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Relevant bibliographies by topics / Fractals Mandelbrot sets

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Academic literature on the topic 'Fractals Mandelbrot sets'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 February 2022

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Journal articles on the topic "Fractals Mandelbrot sets"

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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Fractals Mandelbrot sets"

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Tolmie, Julie. "Visualisation, navigation and mathematical perception : a visual notation for rational numbers mod 1." View thesis entry in Australian Digital Theses Program, 2000. http://thesis.anu.edu.au/public/adt-ANU20020313.101505/index.html.

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Reis, Márcio Vaiz dos. "Conjunto de Mandelbrot." Universidade Federal de Goiás, 2016. http://repositorio.bc.ufg.br/tede/handle/tede/6343.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
The purpose of this dissertation is to present an introductory approach to the complex dynamics and fractal geometry, especially the Mandelbrot set. With the goal to simplify and stimulate the introduction of complex number in high school, the approach adopted was: the definition of the Mandelbrot set and its characteristics; the relationship between the Mandelbrot set and Julia set; how to find by using the Mandelbrot set. One of the tools used to help the teaching was Geogeobra, a dynamic software that allows the student to build the Mandelbrot set. Through this study, it is expected to motivate the learning of complex numbers by using fractal obtained by the study of function ( ) . Obtaining, as a result, a differentiated and motivating way of learning for a better understanding and intellectual development of the students.
Esse trabalho apresenta uma abordagem introdutória para a dinâmica complexa e a geometria fractal, em especial o conjunto de Mandelbrot. Com objetivo de facilitar e motivar a interação dos alunos com o ensino dos números complexos, a abordagem adotada foi: a definição do conjunto de Mandelbrot e suas características; a relação entre o conjunto de Mandelbrot e o conjunto de Julia; a relação do conjunto de Mandelbrot e o número . Uma das ferramentas utilizadas para auxiliar o professor foi o Geogeobra, um software dinâmico que permite o aluno a construção do conjunto de Mandelbrot. Por meio deste trabalho, espera-se motivar o ensino dos números complexos através do fractal obtido pelo estudo da função ( ) . Obtendo assim, como resultado, uma forma diferenciada e motivadora do aprendizado do aluno, garantindo um melhor entendimento e desenvolvimento intelectual.

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Hannah, Walter. "Internal rays of the Mandelbrot set." Thesis, 2006. http://www.ithaca.edu/hs/depts/math/docs/theses/whannahthesis.pdf.

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Wheeler, Jodi Lynette. "Fractals : an exploration into the dimensions of curves and sufaces." Thesis, 2011. http://hdl.handle.net/2152/ETD-UT-2011-08-3825.

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When many people think of fractals, they think of the beautiful images created by Mandelbrot’s set or the intricate dragons of Julia’s set. However, these are just the artistic stars of the fractal community. The theory behind the fractals is not necessarily pretty, but is very important to many areas outside the world of mathematics. This paper takes a closer look at various types of fractals, the fractal dimensionality of surfaces and chaotic dynamical systems. Some of the history and introduction of creating fractals is discussed. The tools used to prevent a modified Koch’s curve from overlapping itself, finding the limit of a curves length and solving for a surfaces dimensional measurement are explored. Lastly, an investigation of the theories of chaos and how they bring order into what initially appears to be random and unpredictable is presented. The practical purposes and uses of fractals throughout are also discussed.
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Silvestri, Stefano. "The Dynamics of Semigroups of Contraction Similarities on the Plane." Thesis, 2019. http://hdl.handle.net/1805/19909.

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Indiana University-Purdue University Indianapolis (IUPUI)
Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M. Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.

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(6983546), Stefano Silvestri. "The Dynamics of Semigroups of Contraction Similarities on the Plane." Thesis, 2019.

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Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M.

Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.

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Books on the topic "Fractals Mandelbrot sets"

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Lesmoir-Gordon, Nigel. The colours of infinity: The beauty and power of fractals. London: Springer Verlag, 2010.

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Mandelbrot, Benoit B. Fractals and chaos: The Mandelbrot set and beyond. New York, NY: Springer, 2004.

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1945-, Kauffman Louis H., and Sandin Daniel J, eds. Hypercomplex iterations: Distance estimation and higher dimensional fractals. River Edge, NJ: World Scientific, 2002.

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Conference Board of the Mathematical Sciences and National Science Foundation (U.S.), eds. Ergodic theory and fractal geometry. Providence, Rhode Island: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 2014.

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Mandelbrot, Benoit B. Fractals and Chaos: The Mandelbrot Set and Beyond. Springer, 2004.

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1945-, Stewart Ian, and Clarke Arthur Charles 1917-, eds. The colours of infinity: The beauty and power of fractals. [S.l.]: Clear Books, 2004.

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Frame, Michael. Fractal worlds: Grown, built, and imagined. 2016.

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Alt.Fractals: A visual guide to fractal geometry and design. Brighton, Uk: Chocolate Tree Books, 2011.

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Book chapters on the topic "Fractals Mandelbrot sets"

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Reeve, Dominic E. "Mandelbrot, Julia Sets and Nonlinear Mappings." In Fractals and Chaos, 35–42. New York, NY: Springer New York, 1991. http://dx.doi.org/10.1007/978-1-4612-3034-2_3.

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Douady, Adrien. "Julia Sets and the Mandelbrot Set." In The Beauty of Fractals, 161–74. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/978-3-642-61717-1_13.

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Peitgen, Heinz-Otto, Hartmut Jürgens, and Dietmar Saupe. "The Mandelbrot Set: Ordering the Julia Sets." In Chaos and Fractals, 783–837. New York, NY: Springer New York, 2004. http://dx.doi.org/10.1007/0-387-21823-8_15.

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Peitgen, Heinz-Otto, Hartmut Jürgens, and Dietmar Saupe. "The Mandelbrot Set: Ordering the Julia Sets." In Chaos and Fractals, 841–901. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-4740-9_15.

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Peitgen, Heinz-Otto, Hartmut Jürgens, and Dietmar Saupe. "The Mandelbrot Set: Ordering the Julia Sets." In Fractals for the Classroom, 415–73. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-4406-6_8.

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Boussoufi, Ouahiba, Kaoutar Lamrini Uahabi, and Mohamed Atounti. "Using Fractal Dimension to Check Similarity Between Mandelbrot and Julia Sets in Misiurewicz Points." In Advances in Intelligent Systems and Computing, 558–66. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-11928-7_50.

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BARNSLEY, MICHAEL F. "Parameter Spaces and Mandelbrot Sets." In Fractals Everywhere, 294–329. Elsevier, 1993. http://dx.doi.org/10.1016/b978-0-12-079061-6.50013-9.

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Falconer, Kenneth J. "4. Julia sets and the Mandelbrot set." In Fractals, 61–84. Oxford University Press, 2013. http://dx.doi.org/10.1093/actrade/9780199675982.003.0004.

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"Complex Fractals : Julia Sets and the Mandelbrot Set." In Encounters with Chaos and Fractals, 327–59. Chapman and Hall/CRC, 2012. http://dx.doi.org/10.1201/b11855-11.

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Calahan, S. Dean, and Jim Flanagan. "Self-Mapping of Mandelbrot Sets by Preiteration." In The Pattern Book: Fractals, Art, and Nature, 166–68. WORLD SCIENTIFIC, 1995. http://dx.doi.org/10.1142/9789812832061_0066.

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Conference papers on the topic "Fractals Mandelbrot sets"

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Dejun, Yan, Yang Rijing, Xin Huijie, and Zheng Jiangchao. "Generalized Mandelbrot Sets and Julia Sets for Non-analytic Complex Maps." In 2010 International Workshop on Chaos-Fractals Theories and Applications (IWCFTA). IEEE, 2010. http://dx.doi.org/10.1109/iwcfta.2010.42.

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Yan, Dejun, Junxing Zhang, Nan Jiang, and Lidong Wang. "General Mandelbrot Sets and Julia Sets Generated from Non-analytic Complex Iteration ⨍m(z)=z^n+c." In 2009 International Workshop on Chaos-Fractals Theories and Applications (IWCFTA 2009). IEEE, 2009. http://dx.doi.org/10.1109/iwcfta.2009.89.

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Shahinpoor, Mohsen. "An Introduction to Smart Fractal Structures and Mechanisms." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0160.

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Abstract Fractal structures are unique in the sense that they are highly expandible or collapsible and yet they are capable of preserving their basic structural geometry in a dynamic fashion. This dynamic geometric invariance opens up a new territory in fractal solids, i.e., fractal structures, mechanisms and robot manipulators. Some of these structure are in the form of highly deployable mechanisms and possibly redundant, multi-axis, multi-arm, multi-finger robot manipulators whose kinematic structure is fractal. Thus, simple fractal structures, such as triadic cantor set, and fractal functions, such as the Weirstraus-Mandelbrot functions, govern the structural branching of such robots and essentially define their kinematic structure. These deployable fractal structures, mechanisms and robot manipulators are shown to be capable of generating unique, and yet unparalleled properties such as computer-controlled microsensing even down to molecular level (micromachining) and computer-controlled dynamics such as the creation of hypervelocity fractons with speeds in the range of hundreds of kilometers per second. A number of structures and mechanisms and their unique properties are presented in this paper and a simple kinematic model is presented and briefly discussed.

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Yan, Dejun, Xiaodan Wei, Hongpeng Zhang, Nan Jiang, and Xiangdong Liu. "Fractal Structures of General Mandelbrot Sets and Julia Sets Generated From Complex Non-Analytic Iteration Fm(Z)=Zm+c." In 2nd International Symposium on Computer, Communication, Control and Automation. Paris, France: Atlantis Press, 2013. http://dx.doi.org/10.2991/isccca.2013.42.

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Michopoulos, John G., and Athanasios Iliopoulos. "High Dimensional Full Inverse Characterization of Fractal Volumes." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-71050.

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The present paper describes a methodology for the inverse identification of the complete set of parameters associated with the Weirstrass-Mandelbrot (W-M) function that can describe any fractal scalar field distribution of measured data defined within a volume. Our effort is motivated by the need to be able to describe a scalar field quantity distribution in a volume in order to be able to represent analytically various non-homogeneous material properties distributions for engineering and science applications. Our method involves utilizing a refactoring of the W-M function that permits defining the characterization problem as a high dimensional singular value decomposition problem for the determination of the so-called phases of the function. Coupled with this process is a second level exhaustive search that enables the determination of the density of the frequencies involved in defining the trigonometric functions involved in the definition of the W-M function. Numerical applications of the proposed method on both synthetic and actual volume data, validate the efficiency and the accuracy of the proposed approach. This approach constitutes a radical departure from the traditional fractal dimension characterization studies and opens the road for a very large number of applications and generalizes the approach developed by the authors for fractal surfaces to that of fractal volumes.

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Dawkins, Jeremy J., David M. Bevly, and Robert L. Jackson. "Multiscale Terrain Characterization Using Fourier and Wavelet Transforms for Unmanned Ground Vehicles." In ASME 2009 Dynamic Systems and Control Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/dscc2009-2718.

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This paper investigates the use of the Fourier transform and Wavelet transform as methods to supplement the more common root mean squared elevation and power spectral density methods of terrain characterization. Two dimensional terrain profiles were generated using the Weierstrass-Mandelbrot fractal equation. The Fourier and Wavelet transforms were used to decompose these terrains into a parameter set. A two degree of freedom quarter car model was used to evaluate the vehicle response before and after the terrain characterization. It was determined that the Fourier transform can be used to reduce the profile into the key frequency components. The Wavelet transform can effectively detect discontinuities of the profile and changes in the roughness of the profile. These two techniques can be added to current methods to yield a more robust terrain characterization.

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Michopoulos, John G., and Athanasios Iliopoulos. "Complete High Dimensional Inverse Characterization of Fractal Surfaces." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-47784.

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The present paper describes a methodology for the inverse identification of the complete set of parameters associated with the Weirstrass-Mandelbrot (W-M) function that can describe any rough surface known by its profilometric or topographic data. Our effort is motivated by the need to determine the mechanical, electrical and thermal properties of contact surfaces between deformable materials that conduct electricity and heat and require an analytical representation of the surfaces involved. Our method involves utilizing a refactoring of the W-M function that permits defining the characterization problem as a high dimensional singular value decomposition problem for the determination of the so-called phases of the function. Coupled with this process is a second level exhaustive search that enables the determination of the density of the frequencies involved in defining the trigonometric functions involved in the definition of the W-M function. Our approach proves that this is the only additional parameter that needs to be determined for full characterization of the W-M function as the rest can be selected arbitrarily. Numerical applications of the proposed method on both synthetic and actual elevation data, validate the efficiency and the accuracy of the proposed approach. This approach constitutes a radical departure from the traditional fractal dimension characterization studies and opens the road for a very large number of applications.

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