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Journal articles on the topic 'Fractal system'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Suny, Vian Hafid, Kosala Dwidja Purnomo, and Firdaus Ubaidillah. "PEMANFAATAN METODE ITERATED FUNCTION SYSTEM (IFS) PADA PEMBANGKITAN KURVA NAGA." Majalah Ilmiah Matematika dan Statistika 20, no. 2 (September 29, 2020): 89. http://dx.doi.org/10.19184/mims.v20i2.15780.

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Fractals have two types, namely fractals sets (artificial fractals) and natural fractals. Each type of fractal has a variety of fractal objects. One of the fractal objects is the Dragon Curve. Fractal objects can be generated through two methods, namely the Lindenmayer System (L-System) and the Iterated Function System (IFS). In previous studies, the Dragon curve can be generated through the L-System approach. The method is to start from determining the rotation angle, then determining the initial string, and the last one, which is determining the production rules. In this study, the Dragon curve is generated using IFS with Affine Transformation. The Affine transformation used in this study is dilation and rotation. Some variation is given on the scale of dilation and rotation angle. The variation is using a fixed angle with a variety of scale and using a fixed scale with a variation of angle. Each variation gives a different effect. This influence results in a varied visualization of the Naga curve. If the scale and angle that is varied approach a scale of one and an angle of 90° then the fractal formed approaches the Dragon curve of a scale of one with an angle of 90°. Conversely, if the scale and angle are varied away from one scale and angle of 90°, the fractal formed away from the Dragon curve of scale one with an angle of 90°. Keywords: Affine transformation, dragon curve, IFS method.

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2

Benguigui, L. "A Fractal Analysis of the Public Transportation System of Paris." Environment and Planning A: Economy and Space 27, no. 7 (July 1995): 1147–61. http://dx.doi.org/10.1068/a271147.

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An analysis of the railway networks of the public transportation system of Paris, based on the notion of fractals, is presented. The two basic networks, (metropolitan and suburban) which have different functions, have also a different fractal dimension: 1.8 for the metropolitan network, and 1.5 for the suburban network. By means of computer simulation, it is concluded that the true dimension of the metro network is probably 2.0. A fractal model of the suburban network, with the same features and the same fractal dimension, is proposed. This supports the conclusion that this network is fractal.

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Zhao, Shanrong, Jin Tan, Jiyang Wang, Xiaohong Xu, and Hong Liu. "A Dendrite with "Sierpinski Gasket" Fractal Morphology in Matt Glaze of LiAlSiO4-SiO2 System." Fractals 11, no. 03 (September 2003): 271–76. http://dx.doi.org/10.1142/s0218348x03001525.

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In this paper, we introduce a dendritic crystal, formed in matt glaze of LiAlSiO 4- SiO 2, having "Sierpinski gasket" fractal morphology. The crystal structure of this "Sierpinski gasket" dendrite is β-quartz. β-quartz can grow two kinds of fractal patterns: snow-shaped dendrite and "Sierpinski gasket" dendrite, depending on different supercooling conditions. These two kinds of fractals can develop together in one dendritic crystal. The evolution of the boundary morphologies between these two kinds of fractal dendrites can be described by another fractal — Koch curve. The "Sierpinski gasket" dendrite is a rather new fractal growth pattern which can introduce new opportunities to fractal growth research of nonlinear sciences.

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4

Giona, Massimiliano. "Chemical Engineering, Fractal and Disordered System Theory." Fractals 05, no. 03 (September 1997): 333–54. http://dx.doi.org/10.1142/s0218348x97000334.

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This article critically discusses the applications of fractal and disordered system theory to chemical engineering problems in order to highlight some promising research directions and the difficulties that may be encountered. Starting from the analysis of transport and reaction kinetics, the question is addressed, with the aid of some examples, of whether and how engineering research could help in the study of complex phenomenologies on fractals and disordered systems. The effects of thermodynamical nonidealities in transport and adsorption, and the influence of nonlinearities in reaction kinetics are discussed in some detail. Examples of typical engineering problems in which fractal analysis may help towards a better understanding of the physical phenomenologies in the presence of complex porous substrata and fluid mixtures are discussed. The role played by the boundary conditions on transport phenomena involving fractal structures is also analyzed. A critical discussion on the perspectives in the characterization of disordered and fractal porous structures, and in the study of turbulent transport and mixing is also developed.

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Cherny, A. Yu, E. M. Anitas, V. A. Osipov, and A. I. Kuklin. "Scattering from surface fractals in terms of composing mass fractals." Journal of Applied Crystallography 50, no. 3 (June 1, 2017): 919–31. http://dx.doi.org/10.1107/s1600576717005696.

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It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensityI(q) ∝ q^{D_{\rm s}-6}, where 2 <Ds< 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝r−τdr, withDs= τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface fractals, such as the fractal iteration number and the scaling factor.

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6

HUYNH, HOAI NGUYEN, and LOCK YUE CHEW. "ARC-FRACTAL AND THE DYNAMICS OF COASTAL MORPHOLOGY." Fractals 19, no. 02 (June 2011): 141–62. http://dx.doi.org/10.1142/s0218348x11005178.

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In this paper, we present an idea of creating fractals by using the geometric arc as the basic element. This approach of generating fractals, through the tuning of just three parameters, gives a universal way to obtain many novel fractals including the classic ones. Although this arc-fractal system shares similar features with the well-known Lindenmayer system, such as the same set of invariant points and the ability to tile the space, they do have different properties. One of which is the generation of pseudo-random number, which is not available in the Lindenmayer system. Furthermore, by assuming that coastline formation is based purely on the processes of erosion and deposition, the arc-fractal system can also serve as a dynamical model of coastal morphology, with each level of its construction corresponds to the time evolution of the shape of the coastal features. Remarkably, our results indicate that the arc-fractal system can provide an explanation on the origin of fractality in real coastline.

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7

Skydan, Oleg, Olga Nykolyuk, Oleksandr Chaikin, and Vasyl Shukalovych. "Concept of fractal organization of organic business systems." Agricultural and Resource Economics: International Scientific E-Journal 7, no. 2 (June 20, 2021): 59–76. http://dx.doi.org/10.51599/are.2021.07.02.04.

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Purpose. The purpose of the article is determining the possibilities of fractal approach, as the one that enables not only flexibility and viability, but also, management efficiency improvement, new competencies of the company formation, self-renewal ability formation and conflicts of interest between structural subdivisions in complex vertically integrated structures elimination, to the organization of implementation of organic business entities. Methodology / approach. The methodological basis of the research is general scientific and specific methods of economic phenomena and processes cognition. Therefore, the following methods have been applied: logical generalization (in determining the properties and benefits of agricultural business systems of the fractal type); comparison (when the practice of functioning of properties of organic products is analyzed); abstract-logical (when features of the functioning of network structures in fractally organized business systems are designed); monographic (in the study of the recent concepts of the functioning of fractal organized business systems); graphic (for visual presentation of the cooperation network of vertically integrated structure members); heuristic (when formulating conclusions and generalizations, as well as when justifying the directions for future research of the business system). Results. The essence of fractal business organization and the properties of fractal type business systems have been identified which include heterarchy, structure complexity, self-organization, self-optimization, openness, as well as autonomy and elements. The fractally organized business systems benefits in agribusiness compared with agrarian business systems with a traditional structure and management system have been determined. The existence of objective prerequisites for organic farms fractalization has been substantiated, which is already inherent in some of fractally organized business systems properties. The properties and features of fractally organized business systems of network structures functioning have been defined. Originality / scientific novelty. For the first time the substantiation of fractal type business systems formation in agriculture is proved, organic production in particular (previously expediency of fractal type business systems was studied only for industrial enterprises use). In particular, potential subjects of fractalization in organic production are identified, which include complex diversified agricultural business systems; the properties and advantages of fractally organized organic farms are identified and formalized, that are defined for a single fractal as well as a business system in general; the network structure of fractally organized organic farms is substantiated, particularly the relationship structure, network interaction rules, properties and values of fractally organized business structures in organic farming. In addition, the identification and formalization of the factors that affect Ukrainian organic production development got further development. Practical value / implications. To ensure the fulfillment of obligations by all parties as well as maintaining the basic principles of fractal organization in the field of goal-setting the function of the institutional environment is proposed. As PE “Gallex-Agro” is the vivid example of interconnections network that corresponds to the features of fractal business systems design, vertically integrated structure of member’s interaction network is designed at its case.

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Bolotov, V. N., and Yu V. Tkach. "Fractal communication system." Technical Physics 53, no. 9 (September 2008): 1192–96. http://dx.doi.org/10.1134/s1063784208090107.

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9

Navascués, M. A. "Fractal Haar system." Nonlinear Analysis: Theory, Methods & Applications 74, no. 12 (August 2011): 4152–65. http://dx.doi.org/10.1016/j.na.2011.03.048.

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Shevchenko, Svitlana, Yulia Zhdanovа, Svitlana Spasiteleva, Olena Negodenko, Nataliia Mazur, and Kateryna Kravchuk. "MATHEMATICAL METHODS IN CYBER SECURITY: FRACTALS AND THEIR APPLICATIONS IN INFORMATION AND CYBER SECURITY." Cybersecurity: Education, Science, Technique, no. 5 (2019): 31–39. http://dx.doi.org/10.28925/2663-4023.2019.5.3139.

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The article deals with the application of modern mathematical apparatus in information and cyber security namely fractal analysis. The choice of fractal modeling for the protection of information in the process of its digital processing is grounded. Based on scientific sources, the basic definitions of the research are analyzed: fractal, its dimension and basic properties used in the process of information protection. The basic types of fractals (geometric, algebraic, statistical) are presented and the most famous of them are described. The historical perspective of the development of fractal theory is conducted. Different approaches to the application of fractal theory in information and cyber security have been reviewed. Among them are: the use of fractal analysis in encryption algorithms; development of a method of protecting documents with latent elements based on fractals; modeling the security system of each automated workplace network using a set of properties that can be represented as fractals. The considered approaches to the application of fractal analysis in information and cyber security can be used in the preparation of specialists in the process of research work or diploma work.

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PAVLOVITCH, BULAEV BORIS. "PHASE-PERIODIC STRUCTURES OF SELF-SIMILAR STAIRCASE FRACTALS." Fractals 08, no. 04 (December 2000): 323–35. http://dx.doi.org/10.1142/s0218348x00000378.

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The possibilities of investigating the self-similar staircase fractals in a discrete coordinate system are clearly very promising. Surprisingly, such fractals are found to have holographic properties. Some geometric shapes e.g. a circle or a quadrate, are produced by a well-defined boundary through generating the staircase fractals in 2D discrete space. The obtained "luminous boundaries" are remembered outer boundaries of any geometric form, regardless of the size. Actually, this process is similar to the photographing of an exterior form. Thus there is destruction of a natural phase-periodical structure of the fractal. The staircase fractals are subject to the same rules that apply in information theory. It is shown that the difference in sharpness of holographic shapes is a direct consequence of spectra informations of the forms. The Hausdorff-Besicovitch dimension for this fractal has been estimated to be 0.5. On this view, the structure of staircase fractals is that of a Cantor Set. As indicated in the picture, such fractal resembles a "goose-wing." Also, the self-similar staircase fractal has much in common with a set of straight lines in a discrete coordinate system.

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12

Gdawiec, Krzysztof, and Diana Domańska. "Partitioned iterated function systems with division and a fractal dependence graph in recognition of 2D shapes." International Journal of Applied Mathematics and Computer Science 21, no. 4 (December 1, 2011): 757–67. http://dx.doi.org/10.2478/v10006-011-0060-8.

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Partitioned iterated function systems with division and a fractal dependence graph in recognition of 2D shapesOne of the approaches in pattern recognition is the use of fractal geometry. The property of self-similarity of fractals has been used as a feature in several pattern recognition methods. All fractal recognition methods use global analysis of the shape. In this paper we present some drawbacks of these methods and propose fractal local analysis using partitioned iterated function systems with division. Moreover, we introduce a new fractal recognition method based on a dependence graph obtained from the partitioned iterated function system. The proposed method uses local analysis of the shape, which improves the recognition rate. The effectiveness of our method is shown on two test databases. The first one was created by the authors and the second one is the MPEG7 CE-Shape-1 PartB database. The obtained results show that the proposed methodology has led to a significant improvement in the recognition rate.

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13

BAK, PER, and MAYA PACZUSKI. "THE DYNAMICS OF FRACTALS." Fractals 03, no. 03 (September 1995): 415–29. http://dx.doi.org/10.1142/s0218348x95000345.

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Fractals are formed by avalanches, driving the system toward a critical state. This critical state is a fractal in d spatial plus one temporal dimension. Long range spatial and temporal properties are described by different cuts in this fractal attractor. We unify the origin of fractals, 1/f noise, Hurst exponents, Levy flights, and punctuated equilibria in terms of avalanche dynamics, and elucidate their relationships through analytical and numerical studies of simple models.

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14

Perham, Arnold E., and Faustine L. Perham. "The Power of L-systems in Fractal Construction and Theory." Mathematics Teacher 98, no. 7 (March 2005): 459–67. http://dx.doi.org/10.5951/mt.98.7.0459.

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In searching the Web for an interactive applet that would help high school geometry students construct line fractals, we discovered a unique method called L–systems. Applets that integrate L–system theory with Turtle Geometry provide students with an easy–to–use graphics tool capable of creating a wide variety of fractals. In our experience, when students construct their own fractal drawings, rather than using prepared ones, they exhibit increased interest, insight, and enthusiasm. Mandelbrot and Frame (2002) make a strong case for fractals in the classroom, noting that the study of fractals heightens interest for non–mathematics/science–bound students.

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15

JAMPOUR, MAHDI, MAHDI YAGHOOBI, MARYAM ASHOURZADEH, and ADEL SOLEIMANI. "A NEW FAST TECHNIQUE FOR FINGERPRINT IDENTIFICATION WITH FRACTAL AND CHAOS GAME THEORY." Fractals 18, no. 03 (September 2010): 293–300. http://dx.doi.org/10.1142/s0218348x10005020.

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Fingerprints are one of the simplest and most reliable human biometric features for identification. Geometry of the fingerprint is fractal and we can classify a fingerprint database with fractal dimension, but one can't identify a fingerprint with fractal dimension uniquely. In this paper we present a new approach for identifying fingerprint uniquely; for this purpose a new fractal is initially made from a fingerprint by using Fractal theory and Chaos Game theory. While making the new fractal, five parameters that can be used in identification process can be achieved. Finally a fractal is made for each fingerprint, and then by analyzing the new fractal and parameters obtained by Chaos Game, fingerprint identification can be performed. We called this method Fingerprint Fractal Identification System (FFIS). The presented method besides having features of fractals such as stability against turning, magnifying, deleting a part of image, etc. also has a desirable speed.

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LANDINI, GABRIEL, and JOHN W. RIPPIN. "FRACTAL FRAGMENTATION IN REPLICATIVE SYSTEMS." Fractals 01, no. 02 (June 1993): 239–46. http://dx.doi.org/10.1142/s0218348x93000241.

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This paper describes a cell growth model formed by two cell types in which the cells are capable of displacing adjacent populations. Evolution of the model gives rise to patches that are fractally distributed (fractal fragmentation). The fragmentation of the system is not highly sensitive to the relative proportions of the two cell types, and it reveals new insights into fractal pattern formation. It is suggested that the fractal fragmentation is the natural outcome of multiple small perturbations in spatial rearrangement of the cells during multiplication. In addition, the model could prove useful in explaining both the development and spread of clones in a population of cells, and pattern formation in mosaic animal organs, in neither of which active movement of cells is implicit.

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Wolff, Rodney C. L., K. D. Lee, and Y. Cohen. "Fractal Attraction: A Fractal Design System for the Macintosh." Journal of the Royal Statistical Society. Series A (Statistics in Society) 156, no. 3 (1993): 508. http://dx.doi.org/10.2307/2983083.

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Ramos, J. l. "Fractal attraction: a fractal design system for the Macintosh." Applied Mathematical Modelling 17, no. 1 (January 1993): 52. http://dx.doi.org/10.1016/0307-904x(93)90130-9.

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You, Fu Cheng, and Yu Jie Chen. "The Development of E-Learning System for Fractal Big Tree Graphics Based on Iterated Function System." Applied Mechanics and Materials 143-144 (December 2011): 233–39. http://dx.doi.org/10.4028/www.scientific.net/amm.143-144.233.

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The traditional method to introduce the design of big tree fractal graphics is to give fixed graphics in black color. In this paper, a new learning environment of triangle fractal graphics based on E-Learning is proposed, development of which is also introduced in detail. In this new learning environment, students can repeat drawing the colorful big tree fractal graphics with single color or random color, which may arouse the students' interests and attract their attentions. By the learning environment, the fractal limitation or fixed point can be seen, and two different kinds of big tree can be got when the IFS code is changed, which will do benefit to students' learning concepts of fractal graphics. By this new learning environment, it is very easy for students to grasp the programming procedure of fractal graphics, and understand the generation procedure and structure of triangle fractal graphics.

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Saeed, Mohammad. "Fractals Analysis of Cardiac Arrhythmias." Scientific World JOURNAL 5 (2005): 691–701. http://dx.doi.org/10.1100/tsw.2005.81.

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Heart rhythms are generated by complex self-regulating systems governed by the laws of chaos. Consequently, heart rhythms have fractal organization, characterized by self-similar dynamics with long-range order operating over multiple time scales. This allows for the self-organization and adaptability of heart rhythms under stress. Breakdown of this fractal organization into excessive order or uncorrelated randomness leads to a less-adaptable system, characteristic of aging and disease. With the tools of nonlinear dynamics, this fractal breakdown can be quantified with potential applications to diagnostic and prognostic clinical assessment. In this paper, I review the methodologies for fractal analysis of cardiac rhythms and the current literature on their applications in the clinical context. A brief overview of the basic mathematics of fractals is also included. Furthermore, I illustrate the usefulness of these powerful tools to clinical medicine by describing a novel noninvasive technique to monitor drug therapy in atrial fibrillation.

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Abed, Salwa Salman, and Anaam Neamah Faraj. "Fixed Point Theorems and Iterative Function System in G-Metric Spaces." JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences 27, no. 2 (April 1, 2019): 329–40. http://dx.doi.org/10.29196/jubpas.v27i2.2228.

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Iterated function space is a method to construct fractals and the results are self-similar. In this paper, we introduce the Hutchinson Barnsley operator (shortly, operator) on a metric space and employ its theory to construct a fractal set as its unique fixed point by using Ciric type generalized -contraction in complete metric space. In addition, some concepts are illustrated by numerical examples.

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Makri, P. K., G. Romanos, T. Steriotis, N. K. Kanellopoulos, and A. Ch Mitropoulos. "Diffusion in a Fractal System." Journal of Colloid and Interface Science 206, no. 2 (October 1998): 605–6. http://dx.doi.org/10.1006/jcis.1998.5740.

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Ames, W. F., and C. Brezinski. "The desktop fractal design system." Mathematics and Computers in Simulation 35, no. 2 (April 1993): 190. http://dx.doi.org/10.1016/0378-4754(93)90025-p.

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NAIR, PRADEEP R., and MUHAMMAD A. ALAM. "KINETIC RESPONSE OF SURFACES DEFINED BY FINITE FRACTALS." Fractals 18, no. 04 (December 2010): 461–76. http://dx.doi.org/10.1142/s0218348x10005032.

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Historically, fractal analysis has been remarkably successful in describing wide ranging kinetic processes on (idealized) scale invariant objects in terms of elegantly simple universal scaling laws. However, as nanostructured materials find increasing applications in energy storage, energy conversion, healthcare, etc., one must reexamine the premise of traditional fractal scaling laws as it only applies to physically unrealistic infinite systems, while all natural/engineered systems are necessarily finite. In this article, we address the consequences of the 'finite-size' problem in the context of time dependent diffusion towards fractal surfaces via the novel technique of Cantor-transforms to (i) illustrate how finiteness modifies its classical scaling exponents; (ii) establish that for finite systems, the diffusion-limited reaction is decelerated below a critical dimension [Formula: see text] and accelerated above it; and (iii) to identify the crossover size-limits beyond which a finite system can be considered (practically) infinite and redefine the very notion of 'finiteness' of fractals in terms of its kinetic response. Our results have broad implications regarding dynamics of systems defined by the same fractal dimension, but differentiated by degree of scaling iteration or morphogenesis, e.g. variation in lung capacity between a child and adult.

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JÄRVENPÄÄ, ESA, MAARIT JÄRVENPÄÄ, ANTTI KÄENMÄKI, HENNA KOIVUSALO, ÖRJAN STENFLO, and VILLE SUOMALA. "Dimensions of random affine code tree fractals." Ergodic Theory and Dynamical Systems 34, no. 3 (January 30, 2013): 854–75. http://dx.doi.org/10.1017/etds.2012.168.

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AbstractWe study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.

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Wang, Ji Zhe, and Qing Jie Guan. "Numerical Simulation of Cell Growth Pattern and Determination of Fractal Dimension of Cell Cluster." Advanced Materials Research 690-693 (May 2013): 1229–33. http://dx.doi.org/10.4028/www.scientific.net/amr.690-693.1229.

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Life system behaves self-similar properties from microcosms to macrostructure. Based on the cell growth roles, the cell cluster growth process is simulated. The sandbox method and box counting are used for determining the fractal dimension of cell associated with the geometrical structure of growing deterministic fractals. The fractal dimension of cell shape is estimated according to the slope of line between the numbers of boxes and box size in double logarithm coordinates.

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Rao, Neeraj, Ankit Malik, Rahul Kumar, Shobhit Goel, and Dinesh Kumar V. "Novel star-shaped fractal antenna for multiband applications." International Journal of Microwave and Wireless Technologies 9, no. 2 (November 23, 2015): 419–25. http://dx.doi.org/10.1017/s1759078715001592.

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Fractals have unique properties such as self-similarity and space-filling. The use of fractal geometry in antenna design provides a good method for achieving the desired miniaturization, multi-band, and wideband properties. In this communication, novel fractal geometry is proposed based on which a multiband antenna is designed. The proposed antenna has fractal patches which are shaped as different iterations of an eight-pointed star. The multiband behavior is in the frequency range from 4.50 to 17.00 GHz. The proposed antenna is designed on a dielectric substrate Roggers RO4003 lossy with a dielectric constant of εr = 3.55. The antenna has applications in commercial and military communication system.

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A. A., Sathakathulla. "Per-tiling, rep-tiling and Penrose tiling: a notion to edge cordial and cordial labeling." International Journal of Applied Mathematical Research 4, no. 2 (March 31, 2015): 308. http://dx.doi.org/10.14419/ijamr.v4i2.4413.

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<p>A fractal is a complex geometric figure that continues to display self-similarity when viewed on all scales. A simple, yet unifying method is provided for the construction of tiling by tiles obtained from the attractor of an iterated function system (IFS). This tiling can be used to extend a fractal transformation on the entire space upon which the IFS acts. There are many in this family of tiling fractals curves but for my study, I have considered each one from the above family of tiling fractals. These fractals have been considered as a graph and the same has been viewed under the scope of cordial and edge cordial labeling to apply this concept for further study.</p>

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Dai, Shengqiu, Kehui Sun, Shaobo He, and Wei Ai. "Complex Chaotic Attractor via Fractal Transformation." Entropy 21, no. 11 (November 14, 2019): 1115. http://dx.doi.org/10.3390/e21111115.

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Based on simplified Lorenz multiwing and Chua multiscroll chaotic systems, a rotation compound chaotic system is presented via transformation. Based on a binary fractal algorithm, a new ternary fractal algorithm is proposed. In the ternary fractal algorithm, the number of input sequences is extended from 2 to 3, which means the chaotic attractor with fractal transformation can be presented in the three-dimensional space. Taking Lorenz system, rotation Lorenz system and compound chaotic system as the seed chaotic systems, the dynamics of the complex chaotic attractors with fractal transformation are analyzed by means of bifurcation diagram, complexity and power spectrum, and the results show that the chaotic sequences with fractal transformation have higher complexity. As the experimental verification, one kind of complex chaotic attractors is implemented by DSP, and the result is consistent with that of the simulation, which verifies the feasibility of digital circuit implement.

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BARNSLEY, MICHAEL, JOHN HUTCHINSON, and ÖRJAN STENFLO. "A FRACTAL VALUED RANDOM ITERATION ALGORITHM AND FRACTAL HIERARCHY." Fractals 13, no. 02 (June 2005): 111–46. http://dx.doi.org/10.1142/s0218348x05002799.

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We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability": at each magnification level any V-variable fractals has at most V key "forms" or "shapes". V-variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IFS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V-variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example ℝ2, and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm, which is highly efficient in terms of memory usage. The algorithm is illustrated by some computed examples. Some variants, special cases, generalizations of the framework, and potential applications are mentioned.

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Abdel-Karim, Benjamin M. "Beautiful Fractals as a Crystal Ball for Financial Markets? - Investment Decision Support System Based on Image Recognition Using Artificial Intelligence." Journal of Prediction Markets 14, no. 2 (December 11, 2020): 27–44. http://dx.doi.org/10.5750/jpm.v14i2.1804.

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The work by Mandelbrot develops a basic understanding of fractals and the artwork of Jackson Pollok to reveal the beauty fractal geometry. The pattern of recurring structures is also reflected in share prices. Mandelbrot himself speaks of the fractal heart of the financial markets. Previous research has shown the potential of image recognition. This paper presents the possibility of using the structure recognition capability of modern machine learning methods to make forecasts based on fractal course information. We generate training data from real and simulated data. These data are represented in images to train a special artificial neural network. Subsequently, real data are presented to the network for use in predicting. The results show that the forecast of time series based on stock price illustration, compared to a benchmark, delivers promising results. This paper makes two essential contributions to research. From a theoretical point of view, fractal geometry shows that it can serve as a means of legitimation for technical analysis. From a practical point of view, highly developed methods from the field of machine learning are able to recognize patterns in data through appropriate data transformation, and that models such as random walk have an informational content that can be used to train machine learning models.

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ZHANG, FANHUI, and YONGXIANG ZHANG. "WADA FRACTAL BASINS OF ATTRACTION IN A ZERO-STIFFNESS VIBRATION ISOLATION SYSTEM." Fractals 28, no. 02 (March 2020): 2050023. http://dx.doi.org/10.1142/s0218348x20500231.

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A fractal basin boundary is a Wada fractal basin boundary if it contains at least three different basins. The corresponding basin is called a Wada fractal basin. Previous results show that some oscillators possess Wada fractal basins with common basin boundaries. Here we find that a nonlinear vibration isolation system can possess abundant coexisting basins and every basin is a Wada fractal basin. These Wada fractal basin boundaries separate different basins in the different regions. A proper classification of these Wada fractal basins is provided according to the order of saddles and Wada numbers. Basin organization is systematic and all basins spiral outward toward the infinity. The entangled basin boundaries are described by the manifolds of saddles and basins (tongues) accumulation.

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WADA, RYOICHI, and KAZUTOSHI GOHARA. "FRACTALS AND CLOSURES OF LINEAR DYNAMICAL SYSTEMS EXCITED STOCHASTICALLY BY TEMPORAL INPUTS." International Journal of Bifurcation and Chaos 11, no. 03 (March 2001): 755–79. http://dx.doi.org/10.1142/s0218127401002602.

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Fractals and closures of two-dimensional linear dynamical systems excited by temporal inputs are investigated. The continuous dynamics defined by the set of vector fields in the cylindrical phase space is reduced to the discrete dynamics defined by the set of iterated functions on the Poincaré section. When all iterated functions are contractions, it has already been shown theoretically that a trajectory in the cylindrical phase space converges into an attractive invariant set with a fractal-like structure. Calculating analytically the Lipschitz constants of iterated functions, we show that, under some conditions, noncontractions often appear. However, we numerically show that, even for noncontractions, an attractive invariant set with a fractal-like structure exists. By introducing the interpolating system, we can also show that the set of trajectories in the cylindrical phase space is enclosed by the tube structure whose initial set is the closure of the fractal set on the Poincaré section.

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Wahjono, Tri Djoko, Syaeful Karim, and Bayu Riyadi. "PERANCANGAN PERANGKAT LUNAK GENERATOR GAMBAR DAN MUSIK FRAKTAL DENGAN METODE ITERATED FUNCTION SYSTEM." CommIT (Communication and Information Technology) Journal 1, no. 2 (October 31, 2007): 140. http://dx.doi.org/10.21512/commit.v1i2.475.

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Article presents analysis and analyze a software that utilize Geometry Fractal, especially Iterated FunctionSystem Fractal, as art. Research method that has been used in this research is by library study and by laboratoriumstudy to test the performance of the software. Result of the research has shown that converted music by GeometryFractal has various results, which depend on the parameters used in it and type of Geometry Fractal image produced.It can be said that usage of fractal in high iteration can produce clear image fractal and complicated music fractal.

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Umami, Riza, Kosala Dwija Purnomo, and Firdaus Ubaidillah. "KAJIAN FRAKTAL i-FIBONACCI WORD GENERALISASI GANJIL DENGAN MENGGUNAKAN L-SYSTEM." Majalah Ilmiah Matematika dan Statistika 19, no. 1 (March 1, 2019): 1. http://dx.doi.org/10.19184/mims.v19i1.17258.

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The i-Fibonacci Words are words over {0,1}. The i-Fibonacci Word can be associated with a fractal curve by using odd-even drawing rule and L-System methods, then also known as an i-Fibonacci Word fractal. L-System is one of methods that is used to create objects with repetitive self-similiarity. Framework of L-System consists of axiom and rules. L-System is a parallel rewriting system with existing rules. The purpose of this research is to look for the LSystem rules of i-Fibonacci Word special for odd i, then look how its characteristics. The LSystem rules for i-Fibonacci Word odd i are divided into two types, the rules for i=1 and the others odd i. The characteristic of i-Fibonacci Word fractal is the more generation and i value of fractal, then the more segments and archs of fractal curve. Next, the words of i-Fibonacci Word fractal segments number is a subwords of the i-Fibonacci Word digit numbers. It is also known that the fractal curve will be stretched as the decreased angle. Keywords: Fractal, i-Fibonacci Word, L-System

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Shang, Xiaoji, Jianguo Wang, and Xiaojun Yang. "Fractal analysis for heat extraction in geothermal system." Thermal Science 21, suppl. 1 (2017): 25–31. http://dx.doi.org/10.2298/tsci17s1025s.

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Heat conduction and convection play a key role in geothermal development. These two processes are coupled and influenced by fluid seepage in hot porous rock. A number of integer dimension thermal fluid models have been proposed to describe this coupling mechanism. However, fluid flow, heat conduction and convection in porous rock are usually non-linear, tortuous and fractal, thus the integer dimension thermal fluid flow models can not well describe these phenomena. In this study, a fractal thermal fluid coupling model is proposed to describe the heat conduction and flow behaviors in fractal hot porous rock in terms of local fractional time and space derivatives. This coupling equation is analytically solved through the fractal travelling wave transformation method. Analytical solutions of Darcy?s velocity, fluid temperature with fractal time and space are obtained. The solutions show that the introduction of fractional parameters is essential to describe the mechanism of heat conduction and convection.

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ZHANG, YONGXIANG. "CHARACTERIZING FRACTAL BASIN BOUNDARIES FOR PLANAR SWITCHED SYSTEMS." Fractals 25, no. 03 (May 18, 2017): 1750031. http://dx.doi.org/10.1142/s0218348x17500311.

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This paper is to introduce some analytical tools to characterize the properties of fractal basin boundaries for planar switched systems (with time-dependent switching). The characterizing methods are based on the view point of limit sets and prime ends. By constructing the auxiliary dynamical system, the fractal basin boundaries of planar switched systems can be proved if every diverging path in the basin of associated auxiliary system has the entire basin boundary as its limit set. Fractal property is also verified if every prime end that is defined in the basin of associated auxiliary system is a prime end of type 3 and all other prime ends are of type 1. Bifurcations of fractal basin boundary are investigated by analyzing what types of prime ends in the basin are involved. The fractal basin boundary of switched system is also described by the indecomposable continuum.

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You, Fu Cheng, and Yu Jie Chen. "The Programming of Snow Flower Fractal Graphics Based on Iterated Function System." Applied Mechanics and Materials 143-144 (December 2011): 765–69. http://dx.doi.org/10.4028/www.scientific.net/amm.143-144.765.

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The traditional method to introduce the design of snow flower fractal graphics is to give fixed graphics in black color. In this paper, a new learning environment of triangle fractal graphics based on E-Learning is proposed, development of which is also introduced in detail. In this new learning environment, students can repeat drawing the colorful snow flower fractal graphics with single color or random color, which may arouse the students' interests and attract their attentions. By the learning environment, the fractal limitation or fixed point can be seen, and two different kinds of snow flower can be got when the IFS code is changed, which will do benefit to students' learning concepts of fractal graphics. By this new learning environment, it is very easy for students to grasp the programming procedure of fractal graphics, and understand the generation procedure and structure of triangle fractal graphics.

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Doungmo Goufo, Emile Franc. "The Proto-Lorenz System in Its Chaotic Fractional and Fractal Structure." International Journal of Bifurcation and Chaos 30, no. 12 (September 30, 2020): 2050180. http://dx.doi.org/10.1142/s0218127420501801.

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It is not common in applied sciences to realize simulations which depict fractal representation in attractors’ dynamics, the reason being a combination of many factors including the nature of the phenomenon that is described and the type of differential operator used in the system. In this work, we use the fractal-fractional derivative with a fractional order to analyze the modified proto-Lorenz system that is usually characterized by chaotic attractors with many scrolls. The fractal-fractional operator used in this paper is a combination of fractal process and fractional differentiation, which is a relatively new concept with most of the properties and features still to be known. We start by summarizing the basic notions related to the fractal-fractional operator. After that, we enumerate the main points related to the establishment of proto-Lorenz system’s equations, leading to the [Formula: see text]th cover of the proto-Lorenz system that contains [Formula: see text] scrolls ([Formula: see text]). The triple and quadric cover of the resulting fractal and fractional proto-Lorenz system are solved using the Haar wavelet methods and numerical simulations are performed. Due to the impact of the fractal-fractional operator, the system is able to maintain its chaotic state of attractor with many scrolls. Additionally, such attractor can self-replicate in a fractal process as the derivative order changes. This result reveals another great feature of the fractal-fractional derivative with fractional order.

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Diebel, Kenneth E., and Peter P. Feret. "Using Fractal Geometry to Quantify Loblolly Pine Seedling Root System Architecture." Southern Journal of Applied Forestry 17, no. 3 (August 1, 1993): 130–34. http://dx.doi.org/10.1093/sjaf/17.3.130.

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Abstract Fractal geometry is a simple method of quantifying and describing complex shapes often found in nature (Mandelbrot 1983). The spatial arrangement of pine seedling roots is complex and not easily quantified. In this paper we report on a method for quantifying seedling roots based on concepts of fractal geometry. Ten 1+0 bareroot seedlings of each of three grades of loblolly pine (Pinus taeda L.) were obtained from two Virginia Department of Forestry nurseries. The fractal dimension (D) was estimated for seedling roots using a computer-based box-count method. The results show that pine seedling roots have a fractal dimension, and the fractal dimension is highly correlated with root morphological traits. We propose that fractal geometry may be a new and efficient method to describe tree seedling root morphology. South. J. Appl. For. 17(3):130-134.

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GUARNIERI, P., S. VINCIGUERRA, and OSSERVATORIO VESUVIANO. "FRACTAL PROPERTIES OF A NEOGENE FAULT SYSTEM IN NORTHWESTERN SICILY, ITALY." Fractals 12, no. 01 (March 2004): 41–48. http://dx.doi.org/10.1142/s0218348x04002392.

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A Neogene transcurrent fault system affecting the Apenninic-Maghrebian thrust belt, located in northwestern Sicily, has been investigated using fractal analysis. The present-day structural setting of the sector between the Palermo and Madonie Mountains is the result of the superimposition, in space and time, of two distinct deformational events, a Miocene southeast-verging thrusting followed by Messinian to Pliocene strike-slip faulting, which cuts obliquely through the compressive fronts. The spatial distribution properties of the fault array were investigated by means of a fractal analysis. Fractal dimension was computed by adopting the correlation integral method. Fractal ranges have been evidenced between 350 and 3000 m. The fractal dimension obtained for the whole array is D=1.66. The scaling spatial distribution property of the fault array has been analyzed by calculating the fractal dimension with a moving window.

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Volchyuk, V. N. "Fractal analysis of the point system." Bulletin of Prydniprovs’ka State Academy of Civil Engineering and Architecture, no. 5 (October 23, 2018): 47–53. http://dx.doi.org/10.30838/j.bpsacea.2312.271118.47.365.

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Fitter, A. H., and T. R. Stickland. "Fractal Characterization of Root System Architecture." Functional Ecology 6, no. 6 (1992): 632. http://dx.doi.org/10.2307/2389956.

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KALDA, JAAN. "FRACTAL MODEL OF BLOOD VESSEL SYSTEM." Fractals 01, no. 02 (June 1993): 191–97. http://dx.doi.org/10.1142/s0218348x93000204.

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A possible way of modeling of self-similar biological tree-like structures is proposed. Special attention is paid to the blood-vessel system, with elaboration on a model with certain spatial arrangement of the vessels and reasonable dependence of the blood pressure on the vessels diameter, such that the organism has a homogeneous oxygen supply. A model of the lung is also presented, which reproduces a qualitatively right dependence of the average diameter of the tubes on their generation number. The model of the blood-vessel system is based on suitably generalized Scheidegger’s model of rivers. The statistical characteristics of the modified Scheidegger’s model are established.

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Yamakita, Masaki, and Toshiyasu Yonemura. "Yet Another Fractal in Pendulum System." IFAC Proceedings Volumes 37, no. 14 (September 2004): 343–46. http://dx.doi.org/10.1016/s1474-6670(17)31127-8.

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Abd-El-Hafiz, Salwa Kamal, Ahmed G. Radwan, Mohamed L. Barakat, and Sherif H. Abdel Haleem. "A fractal-based image encryption system." IET Image Processing 8, no. 12 (December 1, 2014): 742–52. http://dx.doi.org/10.1049/iet-ipr.2013.0570.

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Crişan, Daniela, Catalina Popescu-Mina, and Vasilica Voinea. "FRACTAL TECHNIQUES FOR CLASSIFYING BIOLOGIC SYSTEM." IFAC Proceedings Volumes 40, no. 18 (September 2007): 235–40. http://dx.doi.org/10.3182/20070927-4-ro-3905.00040.

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Lin, Zhifang, and Masaki Goda. "Mobility edge in a fractal system." Physics Letters A 193, no. 3 (October 1994): 305–10. http://dx.doi.org/10.1016/0375-9601(94)90602-5.

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Sun, H. H., and A. Charef. "Fractal system — A time domain approach." Annals of Biomedical Engineering 18, no. 6 (November 1990): 597–621. http://dx.doi.org/10.1007/bf02368450.

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Pérez, Yudier Peña, Ricardo Abreu Blaya, Paul Bosch, and Juan Bory Reyes. "Dirichlet Type Problem for 2D Quaternionic Time-Harmonic Maxwell System in Fractal Domains." Advances in Mathematical Physics 2020 (January 31, 2020): 1–8. http://dx.doi.org/10.1155/2020/4735357.

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We investigate an electromagnetic Dirichlet type problem for the 2D quaternionic time-harmonic Maxwell system over a great generality of fractal closed type curves, which bound Jordan domains in R2. The study deals with a novel approach of h-summability condition for the curves, which would be extremely irregular and deserve to be considered fractals. Our technique of proofs is based on the intimate relations between solutions of time-harmonic Maxwell system and those of the Dirac equation through some nonlinear equations, when both cases are reformulated in quaternionic forms.

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