Fractals最新文献

{"title":"EXACT SOLUTIONS AND BIFURCATION OF A MODIFIED GENERALIZED MULTIDIMENSIONAL FRACTIONAL KADOMTSEV–PETVIASHVILI EQUATION","authors":"MINYUAN LIU, HUI XU, ZENGGUI WANG, GUIYING CHEN","doi":"10.1142/s0218348x24500464","DOIUrl":"https://doi.org/10.1142/s0218348x24500464","url":null,"abstract":"<p>In this paper, we investigate the exact solutions of a modified generalized multidimensional fractional Kadomtsev–Petviashvili (KP) equation by the bifurcation method. First, the equation is converted into a planar dynamical system through fractional complex wave transformation. The phase portraits of the equation and qualitative analysis are presented under different bifurcation conditions. Then, the bounded and unbounded traveling wave solutions, including periodic, kink, anti-kink, dark-solitary, bright-solitary and breaking wave solutions, are acquired by integrating along different orbits. Finally, numerical simulations of the dynamic behaviors of the solutions obtained are graphically illustrated by choosing appropriate parameters.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"88 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140534365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SOME ZAGREB-TYPE INDICES OF VICSEK POLYGON GRAPHS","authors":"ZHIQIANG WU, YUMEI XUE, HUIXIA HE, CHENG ZENG, WENJIE WANG","doi":"10.1142/s0218348x24500658","DOIUrl":"https://doi.org/10.1142/s0218348x24500658","url":null,"abstract":"<p>Chemical graph theory plays an essential role in modeling and designing any chemical structure or chemical network. For a (molecular) graph, the Zagreb indices and the Zagreb eccentricity indices are well-known topological indices to describe the structure of a molecule or graph and can be used to predict properties such as the size and number of rings in a molecule, as well as the thermodynamic stability and reactivity of compounds. In this paper, we introduce a class of molecular models, namely, the Vicsek polygon graphs, which extend the traditional Vicsek networks. We compute the Zagreb indices and the Zagreb eccentricity indices of Vicsek polygon graphs by self-similarity and the recurrence relation based on the construction of graphs.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140534350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"CHAOS THEORY, ADVANCED METAHEURISTIC ALGORITHMS AND THEIR NEWFANGLED DEEP LEARNING ARCHITECTURE OPTIMIZATION APPLICATIONS: A REVIEW","authors":"AKIF AKGUL, YELl̇Z KARACA, MUHAMMED ALI PALA, MURAT ERHAN ÇIMEN, ALI FUAT BOZ, MUSTAFA ZAHID YILDIZ","doi":"10.1142/s0218348x24300010","DOIUrl":"https://doi.org/10.1142/s0218348x24300010","url":null,"abstract":"<p>Metaheuristic techniques are capable of representing optimization frames with their specific theories as well as objective functions owing to their being adjustable and effective in various applications. Through the optimization of deep learning models, metaheuristic algorithms inspired by nature, imitating the behavior of living and non-living beings, have been used for about four decades to solve challenging, complex, and chaotic problems. These algorithms can be categorized as evolution-based, swarm-based, nature-based, human-based, hybrid, or chaos-based. Chaos theory, as a useful approach to understanding neural network optimization, has the basic idea of viewing the neural network optimization as a dynamical system in which the equation schemes are utilized from the space pertaining to learnable parameters, namely optimization trajectory, to itself, which enables the description of the evolution of the system by understanding the training behavior, which is to say the number of iterations over time. The examination of the recent studies reveals the importance of chaos theory, which is sensitive to initial conditions with randomness and dynamical properties that are principally emerging on the complex multimodal landscape. Chaotic optimization, in this regard, accelerates the speed of the algorithm while also enhancing the variety of movement patterns. The significance of hybrid algorithms developed through their applications in different domains concerning real-world phenomena and well-known benchmark problems in the literature is also evident. Metaheuristic optimization algorithms have also been applied to deep learning or deep neural networks (DNNs), a branch of machine learning. In this respect, the basic features of deep learning and DNNs and the extensive use of metaheuristic algorithms are overviewed and explained. Accordingly, the current review aims at providing new insights into the studies that deal with metaheuristic algorithms, hybrid-based metaheuristics, chaos-based metaheuristics as well as deep learning besides presenting recent information on the development of the essence of this branch of science with emerging opportunities, applicability-based optimization aspects and generation of well-informed decisions.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140534369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"VARIATIONAL FORMULATIONS FOR A COUPLED FRACTAL–FRACTIONAL KdV SYSTEM","authors":"YINGZI GUAN, KHALED A. GEPREEL, JI-HUAN HE","doi":"10.1142/s0218348x24500543","DOIUrl":"https://doi.org/10.1142/s0218348x24500543","url":null,"abstract":"<p>Every shallow-water wave propagates along a fractal boundary, and its mathematical model cannot be precisely represented by integer dimensions. In this study, we investigate a coupled fractal–fractional KdV system moving along an irregular boundary within the framework of variational theory, which is commonly employed to derive governing equations. However, not every fractal–fractional differential equation can be formulated using variational principles. The semi-inverse method proves to be challenging in finding an appropriate variational principle for nonlinear problems and eliminating extraneous components from the studied model. We consider the coupled fractal–fractional KdV system with arbitrary coefficients and establish its variational formulation to unveil the remarkable insights into the energy structure of the model and interrelationships among coefficients. Encouraging results are obtained for this coupled KdV system.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140534259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"DECODING OF THE EXTRAOCULAR MUSCLES ACTIVATIONS BY COMPLEXITY-BASED ANALYSIS OF ELECTROMYOGRAM (EMG) SIGNALS","authors":"SRIDEVI SRIRAM, KARTHIKEYAN RAJAGOPAL, ONDREJ KREJCAR, HAMIDREZA NAMAZI","doi":"10.1142/s0218348x24500671","DOIUrl":"https://doi.org/10.1142/s0218348x24500671","url":null,"abstract":"<p>The analysis of extraocular muscles’ activation is crucial for understanding eye movement patterns, providing insights into oculomotor control, and contributing to advancements in fields such as vision research, neurology, and biomedical engineering. Ten subjects went through the experiments, including normal watching, blinking, upward and downward movements of eyes, and eye movements to the left and right while their electromyogram (EMG) signals were recorded. We analyzed the complexity of recorded EMG signals using fractal theory, sample entropy, and approximate entropy (ApEn). The results showed that the techniques are able to decode the changes in the complexity of EMG signals between different eye movements. In other words, we can use these methods to study extraocular muscle activations in different conditions.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140534187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A NEW ROUGH FRACTURE PERMEABILITY MODEL OF COAL WITH INJECTED WATER BASED ON DAMAGED TREE-LIKE BRANCHING NETWORK","authors":"ZHEN LIU, ZHENG LI, HE YANG, JING HAN, MUYAO ZHU, SHUAI DONG, ZEHAN YU","doi":"10.1142/s0218348x24500579","DOIUrl":"https://doi.org/10.1142/s0218348x24500579","url":null,"abstract":"<p>The fracture network structure of coal is very complex, and it has always been a hot issue to characterize the fracture network structure of coal by using a tree-like branching network. In this paper, a new rough fracture permeability model of water injection coal based on a damaged tree-like branching network is proposed. In this model, fractal theory and sine wave model are used to characterize the rough characteristics of fracture structure. In addition, the applicability of the model is verified by the self-developed experimental system, and the sensitivity analysis and weight analysis of the influencing factors in the model are carried out. The results show that the factors affecting the permeability of the established permeability mathematical model are mainly the amplitude factors of sinusoidal fluctuation, length ratio, maximum branch series, number of damaged fractures, opening ratio, opening fractal dimension and initial fracture length. Among them, the sinusoidal fluctuation amplitude factor has the greatest influence on the permeability and the other two are inversely proportional. Therefore, with pulsating water injection, the frequency of changes in the water pressure was altered to improve the water injection efficiency in coal fractures with sinusoidal distribution to increase permeability. The research results provide a theoretical basis for further improving the theoretical models for seepage in porous media and fluid flow analysis of damaged tree-like branching networks.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140534368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"INVESTIGATION ON CONCRETE MICROSTRUCTURAL EVOLUTION AND SLOPE STABILITY BASED ON COUPLED FRACTAL FLUID–STRUCTURE MODEL","authors":"Tingting Yang, Yang Liu, Guannan Liu, Boming Yu, Mingyao Wei","doi":"10.1142/s0218348x24500555","DOIUrl":"https://doi.org/10.1142/s0218348x24500555","url":null,"abstract":"Slope instability is a common type of damage in embankment dams. Analyzing its microstructural changes during water transport is beneficial to identify the critical damage point in more detail. To this end, we closely link both diffused water molecule and damaged concrete. On the basis of the original research on fractal theory, the fractal permeability model for the pore system is established. At the same time, considering the pore structure characteristics of concrete, the models for fluid–solid coupling inside and outside the embankment are established in this work. The simulated results and experimental data agree well, thus verifying the correctness of the proposed models. The changes of concrete pore structures under different water pressure and different initial porosities are simulated by numerical methods. The results show that: (1) with the increase of the safety factor, the slope of the embankment begins to be damaged at the critical point and then completely destabilized; (2) with the increase of the pressure head, the change of the fractal dimension of the embankment’s porosity from the top to the bottom changes from linear to nonlinear. (3) The initial porosity of concrete is proportional to the evolution of concrete pore structure. From a technical point of view, this study can contribute the effective technical guidance for related professional practitioners in the analysis of embankment slope stability from a microscopic point of view.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"76 16","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140355666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"RESEARCH ON FRACTAL DIMENSIONS AND THE HÖLDER CONTINUITY OF FRACTAL FUNCTIONS UNDER OPERATIONS","authors":"BINYAN YU, YONGSHUN LIANG","doi":"10.1142/s0218348x2450052x","DOIUrl":"https://doi.org/10.1142/s0218348x2450052x","url":null,"abstract":"<p>Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140534243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SOME NEW PARAMETRIZED INEQUALITIES ON FRACTAL SET","authors":"HONGYAN XU, ABDELGHANI LAKHDARI, WEDAD SALEH, BADREDDINE MEFTAH","doi":"10.1142/s0218348x24500634","DOIUrl":"https://doi.org/10.1142/s0218348x24500634","url":null,"abstract":"<p>The aim of this study is to examine certain open three-point Newton–Cotes-type inequalities for differentiable generalized <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi></math></span><span></span>-convex functions on a fractal set. To begin, we introduce a novel parametrized identity involving the relevant formula, which yields various new findings as well as previously established ones. Finally, an example is given to demonstrate the accuracy of the new results and their graphical depiction. Moreover, we emphasize the applications of the results obtained.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140310486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"3D RENDERING OF THE QUATERNION MANDELBROT SET WITH MEMORY","authors":"RICARDO FARIELLO, PAUL BOURKE, GABRIEL V. S. ABREU","doi":"10.1142/s0218348x24500610","DOIUrl":"https://doi.org/10.1142/s0218348x24500610","url":null,"abstract":"<p>In this paper, we explore the quaternion equivalent of the Mandelbrot set equipped with memory and apply various visualization techniques to the resulting <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mn>4</mn></math></span><span></span>-dimensional geometry. Three memory functions have been considered, two that apply a weighted sum to only the previous two terms and one that performs a weighted sum of all previous terms of the series. The visualization includes one or two cutting planes for dimensional reduction to either <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span> or <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span> dimensions, respectively, as well as employing an intersection with a half space to trim the <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span>D solids in order to reveal the interiors. Using various metrics, we quantify the effect of each memory function on the structure of the quaternion Mandelbrot set.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140310805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}