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Journal articles on the topic 'Fractals Mandelbrot sets'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 February 2022

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1

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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2

Zhou, Hao, Muhammad Tanveer, and Jingjng Li. "Comparative Study of Some Fixed-Point Methods in the Generation of Julia and Mandelbrot Sets." Journal of Mathematics 2020 (July 20, 2020): 1–15. http://dx.doi.org/10.1155/2020/7020921.

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Fractal is a geometrical shape with property that each point of the shape represents the whole. Having this property, fractals procured the attention in computer graphics, engineering, biology, mathematics, physics, art, and design. The fractals generated on highest priorities are the Julia and Mandelbrot sets. So, in this paper, we develop some necessary conditions for the convergence of sequences established for the orbits of M, M∗, and K-iterative methods to generate these fractals. We adjust algorithms according to the develop conditions and draw some attractive Julia and Mandelbrot sets with sequences of iterates from proposed fixed-point iterative methods. Moreover, we discuss the self-similarities with input parameters in each graph and present the comparison of images with proposed methods.

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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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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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4

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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Жихарев and L. Zhikharev. "Generalization to Three-Dimensional Space Fractals of Pythagoras and Koch. Part I." Geometry & Graphics 3, no. 3 (November 30, 2015): 24–37. http://dx.doi.org/10.12737/14417.

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Fractals are geometric objects, each part of which is similar to the whole object, so that if we take a part and increase its size to the size of the whole object, it would be impossible to notice a difference. In other words, fractals are sets having scale invariance. In mathematics, they are associated primarily with non-differentiable functions. The concept of "fractal" (from the Latin "Fractus" meaning «broken») had been introduced by Benoit Mandelbrot (1924–2010), French and American mathematician, physicist, and economist. Mandelbrot had found that seemingly arbitrary fluctuations in price of goods have a certain tendency to change: it turned out that daily fluctuations are symmetrical with long-term price fluctuations. In fact, Benoit Mandelbrot applied his recursive (fractal) method to solve the problem. Since the last quarter of the nineteenth century, a large number of fractal curves and flat objects have been created; and methods for their application have been developed. From geometrical point of view, the most interesting fractals are "Koch snowflake" and "Pythagoras Tree". Two classes of analogues of the volumetric fractals were created with modern three-dimensional modeling program: "Fractals of growth” – like Pythagoras Tree, “Fractals of separation” – like Koch snowflake; the primary classification was developed, their properties were studied. Empiric data was processed with basic arithmetic calculations as well as with computer software. Among other things, for fractals of separation the task was to create an object with an infinite surface area, which in the future might acquire great importance for the development of the chemical and other industries.

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6

MARTINEAU, ÉTIENNE, and DOMINIC ROCHON. "ON A BICOMPLEX DISTANCE ESTIMATION FOR THE TETRABROT." International Journal of Bifurcation and Chaos 15, no. 09 (September 2005): 3039–50. http://dx.doi.org/10.1142/s0218127405013873.

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In this article, we present some distance estimation formulas that can be used to ray traced slices of the bicomplex Mandelbrot set and the bicomplex filled-Julia sets in dimension three. We also present a simple method to explore and infinitely approach these 3D fractals. Because of its rich fractal structure and symmetry, we emphasize our work on the generalized Mandelbrot set for bicomplex numbers in dimension three: the Tetrabrot.

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Rani, Mamta, and Sanjeev Kumar Prasad. "Superior Cantor Sets and Superior Devil Staircases." International Journal of Artificial Life Research 1, no. 1 (January 2010): 78–84. http://dx.doi.org/10.4018/jalr.2010102106.

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Mandelbrot, in 1975, coined the term fractal and included Cantor set as a classical example of fractals. The Cantor set has wide applications in real world problems from strange attractors of nonlinear dynamical systems to the distribution of galaxies in the universe (Schroder, 1990). In this article, we obtain superior Cantor sets and present them graphically by superior devil’s staircases. Further, based on their method of generation, we put them into two categories.

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8

JONES, MARK P. "FUNCTIONAL PEARL Composing fractals." Journal of Functional Programming 14, no. 6 (October 27, 2004): 715–25. http://dx.doi.org/10.1017/s0956796804005167.

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This paper describes a simple but flexible family of Haskell programs for drawing pictures of fractals such as Mandelbrot and Julia sets. Its main goal is to showcase the elegance of a compositional approach to program construction, and the benefits of a clean separation between different aspects of program behavior. Aimed at readers with relatively little experience of functional programming, the paper can be used as a tutorial on functional programming, as an overview of the Mandelbrot set, or as a motivating example for studies in computability.

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9

Kang, Shin Min, Waqas Nazeer, Muhmmad Tanveer, and Abdul Aziz Shahid. "New Fixed Point Results for Fractal Generation in Jungck Noor Orbit withs-Convexity." Journal of Function Spaces 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/963016.

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We establish new fixed point results in the generation of fractals (Julia sets, Mandelbrot sets, and Tricorns and Multicorns for linear or nonlinear dynamics) by using Jungck Noor iteration withs-convexity.

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10

GALEEVA, R., and A. VERJOVSKY. "QUATERNION DYNAMICS AND FRACTALS IN ℝ4." International Journal of Bifurcation and Chaos 09, no. 09 (September 1999): 1771–75. http://dx.doi.org/10.1142/s0218127499001255.

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In this paper we study the Fatou–Julia theory for some quaternionic rational functions in the quaternion skew-field ℍ. We obtain new dynamically-defined fractals in ℝ4 as the corresponding Julia sets. We also define the quaternionic Mandelbrot set.

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Chen, Zhihua, Muhammad Tanveer, Waqas Nazeer, and Jing Wu. "Fractals via Generalized Jungck–S Iterative Scheme." Discrete Dynamics in Nature and Society 2021 (March 1, 2021): 1–12. http://dx.doi.org/10.1155/2021/8886056.

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The purpose of this research is to introduce a Jungck–S iterative method with m , h 1 , h 2 –convexity and hence unify different comparable iterative schemes existing in the literature. A Jungck–S orbit is constructed, and escape radius is derived with our scheme. A new escape radius is also obtained for generating the fractals. Julia and Mandelbrot set are visualized with the help of proposed algorithms based on our iterative scheme. Moreover, we present some complex graphs of Julia and Mandelbrot sets using the derived orbit and discuss their nature in detail.

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12

Shahid, Abdul Aziz, Waqas Nazeer, and Krzysztof Gdawiec. "The Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets." Monatshefte für Mathematik 195, no. 4 (July 1, 2021): 565–84. http://dx.doi.org/10.1007/s00605-021-01591-z.

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AbstractIn recent years, researchers have studied the use of different iteration processes from fixed point theory in the generation of complex fractals. For instance, the Mann, Ishikawa, Noor, Jungck–Mann and Jungck–Ishikawa iterations have been used. In this paper, we study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. We prove the escape criterion for the $$(k+1)$$ ( k + 1 ) st degree complex polynomial. Moreover, we present some graphical and numerical examples regarding Mandelbrot and Julia sets generated using the proposed iteration.

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13

FRASER, JONATHAN M., JUN JIE MIAO, and SASCHA TROSCHEIT. "The Assouad dimension of randomly generated fractals." Ergodic Theory and Dynamical Systems 38, no. 3 (September 22, 2016): 982–1011. http://dx.doi.org/10.1017/etds.2016.64.

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We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either thealmost sureor theBaire typicalAssouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure-theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.

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Mork, L. K., Keith Sullivan, and Darin J. Ulness. "Lacunary Möbius Fractals on the Unit Disk." Symmetry 13, no. 1 (January 6, 2021): 91. http://dx.doi.org/10.3390/sym13010091.

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Centered polygonal lacunary functions are a type of lacunary function that exhibit behaviors that set them apart from other lacunary functions, this includes rotational symmetry. This work will build off of earlier studies to incorporate the automorphism group of the open unit disk D, which is a subgroup of the Möbius transformations. The behavior, dimension, dynamics, and sensitivity of filled-in Julia sets and Mandelbrot sets to variables will be discussed in detail. Additionally, several visualizations of this three-dimensional parameter space will be presented.

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15

RAMS, MICHAŁ, and KÁROLY SIMON. "Projections of fractal percolations." Ergodic Theory and Dynamical Systems 35, no. 2 (September 11, 2013): 530–45. http://dx.doi.org/10.1017/etds.2013.45.

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AbstractIn this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset { \mathbb{R} }^{2} $, which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than $1$ then the orthogonal projection to every line, the radial projection with every centre, and the distance set from every point contain intervals.

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16

GARANT–PELLETIER, V., and D. ROCHON. "ON A GENERALIZED FATOU–JULIA THEOREM IN MULTICOMPLEX SPACES." Fractals 17, no. 03 (September 2009): 241–55. http://dx.doi.org/10.1142/s0218348x09004326.

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In this article we introduce the hypercomplex 3D fractals generated from Multicomplex Dynamics. We generalize the well known Mandelbrot and filled-in Julia sets for the multicomplex numbers (i.e. bicomplex, tricomplex, etc.). In particular, we give a multicomplex version of the so-called Fatou-Julia theorem. More precisely, we present a complete topological characterization in ℝ2n of the multicomplex filled-in Julia set for a quadratic polynomial in multicomplex numbers of the form w2 + c. We also point out the symmetries between the principal 3D slices of the generalized Mandelbrot set for tricomplex numbers.

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17

Жихарев, Л., and L. Zhikharev. "Fractal Dimensionalities." Geometry & Graphics 6, no. 3 (November 14, 2018): 33–48. http://dx.doi.org/10.12737/article_5bc45918192362.77856682.

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One of the most important characteristics of a fractal is its dimensionality. In general, there are several options for mathematical definition of this value, but usually under the object dimensionality is understood the degree of space filling by it. It is necessary to distinguish the dimensionality of space and the dimension of multitude. Segment, square and cube are objects with dimensionality 1, 2 and 3, which can be in respective spaces: on a straight line, plane or in a 3D space. Fractals can have a fractional dimensionality. By definition, proposed by Bernois Mandelbrot, this fractional dimensionality should be less than the fractal’s topological dimension. Abram Samoilovich Bezikovich (1891–1970) was the author of first mathematical conclusions based on Felix Hausdorff (1868–1942) arguments and allowing determine the fractional dimensionality of multitudes. Bezikovich – Hausdorff dimensionality is determined through the multitude covering by unity elements. In practice, it is more convenient to use Minkowsky dimensionality for determining the fractional dimensionalities of fractals. There are also numerical methods for Minkowsky dimensionality calculation. In this study various approaches for fractional dimensionality determining are tested, dimensionalities of new fractals are defined. A broader view on the concept of dimensionality is proposed, its dependence on fractal parameters and interpretation of fractal sets’ structure are determined. An attempt for generalization of experimental dependences and determination of general regularities for fractals structure influence on their dimensionality is realized. For visualization of three-dimensional geometrical constructions, and plain evidence of empirical hypotheses were used computer models developed in the software for three-dimensional modeling (COMPASS, Inventor and SolidWorks), calculations were carried out in mathematical packages such as Wolfram Mathematica.

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18

Zhang, Yi, and Da Wang. "Fractals Parrondo’s Paradox in Alternated Superior Complex System." Fractal and Fractional 5, no. 2 (April 28, 2021): 39. http://dx.doi.org/10.3390/fractalfract5020039.

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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.

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FERNÁNDEZ-GUASTI, M. "FRACTALS WITH HYPERBOLIC SCATORS IN 1 + 2 DIMENSIONS." Fractals 23, no. 02 (May 28, 2015): 1550004. http://dx.doi.org/10.1142/s0218348x15500048.

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A nondistributive scator algebra in 1 + 2 dimensions is used to map the quadratic iteration. The hyperbolic numbers square bound set reveals a rich structure when taken into the three-dimensional (3D) hyperbolic scator space. Self-similar small copies of the larger set are obtained along the real axis. These self-similar sets are located at the same positions and have equivalent relative sizes as the small M-set copies found between the Myrberg-Feigenbaum (MF) point and -2 in the complex Mandelbrot set. Furthermore, these small copies are self similar 3D copies of the larger 3D bound set. The real roots of the respective polynomials exhibit basins of attraction in a 3D space. Slices of the 3D confined scator set, labeled [Formula: see text](s;x,y), are shown at different planes to give an approximate idea of the 3D objects highly complicated boundary.

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Sallow, Amira Bibo. "Implementation and Analysis of Fractals Shapes using GPU-CUDA Model." Academic Journal of Nawroz University 10, no. 2 (April 28, 2021): 1. http://dx.doi.org/10.25007/ajnu.v10n2a1030.

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The rapid evolution of floating-point computing capacity and memory in recent years has resulted graphics processing units (GPUs) an increasingly attractive platform to speed scientific applications and are popular rapidly due to the large amount of data that processes the data on time. Fractals have many implementations that involve faster computation and massive amounts of floating-point computation. In this paper, constructing the fractal image algorithm has been implemented both sequential and parallel versions using fractal Mandelbrot and Julia sets. CPU was used for the execution in sequential mode while GPUarray and CUDA kernel was used for the parallel mode. The evaluation of the performance of the constructed algorithms for sequential structure using CPUs (2.20 GHz and 2.60 GHz) and parallelism structure for various models of GPU (GeForce GTX 1060 and GeForce GTX 1660 Ti ) devices, calculated in terms of execution time and speedup to compare between CPU and GPU maximum ability. The results showed that the execution on GPU using GPUArray or GUDA kernel is faster than its sequential implementation using CPU. And the execution using the GUDA kernel is faster than the execution using GPUArray, and the execution time between GPU devices was different, GPU with (Ti) series execute faster than the other models.

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Klein, Moshe, and Oded Maimon. "The Dynamics in the Soft Numbers Coordinate System." JOURNAL OF ADVANCES IN MATHEMATICS 18 (January 4, 2020): 1–17. http://dx.doi.org/10.24297/jam.v18i.8531.

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"Soft Logic" extends the number 0 from a single point to a continuous line, which we term "The zero axis". One of the modern science challenges is finding a bridge between the real world outside the observer and the observer's inner world. In “Soft Logic” we suggested a constructive model of bridging the two worlds by defining, on the base of the zero axis, a new kind of numbers, which we called ‘Soft Numbers’. Inspired by the investigation and visualization of fractals by Mandelbrot, within the investigation of the dynamics of some special function of a complex variable on the complex plane, we investigate in this paper the dynamics of soft functions on the plane strip with a special coordinate system. The recursive process that creates this soft dynamics allows us to discover new dynamics sets in a plane.

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Mohanty, S., and S. N. Nayak. "Fractal Geometry of Helicity Amplitude." Fractals 05, no. 02 (June 1997): 229–35. http://dx.doi.org/10.1142/s0218348x9700022x.

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Fractal geometry is in use to get lucid and visual explanation of many a natural and complex phenomena. High Energy Scattering process is quite complex a happening in science. Several methods are available to explain the events linked with high energy scattering. Mohanty et al.4 have tried using a method named Optimized Polynomial Expansion Technique to study the characteristics of hyperon-nucleon scattering. The present paper attempts to study the same in analysing its mathematical function as a fractal exposition. Here the analyticity property of helicity amplitude is treated using fractal theory and it is found that the function is analytic and the domain of analyticity is also an ellipse. The fractal geometry of this function presents a typical Julia Dragon set which looks like a San Marcos Dragon as per Mandelbrot6 and has Mandelbrot-like sets as well as Julia sets carved inside it.

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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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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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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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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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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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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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Ojha, D. B., Ms Shree, A. Dwivedi, and A. Mishra. "An approach for Embedding Elliptic Curve in Fractal Based Digital Signature Scheme." Journal of Scientific Research 3, no. 1 (December 19, 2010): 75. http://dx.doi.org/10.3329/jsr.v3i1.4694.

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We established a new approach for cryptographic digital signature scheme based on Mandelbrot and Julia fractal sets. We have embedded the features of ECC (elliptic curve cryptography) to the digital signature scheme based on Mandelbrot and Julia fractal sets. We offered a digital signature that has advantages of both the fractal based digital signature as well as of elliptic curve digital signature.Keywords: Fractal; ECC; Digital signature.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i1.4694 J. Sci. Res. 3 (1), 75-79 (2011)

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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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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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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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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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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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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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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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Cheng, Chia-Chin, and Sy-Sang Liaw. "Similarity between the Dynamic and Parameter Spaces in Cubic Mappings." Fractals 06, no. 03 (September 1998): 293–99. http://dx.doi.org/10.1142/s0218348x98000341.

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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.

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Troscheit, Sascha. "The quasi-Assouad dimension of stochastically self-similar sets." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 1 (January 24, 2019): 261–75. http://dx.doi.org/10.1017/prm.2018.112.

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AbstractThe class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions.

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Ahmad Alia, Mohammad, and Azman Bin Samsudin. "A New Digital Signature Scheme Based on Mandelbrot and Julia Fractal Sets." American Journal of Applied Sciences 4, no. 11 (November 1, 2007): 848–56. http://dx.doi.org/10.3844/ajassp.2007.848.856.

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WANG, XINGYUAN, and CHAO LUO. "BIFURCATION AND FRACTAL OF THE COUPLED LOGISTIC MAP." International Journal of Modern Physics B 22, no. 24 (September 30, 2008): 4275–90. http://dx.doi.org/10.1142/s0217979208038971.

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The nature of the fixed points of the coupled Logistic map is researched, and the boundary equation of the first bifurcation of the coupled Logistic map in the parameter space is given out. Using the quantitative criterion and rule of system chaos, i.e., phase graph, bifurcation graph, power spectra, the computation of the fractal dimension, and the Lyapunov exponent, the paper reveals the general characteristics of the coupled Logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the coupled Logistic map may emerge out of double-periodic bifurcation and Hopf bifurcation, respectively; (2) during the process of double-period bifurcation, the system exhibits self-similarity and scale transform invariability in both the parameter space and the phase space. From the research of the attraction basin and Mandelbrot–Julia set of the coupled Logistic map, the following conclusions are indicated: (1) the boundary between periodic and quasiperiodic regions is fractal, and that indicates the impossibility to predict the moving result of the points in the phase plane; (2) the structures of the Mandelbrot–Julia sets are determined by the control parameters, and their boundaries have the fractal characteristic.

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PIACQUADIO, M., and E. CESARATTO. "MULTIFRACTAL SPECTRUM AND THERMODYNAMICAL FORMALISM OF THE FAREY TREE." International Journal of Bifurcation and Chaos 11, no. 05 (May 2001): 1331–58. http://dx.doi.org/10.1142/s0218127401002754.

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Let (Ω, μ) be a set of real numbers to which we associate a measure μ. Let α≥0, let Ωα={x∈Ω/α(x)=α}, where α is the concentration index defined by Halsey et al. [1986]. Let fH(α) be the Hausdorff dimension of Ωα. Let fL(α) be the Legendre spectrum of Ω, as defined in [Riedi & Mandelbrot, 1998]; and fC(α) the classical computational spectrum of Ω, defined in [Halsey et al., 1986]. The task of comparing fH, fC and fL for different measures μ was tackled by several authors [Cawley & Mauldin, 1992; Mandelbrot & Riedi, 1997; Riedi & Mandelbrot, 1998] working, mainly, on self-similar measures μ. The Farey tree partition in the unit segment induces a probability measure μ on an universal class of fractal sets Ω that occur in physics and other disciplines. This measure μ is the Hyperbolic measure μℍ, fundamentally different from any self-similar one. In this paper we compare fH, fC and fL for μℍ.

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WANG, XINGYUAN, WEI LIU, and XUEJING YU. "RESEARCH ON BROWNIAN MOVEMENT BASED ON GENERALIZED MANDELBROT–JULIA SETS FROM A CLASS COMPLEX MAPPING SYSTEM." Modern Physics Letters B 21, no. 20 (August 30, 2007): 1321–41. http://dx.doi.org/10.1142/s0217984907013560.

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This article analyzes the typical Langevin problem, i.e. the dynamics of a charged particle under the circularly successive influence of several simple impulse functions moving in a double-well potential and a time-dependent magnetic field. Using the stroboscopic sampling, by selecting an appropriate magnetic intensity and time interval, we reduce the Langevin equation to a class complex mapping system. Through an experimental mathematical method, the authors study the structures of generalized M–J (Mandelbrot–Julia) sets generated by the complex mapping system, and expatiate the theory of Brownian movement. The authors find that:.(1) This paper extends Shirriff constructed M sets by combining two simple complex mappings;.(2) The fractal structure of the generalized M–J sets may visually depict the rule of Brownian movement, and the infinite overlapping embedment self-similar structure reflects the complexity of Brownian movement;.(3) Whether the selected time interval is significant or not determines the continuity of the fractal structure for the generalized M–J sets;.(4) The changing rule of particle velocity depends on the different choices of the principal range of phase angle;.(5) If we change the choices of the magnetic intensity and time interval, for example, we choose a randomly fluctuating magnetic field, the generalized J sets may emerge the interior-filling structure feature, i.e. "explosion" phenomena appear in the closure of the unstable periodic orbits of the particle in the velocity space.

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TABORDA, J. A., F. ANGULO, and G. OLIVAR. "MANDELBROT-LIKE BIFURCATION STRUCTURES IN CHAOS BAND SCENARIO OF SWITCHED CONVERTER WITH DELAYED-PWM CONTROL." International Journal of Bifurcation and Chaos 20, no. 01 (January 2010): 99–119. http://dx.doi.org/10.1142/s0218127410025430.

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In this paper, we report Mandelbrot-like bifurcation structures in a one-dimensional parameter space of real numbers corresponding to a dc-dc power converter modeled as a piecewise-smooth system with three zones. These fractal patterns have been studied in two-dimensional parameter space for smooth systems, but for nonsmooth systems has not been reported yet. The Mandelbrot-like sets we found are created in transition from the torus band to chaos band scenarios exhibited by a dc-dc buck power converter controlled by Delayed Pulse-Width Modulator (PWM) based on Zero Average Dynamics (or ZAD strategy), which corresponds to a piecewise-smooth system (PWS). The real parameter is provided by the PWM control strategy, namely ZAD strategy, and it can be varied in a large range, ideally (-∞, +∞). At -∞ and +∞ the dynamical behavior is the same, and thus we will describe the synamics in an ring-like parameter space. Mandelbrot-like borders are built by four chaotic bands, therefore these structures can be thought as instability islands where the state variables cannot be located. Using the Poincaré map approach we characterize the bifurcation structures and we describe recurrent patterns in different scales.

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Ortega, Alfonso, Marina de la Cruz, and Manuel Alfonseca. "Parametric 2-dimensional L systems and recursive fractal images: Mandelbrot set, Julia sets and biomorphs." Computers & Graphics 26, no. 1 (February 2002): 143–49. http://dx.doi.org/10.1016/s0097-8493(01)00162-5.

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HIMEKI, YUTARO, and YUTAKA ISHII. "is regular-closed." Ergodic Theory and Dynamical Systems 40, no. 1 (April 10, 2018): 213–20. http://dx.doi.org/10.1017/etds.2018.27.

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For each $n\geq 2$, we investigate a family of iterated function systems which is parameterized by a common contraction ratio $s\in \mathbb{D}^{\times }\equiv \{s\in \mathbb{C}:0<|s|<1\}$ and possesses a rotational symmetry of order $n$. Let ${\mathcal{M}}_{n}$ be the locus of contraction ratio $s$ for which the corresponding self-similar set is connected. The purpose of this paper is to show that ${\mathcal{M}}_{n}$ is regular-closed, that is, $\overline{\text{int}\,{\mathcal{M}}_{n}}={\mathcal{M}}_{n}$ holds for $n\geq 4$. This gives a new result for $n=4$ and a simple geometric proof of the previously known result by Bandt and Hung [Fractal $n$-gons and their Mandelbrot sets. Nonlinearity 21 (2008), 2653–2670] for $n\geq 5$.

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Alia, Mohammad, and Khaled Suwais. "Improved Steganography Scheme based on Fractal Set." International Arab Journal of Information Technology 17, no. 1 (January 1, 2019): 128–36. http://dx.doi.org/10.34028/iajit/17/1/15.

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Steganography is the art of hiding secret data inside digital multimedia such as image, audio, text and video. It plays a significant role in current trends for providing secure communication and guarantees accessibility of secret information by authorised parties only. The Least-Significant Bit (LSB) approach is one of the important schemes in steganography. The majority of LSB-based schemes suffer from several problems due to distortion in a limited payload capacity for stego-image. In this paper, we have presented an alternative steganographic scheme that does not rely on cover images as in existing schemes. Instead, the image which includes the secure hidden data is generated as an image of a curve. This curve is resulted from a series of computation that is carried out over the mathematical chaotic fractal sets. The new scheme aims at improving the data concealing capacity, since it achieves limitless concealing capacity and disposes of the likelihood of the attackers to realise any secret information from the resulted stego-image. From the security side, the proposed scheme enhances the level of security as the scheme depends on the exact matching between secret information and the generated fractal (Mandelbrot-Julia) values. Accordingly, a key stream is created based on these matches. The proposed scheme is evaluated and tested successfully from different perspectives

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Liu, Zhifeng, Tao Zhang, Yongsheng Zhao, and Shuxin Bi. "Time-varying stiffness model of spur gear considering the effect of surface morphology characteristics." Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 233, no. 2 (May 12, 2018): 242–53. http://dx.doi.org/10.1177/0954408918775955.

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The nonuniform cantilever beam and Hertzian contact model have been widely used to derive the mesh stiffness of spur gear assuming that the contact surface is absolutely frictionless. However, studies have confirmed that machined surfaces are rough in microscale and can be simulated by the Weierstrass–Mandelbort function. In order to get a reasonable and precise mesh stiffness model, the M-B contact model and finite element method are combined to express the local contact stiffness Kh. Through the simulation and comparison, the analytical finite element method is proved to be consistent with the traditional models and introduces the roughness parameters of machined tooth surface into the meshing process. Furthermore, the results also show that it is advantageous to improve the total mesh stiffness by increasing the fractal dimension D and input torque T as well as decreasing the roughness parameter G. In this paper, a relationship is built between the total mesh stiffness of gear sets with tooth surface characters and input torque, which can be a guidance in the design of the tooth surface parameters and the choice of the processing method in the future.

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BOUNIAS, M., and A. BONALY. "TOPOLOGICAL AND NONLINEAR PROPERTIES OF LIGAND-RECEPTOR SYSTEMS." Journal of Biological Systems 04, no. 03 (September 1996): 315–52. http://dx.doi.org/10.1142/s0218339096000235.

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Ligand-receptor (L-R) interactions involving substrate-enzyme and hormone-receptor systems, obey complex processes. Many parameters are not easily addressed in classical kinetic approaches; most are generally based on the probabilistic mass-action law and equilibrium constants. We describe the system in terms of tessellation of the reactional space with balls B(x, Γ) centered on the reactive species and whose radius (Γ), used as a scaling unit, is derived from the Hausdorff distance η=dist(L, R). This value is altered by a set of corrective terms representing interactions with inert and non-reactive species present in the medium, the influence of particular cell factors including heterogeneous phase conditions, and metabolic dissipation of the product. Topological properties have been studied for nonlinear pairing function governing the distance between two species, and the fractal dimension of the system is linked with its Bouligand-Minkowski dimension. The set of instant state equations of the system includes a chain of transfer matrices accounting for the linear phase of catalytic functions. An alternative set of iterative functions providing a complete metric description of the system in its topological definition space, exhibits similarities with the equations of the Mandelbrot sets family. Experimental confirmation that the major parameters (Vmax, S50 and Hill coefficient) are polyphasic functions of time rather than constants was obtained for honeybees haemolymph α-glucosidases.

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Rama, Bulusu, and Jibitesh Mishra. "Generation of 3D Fractal Images for Mandelbrot and Julia Sets." International Journal of Computer and Communication Technology, January 2011, 14–18. http://dx.doi.org/10.47893/ijcct.2011.1064.

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Fractals provide an innovative method for generating 3D images of real-world objects by using computational modelling algorithms based on the imperatives of self-similarity, scale invariance, and dimensionality. Images such as coastlines, terrains, cloud mountains, and most interestingly, random shapes composed of curves, sets of curves, etc. present a multivaried spectrum of fractals usage in domains ranging from multi-coloured, multi-patterned fractal landscapes of natural geographic entities, image compression to even modelling of molecular ecosystems. Fractal geometry provides a basis for modelling the infinite detail found in nature. Fractals contain their scale down, rotate and skew replicas embedded in them. Many different types of fractals have come into limelight since their origin. This paper explains the generation of two famous types of fractals, namely the Mandelbrot Set and Julia Set, the3D rendering of which gives a real-world look and feel in the world of fractal images.

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