING THREE ALGORITHMS." Fractals 16, no. 01 (March 2008): 79–88. http://dx.doi.org/10.1142/s0218348x08003764.

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Анотація:

Extreme modulus escaping time algorithm, decomposition algorithm and fisheye algorithm are analyzed in this thesis, and we construct a series of generalized Mandelbrot-Julia (M-J) sets using these three algorithms. By studying the structural character of generalized M-J sets, we find: (1) extreme modulus escaping time algorithm and decomposition algorithm are simple modifications of classic escaping time algorithm, they can both construct the structure of non-boundary areas of generalized M-J sets; (2) non-boundary areas of generalized M-J sets have fractal characters; (3) generalized M-J sets have symmetry, and the process of evolvement depends on range of phase angle; (4) we can observe not only the whole structure of generalized Mandelbrot sets but also the details of some parts by using fisheye algorithm.

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49

Mork, L. K., Trenton Vogt, Keith Sullivan, Drew Rutherford, and Darin J. Ulness. "Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions." Fractal and Fractional 3, no. 3 (July 12, 2019): 42. http://dx.doi.org/10.3390/fractalfract3030042.

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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.

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50

ROCHON, DOMINIC. "A GENERALIZED MANDELBROT SET FOR BICOMPLEX NUMBERS." Fractals 08, no. 04 (December 2000): 355–68. http://dx.doi.org/10.1142/s0218348x0000041x.

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We use a commutative generalization of complex numbers called bicomplex numbers to introduce bicomplex dynamics. In particular, we give a generalization of the Mandelbrot set and of the "filled-Julia" sets in dimensions three and four. Also, we establish that our version of the Mandelbrot set with quadratic polynomial in bicomplex numbers of the form w2 + c is identically the set of points where the associated generalized "filled-Julia" set is connected. Moreover, we prove that our generalized Mandelbrot set of dimension four is connected.

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