Search results for: Fractal Geometry

Authors: Adnan H. M. Al-Helali, Hamza A. Ali

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The growth of censorship and pervasive monitoring on the Internet, Steganography arises as a new means of achieving secret communication. Steganography is the art and science of embedding information within electronic media used by common applications and systems. Generally, hiding information of multimedia within images will change some of their properties that may introduce few degradation or unusual characteristics. This paper presents a new image steganography approach for hiding information of multimedia (images, text, and audio) using generated Mandelbrot Fractal image as a cover. The proposed technique has been extensively tested with different images. The results show that the method is a very secure means of hiding and retrieving steganographic information. Experimental results demonstrate that an effective improvement in the values of the Peak Signal to Noise Ratio (PSNR), Mean Square Error (MSE), Normalized Cross Correlation (NCC), and Image Fidelity (IF) over the pervious techniques.

Keywords: fractal image, information hiding, Mandelbrot set fractal, steganography

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Authors: Harsha Gopalakrishnan, Srijanani Anurag Prasad

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Quadrilateral labyrinth fractals are a new type of fractals that are introduced in this paper. They belong to a unique class of fractals on any plane quadrilateral. The previously researched labyrinth fractals on the unit square and triangle inspire this form of fractal. This work describes how to construct a quadrilateral labyrinth fractal and looks at the circumstances in which it can be understood as the attractor of an iterated function system. Furthermore, some of its topological properties and the Hausdorff and box-counting dimensions of the quadrilateral labyrinth fractals are studied.

Keywords: fractals, labyrinth fractals, dendrites, iterated function system, Haus-Dorff dimension, box-counting dimension, non-self similar, non-self affine, connected, path connected

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Authors: C. Elavarasi, T. Shanmuganantham

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A coplanar waveguide (CPW) feed is presented, and comprising a split ring resonator (SRR) loaded fractal with water lily shape is used for multi band applications. The impedance matching of the antenna is determined by the number of Koch curve fractal unit cells. The antenna is designed on a FR4 substrate with a permittivity of ε_{r =} 4.4 and size of 14 x 16 x 1.6 mm³ to generate multi resonant mode at 3.8 GHz covering S band, 8.68 GHz at X band, 13.96 GHz at Ku band, and 19.74 GHz at K band with reflection coefficient better than -10 dB. Simulation results show that the antenna exhibits the desired voltage standing wave ratio (VSWR) level and radiation patterns across the wide frequency range. The fundamental parameters of the antenna such as return loss, VSWR, good radiation pattern with reasonable gain across the operating bands are obtained.

Keywords: fractal, metamaterial, split ring resonator, waterlily shape

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Authors: Fatima Achouri, Kaddour Chouicha, Abdelwahab Khatir

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It is clear that concrete of quality must be made with selected materials chosen in optimum proportions that remain after implementation, a minimum of voids in the material produced. The different methods of formulations what we use, are based for the most part on a granular curve which describes an ‘optimal granularity’. Many authors have engaged in fundamental research on granular arrangements. A comparison of mathematical models reproducing these granular arrangements with experimental measurements of compactness have to verify that the minimum porosity P according to the following extent granular exactly a power law. So the best compactness in the finite medium are obtained with power laws, such as Furnas, Fuller or Talbot, each preferring a particular setting between 0.20 and 0.50. These considerations converge on the assumption that the optimal granularity Caquot approximates by a power law. By analogy, it can then be analyzed as a granular structure of fractal-type since the properties that characterize the internal similarity fractal objects are reflected also by a power law. Optimized mixtures may be described as a series of installments falling granular stuff to better the tank on a regular hierarchical distribution which would give at different scales, by cascading effects, the same structure to the mix. Likely this model may be appropriate for the entire extent of the size distribution of the components, since the cement particles (and silica fume) correctly deflocculated, micrometric dimensions, to chippings sometimes several tens of millimeters. As part of this research, the aim is to give an illustration of the application of fractal analysis to characterize the granular concrete mixtures optimized for a so-called fractal dimension where different concretes were studying that we proved a fractal structure of their granular mixtures regardless of the method of formulation or the type of concrete.

Keywords: concrete formulation, fractal character, granular packing, method of formulation

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Authors: Abraham Terán Salcedo, Didier Samayoa Ochoa

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This work presents a numerical implementation of a method for estimating the box-counting dimension of self-avoiding curves on a planar space, fractal objects captured on digital images; this method is named fractioning estimator. Classical methods of digital image processing, such as noise filtering, contrast manipulation, and thresholding, among others, are used in order to obtain binary images that are suitable for performing the necessary computations of the fractioning estimator. A user interface is developed for performing the image processing operations and testing the fractioning estimator on different captured images of real-life fractal objects. To analyze the results, the estimations obtained through the fractioning estimator are compared to the results obtained through other methods that are already implemented on different available software for computing and estimating the box-counting dimension.

Keywords: box-counting, digital image processing, fractal dimension, numerical method

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Authors: Darsh N. Patel, Eric Olson

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Fractals can be found everywhere, whether it be the shape of a leaf or a system of blood vessels. Fractals are used to help study and understand different physical and mathematical processes; however, their artistic nature is also beautiful to simply explore. This project explores fractals generated by a cubically convergent extension to Newton's method. With this iteration as a starting point, a complex plane spanning from -2 to 2 is created with a color wheel mapped onto it. Next, the polynomial whose roots the fractal will generate from is established. From the Fundamental Theorem of Algebra, it is known that any polynomial has as many roots (counted by multiplicity) as its degree. When generating the fractals, each root will receive its own color. The complex plane can then be colored to indicate the basins of attraction that converge to each root. From a computational point of view, this project’s code identifies which points converge to which roots and then obtains fractal images. A zoom path into the fractal was implemented to easily visualize the self-similar structure. This path was obtained by selecting keyframes at different magnifications through which a path is then interpolated. Using parallel processing, many images were generated and condensed into a video. This project illustrates how practical techniques used for scientific visualization can also have an artistic side.

Keywords: fractals, cubic method, Julia programming language, basin of attraction

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Authors: Yifan Dong, Xincheng Pu

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Rural settlements originate from the accumulation of residential building elements, and their agglomeration forms the settlement pattern and defines the relationship between the settlement and the inside and outside. The settlement boundary is an important part of the settlement pattern. Compared with the simplification of the urban settlement boundary, the settlement of the country is more complex, fuzzy and uncertain, and then presents a rich and diverse boundary morphological phenomenon. In this paper, China traditional rural settlements plane boundary as the research object, using fractal theory and fractal dimension method, quantitative analysis of planar shape boundary settlement, and expounds the research for the architectural design, ancient architecture protection and renewal and development and the significance of the protection of settlements.

Keywords: rural settlement, border, fractal, quantification

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Authors: Eman A. Al-Hilo, Hawraa H. Al-Waelly

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In this work, fractal compression (FIC) technique is introduced based on using moment features to block indexing the zero-mean range-domain blocks. The moment features have been used to speed up the IFS-matching stage. Its moments ratio descriptor is used to filter the domain blocks and keep only the blocks that are suitable to be IFS matched with tested range block. The results of tests conducted on Lena picture and Cat picture (256 pixels, resolution 24 bits/pixel) image showed a minimum encoding time (0.89 sec for Lena image and 0.78 of Cat image) with appropriate PSNR (30.01dB for Lena image and 29.8 of Cat image). The reduction in ET is about 12% for Lena and 67% for Cat image.

Keywords: fractal gray level image, fractal compression technique, iterated function system, moments feature, zero-mean range-domain block

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Authors: Narek Margaryan, Zhozef Panosyan

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Due to their unique properties, carbon nanofilms have become the object of general attention and intensive research. In this case it plays a very important role to study surface properties of these films. It is also important to study processes of forming of this films, which is accompanied by a process of self-organization at the nano and micro levels. For more detailed investigation, we examined diamond-like carbon (DLC) layers deposited by chemical vapor deposition (CVD) method on Ge substrate and hydro-generated grapheme layers obtained on surface of colloidal solution using grouping method. In this report surface transformation of these CVD nanolayers is studied by atomic force microscopy (AFM) upon deposition time. Also, it can be successfully used to study surface properties of self-assembled grapheme layers. In turn, it is possible to sketch out their boundary line, which enables one to draw an idea of peculiarities of formation of these layers. Images obtained by AFM are investigated as a mathematical set of numbers and fractal and roughness analysis were done. Fractal dimension, Regne’s fractal coefficient, histogram, Fast Fourier transformation, etc. were obtained. The dependence of fractal parameters on the deposition duration for CVD films and on temperature of solution tribolayers was revealed. As an important surface parameter for our carbon films, surface energy was calculated as function of Regne’s fractal coefficient. Surface potential was also measured with Kelvin probe method using semi-contacting AFM. The dependence of surface potential on the deposition duration for CVD films and on temperature of solution for hydro-generated graphene was found as well. Results obtained by fractal analysis method was related with purly esperimental results for number of samples.

Keywords: nanostructured films, self-assembled grapheme, diamond-like carbon, surface potential, Kelvin probe method, fractal analysis

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Authors: Phyo Wai Aung, Sysoev Oleg Evgenevich, Boris Necolavet Maryin

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The article deals with theoretical problems of correlation of processes of microstructure changes of structural materials under cyclic loading and acoustic emission. The ways of the evolution of a microstructure under the influence of cyclic loading are shown depending on the structure of the initial crystal structure of the material. The spectra of the frequency characteristics of acoustic emission signals are experimentally obtained when testing titanium samples for cyclic loads. Changes in the fractal dimension of the acoustic emission signals in the selected frequency bands during the evolution of the microstructure of structural materials from the action of cyclic loads, as well as in the destruction of samples, are studied. The experimental samples were made of VT-20 structural material widely used in aircraft and rocket engineering. The article shows the striving of structural materials for synergistic stability and reduction of the fractal dimension of acoustic emission signals, in accordance with the degradation of the microstructure, which occurs as a result of fatigue processes from the action of low cycle loads. As a result of the research, the frequency range of acoustic emission signals of 100-270 kHz is determined, in which the fractal dimension of the signals, it is possible to most reliably predict the durability of structural materials.

Keywords: cyclic loadings, material structure changing, acoustic emission, fractal dimension

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Authors: Youness Laaroussi, Zine Elabidine Guennoun, Amine Amar

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Modeling and forecasting dynamics of rainfall occurrences constitute one of the major topics, which have been largely treated by statisticians, hydrologists, climatologists and many other groups of scientists. In the same issue, we propose in the present paper a new hybrid method, which combines Extreme Values and fractal theories. We illustrate the use of our methodology for transformed Emberger Index series, constructed basing on data recorded in Oujda (Morocco). The index is treated at first by Peaks Over Threshold (POT) approach, to identify excess observations over an optimal threshold u. In the second step, we consider the resulting excess as a fractal object included in one dimensional space of time. We identify fractal dimension by the box counting. We discuss the prospect descriptions of rainfall data sets under Generalized Pareto Distribution, assured by Extreme Values Theory (EVT). We show that, despite of the appropriateness of return periods given by POT approach, the introduction of fractal dimension provides accurate interpretation results, which can ameliorate apprehension of rainfall occurrences.

Keywords: extreme values theory, fractals dimensions, peaks Over threshold, rainfall occurrences

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Authors: Ali Albadri

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The logistics equation has never been used or studied in scientific fields outside the field of ecology. It has never been used to understand the behavior of a dynamic system of mechanical machines, like an escalator. We have studied the compatibility of the logistic map against real measurements from an escalator. This study has proven that there is good compatibility between the logistics equation and the experimental measurements. It has discovered the potential of a relationship between the fractal dimension and the non-linearity parameter, R, in the logistics equation. The fractal dimension increases as the R parameter (non-linear parameter) increases. It implies that the fractal dimension increases as the phase of the life span of the machine move from the steady/stable phase to the periodic double phase to a chaotic phase. The fractal dimension and the parameter R can be used as a tool to verify and check the health of machines. We have come up with a theory that there are three areas of behaviors, which they can be classified during the life span of a machine, a steady/stable stage, a periodic double stage, and a chaotic stage. The level of attention to the machine differs depending on the stage that the machine is in. The rate of faults in a machine increases as the machine moves through these three stages. During the double period and the chaotic stages, the number of faults starts to increase and become less predictable. The rate of predictability improves as our monitoring of the changes in the fractal dimension and the parameter R improves. The principles and foundations of our theory in this work have and will have a profound impact on the design of systems, on the way of operation of systems, and on the maintenance schedules of the systems. The systems can be mechanical, electrical, or electronic. The discussed methodology in this paper will give businesses the chance to be more careful at the design stage and planning for maintenance to control costs. The findings in this paper can be implied and used to correlate the three stages of a mechanical system to more in-depth mechanical parameters like wear and fatigue life.

Keywords: logistcs map, bifurcation map, fractal dimension, logistics equation

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Authors: Ali Albadri

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Step band in an escalator moves in a cyclic periodic pattern. Similarly, most if not all of the components and sub-assemblies in the escalator operate in the same way. If you mark up one step in the step band of an escalator and stand next to the escalator, on the incline, to watch the marked-up step when it passes by, you ask yourself, does the marked up step behaves exactly the same way during each revolution when it passes you by again and again? We can say that; there is some similarity in this example and the example when an astronomer watches planets in the sky, and he or she asks himself or herself, does each planet intersects the plan of observation in the same position for every pantry rotation? For a fact, we know for the answer to the second example is no, because scientist, astronomers, and mathematicians have proven that planets deviate from their paths to take new paths during their planetary moves, albeit with minimal change. But what about the answer to the question in the first example? considering that there is increase in the wear and tear of components with time in the step, in the step band, in the tracks and in many other places in the escalator. There is also the accumulation of fatigue in the components and sub-assemblies. This research is part of many studies which we are conducting to address the answer for the question in the first example. We have been using the fractal dimension as a quantities tool and the Poincare map as a qualitative tool. This study has shown that the fractal dimension value and the shape and distribution of the orbits in the Poincare map has significant correlation with the quality of the mechanical components and sub-assemblies in the escalator.

Keywords: fractal dimension, Poincare map, rugby ball orbit, worm orbit

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Authors: Khalid Salhi, El Miloud Jaara, Mohammed Talibi Alaoui

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In this paper, we present a neural approach for unsupervised natural color-texture image segmentation, which is based on both Kohonen maps and mathematical morphology, using a combination of the texture and the image color information of the image, namely, the fractal features based on fractal dimension are selected to present the information texture, and the color features presented in RGB color space. These features are then used to train the network Kohonen, which will be represented by the underlying probability density function, the segmentation of this map is made by morphological watershed transformation. The performance of our color-texture segmentation approach is compared first, to color-based methods or texture-based methods only, and then to k-means method.

Keywords: segmentation, color-texture, neural networks, fractal, watershed

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Authors: Reda Abdel Azim, Tariq Shehab

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The main objective of this study is to develop a subsurface fracture map of naturally fractured reservoirs by overcoming the limitations associated with different data sources in characterising fracture properties. Some of these limitations are overcome by employing a nested neuro-stochastic technique to establish inter-relationship between different data, as conventional well logs, borehole images (FMI), core description, seismic attributes, and etc. and then characterise fracture properties in terms of fracture density and fractal dimension for each data source. Fracture density is an important property of a system of fracture network as it is a measure of the cumulative area of all the fractures in a unit volume of a fracture network system and Fractal dimension is also used to characterize self-similar objects such as fractures. At the wellbore locations, fracture density and fractal dimension can only be estimated for limited sections where FMI data are available. Therefore, artificial intelligence technique is applied to approximate the quantities at locations along the wellbore, where the hard data is not available. It should be noted that Artificial intelligence techniques have proven their effectiveness in this domain of applications.

Keywords: naturally fractured reservoirs, artificial intelligence, fracture intensity, fractal dimension

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Authors: Ibrahim Alsendid, Rob Sturman, Benjamin Sharp

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In this paper, we consider a family of 2-dimensional nonlinear area-preserving transformations on the torus. A single parameter η varies between 0 and 1, taking the transformation from a hyperbolic toral automorphism to the “Cerbelli-Giona” map, a system known to exhibit multifractal properties. Here we study the multifractal properties of the family of maps. We apply a box-counting method by defining a grid of boxes Bi(δ), where i is the index and δ is the size of the boxes, to quantify the distribution of stable and unstable manifolds of the map. When the parameter is in the range 0.51< η <0.58 and 0.68< η <1 the map is ergodic; i.e., the unstable and stable manifolds eventually cover the whole torus, although not in a uniform distribution. For accurate numerical results, we require correspondingly accurate construction of the stable and unstable manifolds. Here we use the piecewise linearity of the map to achieve this, by computing the endpoints of line segments that define the global stable and unstable manifolds. This allows the generalized fractal dimension Dq, and spectrum of dimensions f(α), to be computed with accuracy. Finally, the intersection of the unstable and stable manifold of the map will be investigated and compared with the distribution of periodic points of the system.

Keywords: Discrete-time dynamical systems, Fractal geometry, Multifractal behaviour of the Perturbed map, Multifractal of Dynamical systems

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Authors: Abhilash Sahu, Amit Priyadarshi

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In this paper, we will discuss the existence of weak solutions of the Kirchhoff type boundary value problem on the Sierpinski gasket. Where S denotes the Sierpinski gasket in R² and S₀ is the intrinsic boundary of the Sierpinski gasket. M: R → R is a positive function and h: S × R → R is a suitable function which is a part of our main equation. ∆p denotes the p-Laplacian, where p > 1. First of all, we will define a weak solution for our problem and then we will show the existence of at least two solutions for the above problem under suitable conditions. There is no well-known concept of a generalized derivative of a function on a fractal domain. Recently, the notion of differential operators such as the Laplacian and the p-Laplacian on fractal domains has been defined. We recall the result first then we will address the above problem. In view of literature, Laplacian and p-Laplacian equations are studied extensively on regular domains (open connected domains) in contrast to fractal domains. In fractal domains, people have studied Laplacian equations more than p-Laplacian probably because in that case, the corresponding function space is reflexive and many minimax theorems which work for regular domains is applicable there which is not the case for the p-Laplacian. This motivates us to study equations involving p-Laplacian on the Sierpinski gasket. Problems on fractal domains lead to nonlinear models such as reaction-diffusion equations on fractals, problems on elastic fractal media and fluid flow through fractal regions etc. We have studied the above p-Laplacian equations on the Sierpinski gasket using fibering map technique on the Nehari manifold. Many authors have studied the Laplacian and p-Laplacian equations on regular domains using this Nehari manifold technique. In general Euler functional associated with such a problem is Frechet or Gateaux differentiable. So, a critical point becomes a solution to the problem. Also, the function space they consider is reflexive and hence we can extract a weakly convergent subsequence from a bounded sequence. But in our case neither the Euler functional is differentiable nor the function space is known to be reflexive. Overcoming these issues we are still able to prove the existence of at least two solutions of the given equation.

Keywords: Euler functional, p-Laplacian, p-energy, Sierpinski gasket, weak solution

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Authors: Srijanani Anurag Prasad

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The Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS) was built by combining interpolation data from the Iterated Function System (IFS). The interpolation data in a CHFIS comprises a row and/or column of uncertain values when a single point is entered. Alternatively, a row and/or column of additional points are placed in the given interpolation data to demonstrate the node added CHFIS. There are three techniques for inserting new points that correspond to the row and/or column of nodes inserted, and each method is further classified into four types based on the values of the inserted nodes. As a result, numerous forms of node insertion can be found in a CHFIS.

Keywords: fractal, interpolation, iterated function system, coalescence, node insertion, knot insertion

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Authors: Leonard Mike McNelly Longwa, Divine Kusosa Musiku, D. Nahum Kabeya

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In this paper the bifurcation analysis of oilfield in Democratic Republic of Congo is presented in order to enhance petroleum production in an intense tectonic evolution characterized by distinct compressive and extensive phases and the digenetic transformation in the reservoirs during burial geological configuration. The use of porous media in Makelekese MS-25 field has been established to simulate the boundaries within 3 sedimentary basins open to exploration including the coastal basin with an area of 5992 km2, a central basin with an area of 800,000 km2, the western branch of the East African Rift in which there are 50,000 km2. The fractal characterization of complex hydro-dynamic fractures in oilfield is developed to facilitate oil production process based on reservoirs bifurcation model.

Keywords: reservoir bifurcation, fractal characterisation, permeability, conductivity, skin effect

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Authors: Leonard Mike McNelly Longwa, Divine Kusosa Musiku, Dieudonne Nahum Kabeya

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In this paper, the bifurcation analysis of oilfields in the Democratic Republic of Congo is presented in order to enhance petroleum production in an intense tectonic evolution characterized by distinct compressive and extensive phases and the digenetic transformation in the reservoirs during burial geological configuration. The use of porous media in the Makelekese MS-25 field has been established to simulate the boundaries within 3 sedimentary basins open to exploration including the coastal basin with an area of 5992 km², a central basin with an area of 800,000 km², the western branch of the East African Rift in which there are 50,000 km². The fractal characterization of complex hydro-dynamic fractures in oilfields is developed to facilitate the oil production process based on the reservoirs bifurcation model.

Keywords: reservoir bifurcation, fractal characterization, permeability, conductivity, skin effect

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Authors: Manijeh Ghahroudi Tali, Ladan Khedri Gharibvand

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Gavkhouni lagoon, in the South East of Isfahan (Iran), is one of the pluvial lakes and legacy of Quaternary era which has emerged during periods with more precipitation and less evaporation. Climate change, lack of water resources and dried freshwater of Zayandehrood resulted in increased entropy and activated a dynamic which in turn is converted to Playa. The morphometry of 61 polygonal clay microforms in wet zone soil, 52 polygonal clay microforms in pediplain zone soil and 63 microforms in sulfate soil, is evaluated by fractal model. After calculating the microforms’ area–perimeter fractal dimension, their turbulence level was analyzed. Fractal dimensions (DAP) obtained from the microforms’ analysis of pediplain zone, wet zone, and sulfate soils are 1/21-1/39, 1/27-1/44 and 1/29-1/41, respectively, which is indicative of turbulence in these zones. Logarithmic graph drawn for each region also shows that there is a linear relationship between logarithm of the microforms’ area and perimeter so that correlation coefficient (R2) obtained for wet zone is larger than 0.96, for pediplain zone is larger than 0.99 and for sulfated zone is 0.9. Increased turbulence in this region suggests morphological transformation of the system and lagoon’s conversion to a new ecosystem which can be accompanied with serious risks.

Keywords: fractal, Gavkhouni, microform, Iran

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Authors: Kaoutar Lamrini Uahabi, Mohamed Atounti

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In the present work, we consider one category of curves denoted by L(p, k, r, n). These curves are continuous arcs which are trajectories of roots of the trinomial equation zn = αzk + (1 − α), where z is a complex number, n and k are two integers such that 1 ≤ k ≤ n − 1 and α is a real parameter greater than 1. Denoting by L the union of all trinomial curves L(p, k, r, n) and using the box counting dimension as fractal dimension, we will prove that the dimension of L is equal to 3/2.

Keywords: feasible angles, fractal dimension, Minkowski sausage, trinomial curves, trinomial equation

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Authors: Achouri Fatima, Chouicha Kaddour

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The main parameters affecting the workability are the water content, particle size, and the total surface of the grains, as long as the mixing water begins by wetting the surface of the grains and then fills the voids between the grains to form entrapped water, the quantity of water remaining is called free water. The aim is to undertake a fractal approach through the relationship between the concrete formulation parameters and workability, to develop this approach a series of concrete taken from the literature was investigated by varying formulation parameters such as G / S, the quantity of cement C and the quantity of mixing water E. We also call on other model as the model for the thickness of the water layer and model of the thickness of the paste layer to judge their relevance, hence the following results : the relevance of the model of the thickness of the water layer is considered relevant when there is a variation in the water quantity, the model of the thickness of the layer of the paste is only applicable if we consider that the paste is made with the grain value Dmax = 2.85: value from which we see a stable model.

Keywords: concrete, fractal method, paste thickness, water thickness, workability

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Authors: Zakiya Shireen, Sujin B. Babu

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Irreversible diffusion-limited cluster aggregation (DLCA) of binary sticky spheres was simulated by modifying the Brownian Cluster Dynamics (BCD). We randomly distribute N spheres in a 3D box of size L, the volume fraction is given by Φtot = (π/6)N/L³. We identify NA and NB number of spheres as species A and B in our system both having identical size. In these systems, both A and B particles undergo Brownian motion. Irreversible bond formation happens only between intra-species particles and inter-species interact only through hard-core repulsions. As we perform simulation using BCD we start to observe binary gels. In our study, we have observed that species B always percolate (cluster size equal to L) as expected for the monomeric case and species A does not percolate below a critical ratio which is different for different volume fractions. We will also show that the accessible volume of the system increases when compared to the monomeric case, which means that species A is aggregating inside the cage created by B. We have also observed that for moderate Φtot the system undergoes a transition from flocculation region to percolation region indicated by the change in fractal dimension from 1.8 to 2.5. For smaller ratio of A, it stays in the flocculation regime even though B have already crossed over to the percolation regime. Thus, we observe two fractal dimension in the same system.

Keywords: BCD, fractals, percolation, sticky spheres

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Authors: Y. Lasbet, A. L. Boukhalkhal, K. Loubar

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The study of mixed convection is, usually, focused on the straight channels in which the onset of the mixed convection is well defined as function of the ratio between Grashof number and Reynolds number, Gr/Re. This is not the case for a complex channel wherein the mixed convection is not sufficiently examined in the literature. Our paper focuses on the study of the mixed convection in a complex geometry in which our main contribution reveals that the critical value of the ratio Gr/Re for the onset of the mixed convection increases highly in the type of geometry contrary to the straight channel. Furthermore, the accentuated secondary flow in this geometry prevents the thermal stratification in the flow and consequently the buoyancy driven becomes negligible. To perform these objectives, a numerical study in complex geometry for several values of the ratio Gr/Re with prescribed wall heat flux (H2), was realized by using the CFD code.

Keywords: complex geometry, heat transfer, laminar flow, mixed convection, Nusselt number

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Authors: Lana Horvat Dmitrović

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The main purpose of this paper is to study the box dimension as fractal property of bifurcations of discrete dynamical systems in higher dimensions. The paper contains the fractal analysis of the orbits near the hyperbolic and non-hyperbolic fixed points in discrete dynamical systems. It is already known that in one-dimensional case the orbit near the hyperbolic fixed point has the box dimension equal to zero. On the other hand, the orbit near the non-hyperbolic fixed point has strictly positive box dimension which is connected to the non-degeneracy condition of certain bifurcation. One of the main results in this paper is the generalisation of results about box dimension near the hyperbolic and non-hyperbolic fixed points to higher dimensions. In the process of determining box dimension, the restriction of systems to stable, unstable and center manifolds, Lipschitz property of box dimension and the notion of projective box dimension are used. The analysis of the bifurcations in higher dimensions with one multiplier on the unit circle is done by using the normal forms on one-dimensional center manifolds. This specific change in box dimension of an orbit at the moment of bifurcation has already been explored for some bifurcations in one and two dimensions. It was shown that specific values of box dimension are connected to appropriate bifurcations such as fold, flip, cusp or Neimark-Sacker bifurcation. This paper further explores this connection of box dimension as fractal property to some specific bifurcations in higher dimensions, such as fold-flip and flip-Neimark-Sacker. Furthermore, the application of the results to the unit time map of continuous dynamical system near hyperbolic and non-hyperbolic singularities is presented. In that way, box dimensions which are specific for certain bifurcations of continuous systems can be obtained. The approach to bifurcation analysis by using the box dimension as specific fractal property of orbits can lead to better understanding of bifurcation phenomenon. It could also be useful in detecting the existence or nonexistence of bifurcations of discrete and continuous dynamical systems.

Keywords: bifurcation, box dimension, invariant manifold, orbit near fixed point

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Authors: Valeria Bondarenko

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In this paper, we propose two problems related to fractal Brownian motion. First problem is simultaneous estimation of two parameters, Hurst exponent and the volatility, that describe this random process. Numerical tests for the simulated fBm provided an efficient method. Second problem is approximation of the increments of the observed time series by a power function by increments from the fractional Brownian motion. Approximation and estimation are shown on the example of real data, daily deposit interest rates.

Keywords: fractional Brownian motion, Gausssian processes, approximation, time series, estimation of properties of the model

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Authors: Amar Partap Singh Pharwaha, Shweta Rani

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To develop a reliable and cost effective communication platform for the telemedicine applications, novel antenna design has been presented using bacterial foraging optimization (BFO) technique. The proposed antenna geometry is achieved by etching a modified Koch curve fractal shape at the edges and a square shape slot at the center of the radiating element of a patch antenna. It has been found that the new antenna has achieved 43.79% size reduction and better resonating characteristic than the original patch. Representative results for both simulations and numerical validations are reported in order to assess the effectiveness of the developed methodology.

Keywords: BFO, electrical permittivity, fractals, Koch curve

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Authors: Amar Partap Singh Pharwaha, Shweta Rani

Abstract:

This paper is aimed at proposing a rhombus shaped wearable fractal antenna for wireless communication systems. The geometrical descriptors of the antenna have been obtained using bacterial foraging optimization (BFO) for wide band operation. The method of moment based IE3D software has been used to simulate the antenna and observed that miniaturization of 13.08% has been achieved without degrading the resonating properties of the proposed antenna. An analysis with different substrates has also been done in order to evaluate the effectiveness of electrical permittivity on the presented structure. The proposed antenna has low profile, light weight and has successfully demonstrated wideband and multiband characteristics for wearable electronic applications.

Keywords: BFO, bandwidth, electrical permittivity, fractals, wearable antenna

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Authors: Anca Radulescu, Danae Evans

Abstract:

The relationship between a network’s hardwiring and its emergent dynamics are central to neuroscience. We study the principles of this correspondence in a canonical setup (in which network nodes exhibit well-studied complex quadratic dynamics), then test their universality in biological networks. By extending methods from discrete dynamics, we study the effects of network connectivity on temporal patterns, encapsulating long-term behavior into the rich topology of network Mandelbrot sets. Then elements of fractal geometry can be used to predict and classify network behavior.

Keywords: canonical model, complex dynamics, dynamic networks, fractals, Mandelbrot set, network connectivity

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