We are proudly a Ukrainian website. Our country was attacked by Russian Armed Forces on Feb. 24, 2022.
You can support the Ukrainian Army by following the link: https://u24.gov.ua/.
Even the smallest donation is hugely appreciated!
Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles
Consult the lists of relevant articles, books, theses, conference reports, and other scholarly sources on the topic 'Mandelbrot sets.'
Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.
You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.
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.
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).
2
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.
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.
3
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.
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.
4
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.
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 $z^2 + c$.
5
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 Fm(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.
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.
6
Kauko, Virpi. "Shadow trees of Mandelbrot sets." Fundamenta Mathematicae 180, no. 1 (2003): 35–87. http://dx.doi.org/10.4064/fm180-1-4.
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.
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.
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.
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.
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.
Dissertations / Theses on the topic "Mandelbrot sets":
1
Tingen, Larry L. "The Julia and Mandelbrot sets for the Hurwitz zeta function." View electronic thesis (PDF), 2009. http://dl.uncw.edu/etd/2009-3/tingenl/larrytingen.pdf.
Jones, Rafe. "Galois martingales and the hyperbolic subset of the p-adic Mandelbrot set /." View online version; access limited to Brown University users, 2005. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:3174623.
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.
Poirier, Schmitz Alfredo. "Invariant measures on polynomial quadratic Julia sets with no interior." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/96022.
We characterize invariant measures for quadratic polynomial Julia sets with no interior. We prove that besides the harmonic measure —the only one that is even and invariant—, all others are generated by a suitable odd measure. En este artículo caracterizamos medidas invariantes sobre conjuntos de Julia sin interior asociados con polinomios cuadráticos. Probamos que más allá de la medida armónica —la única par e invariante—, el resto son generadas por su parte impar.
5
Kuo, Li-Feng, and 郭立峰. "Mandelbrot Sets, Julia Sets and Their Algorithms." Thesis, 2019. http://ndltd.ncl.edu.tw/handle/6n28d7.
碩士 國立中央大學 數學系 107 In this thesis, we survey the big theme of fractals － Mandelbrot sets. We start to study Julia sets before study Mandelbrot sets, and the goal is generating figures of fractals and applying to arts.
Hence, we introduce the definition and properties of Julia sets firstly, and use this theory to arrange some useful algorithms for generating the figures of Julia sets. After we survey Julia sets, we can study Mandelbrot sets, since the definition of Mandelbrot sets is all of the points such that the Julia set is onnected. However, we obtain the obstacle when generating andelbrot sets, that is, how to check the Julia set is connected or not? The answer of this question is － the fundamental theorem of Mandelbrot sets, we can generate the figures of Mandelbrot sets by this theorem.
Finally, we give some examples of Mandelbrot sets and Julia sets, and introduce 3-dimensional Mandelbrot sets and Julia sets.
6
Hannah, Walter. "Internal rays of the Mandelbrot set." Thesis, 2006. http://www.ithaca.edu/hs/depts/math/docs/theses/whannahthesis.pdf.
Fitzgibbon, Elizabeth Laura. "Rational maps: the structure of Julia sets from accessible Mandelbrot sets." Thesis, 2014. https://hdl.handle.net/2144/15111.
For the family of complex rational maps F_λ(z)=z^n+λ/z^d, where λ is a complex parameter and n, d ≥ 2 are integers, many small copies of the well-known Mandelbrot set are visible in the parameter plane. An infinite number of these are located around the boundary of the connectedness locus and are accessible by parameter rays from the Cantor set locus. Maps taken from main cardioid of these accessbile Mandelbrot sets have attracting periodic cycles. A method for constructing models of the Julia sets corresponding to such maps is described. These models are then used to explore the existence of dynamical conjugacies between maps taken from distinct accessible Mandelbrot sets in the upper halfplane.
8
Lauber, Arnd. "On the Stability of Julia Sets of Functions having Baker Domains." Doctoral thesis, 2004. http://hdl.handle.net/11858/00-1735-0000-0006-B3DE-F.
Banaś, Marian. Analiza teoretyczna i badania właściwości zawiesin nieziarnistych w zastosowaniu do projektowsnia i eksploatacji wielostrumieniowych urządzeń sedymentacyjnych: Theoretical analysis and investigations of the properties of the non-grainy suspensions in terms to design and use of the lamella settling devices. Kraków: Wydawnictwa AGH, 2012.
Devaney, Robert, ed. Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets. Providence, Rhode Island: American Mathematical Society, 1995. http://dx.doi.org/10.1090/psapm/049.
AMS-IMS-SIAM Joint Summer Research Conference on Complex Dynamics: Twenty-five Years after the Appearance of the Mandelbrot Set (2004 Snowbird, Utah). Complex dynamics: Twenty-five years after the appearance of the Mandelbrot set : proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Complex Dynamics--Twenty-five Years after the Appearance of the Mandelbrot Set, June 13-17, 2004, Snowbird, Utah. Edited by Devaney Robert L. 1948- and Keen Linda. Providence, R.I: American Mathematical Society, 2006.
Furstenberg, Harry. Ergodic theory and fractal geometry. Providence, Rhode Island: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 2014.
Agarwal, Ravi P., Kanishka Perera, and Sandra Pinelas. "Julia and Mandelbrot Sets." In An Introduction to Complex Analysis, 316–20. Boston, MA: Springer US, 2011. http://dx.doi.org/10.1007/978-1-4614-0195-7_49.
Korsch, H. J., and H. J. Jodl. "Mandelbrot and Julia Sets." In Chaos, 227–48. Berlin, Heidelberg: Springer Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/978-3-662-03866-6_11.
Korsch, H. J., and H. J. Jodl. "Mandelbrot and Julia Sets." In Chaos, 227–48. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/978-3-662-02991-6_11.
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.
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.
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.
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.
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.
McClure, Mark. "Complex Dynamics:Julia Sets and the Mandelbrot Set." In Mathematica in Action, 277–300. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-0-387-75477-2_12.
Ochkov, Valery, Alan Stevens, and Anton Tikhonov. "Iterations and Fractal Sets of Mandelbrot and Julia." In STEM Problems with Mathcad and Python, 263–91. Boca Raton: Chapman and Hall/CRC, 2022. http://dx.doi.org/10.1201/9781003228356-14.
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.
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.
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.
Ganikhodzhayev, Rasul, and Shavkat Seytov. "An analytical description of mandelbrot and Julia sets for some multi-dimensional cubic mappings." In INTERNATIONAL UZBEKISTAN-MALAYSIA CONFERENCE ON “COMPUTATIONAL MODELS AND TECHNOLOGIES (CMT2020)”: CMT2020. AIP Publishing, 2021. http://dx.doi.org/10.1063/5.0058341.
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.
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.
7
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.
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.
8
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.
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.
9
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.
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.
