Pub Date : 2023-10-19DOI: 10.1142/s0218348x23401977
M. M. Abelazeem, Raghda A. M. Attia
This study employs three advanced computational and numerical techniques to solve the nonlinear fractional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV–ZK) equation in magnetized plasma. The reductive perturbation approach is utilized to investigate the dynamics of various components, namely isothermal species, immobile background species, and warm adiabatic fluid, in magnetized plasma. Emphasis is placed on unraveling the asymmetrical propagation characteristics of nonlinear electrostatic waves. The model’s solutions encompass diverse types of solitons, including ion-acoustic, dust acoustic, and electron acoustic solitons. Analytical solutions are obtained using a variety of mathematical functions, such as exponents, trigonometry, and hyperbolas. Two- and three-dimensional density graphs illustrate the practical behavior of a single soliton. The primary objective of employing numerical schemes is to assess the accuracy of the derived solutions, and the outcomes demonstrate the efficacy of the analytical method in solving nonlinear mathematical and physical problems. Several techniques are employed to validate the consistency between calculated and estimated results, ensuring the study’s accuracy and reliability. Overall, this investigation underscores the effectiveness of numerical and analytical techniques in tackling complex mathematical models, offering a promising avenue for future research in the field. The findings carry significant implications for comprehending nonlinear phenomena in magnetized plasma and contribute to advancing the field.
{"title":"Investigation of the Elaborate Dynamics of Weakly Nonlinear Fractional Ion-Acoustic Waves in Magnetized Electron-Positron Plasma","authors":"M. M. Abelazeem, Raghda A. M. Attia","doi":"10.1142/s0218348x23401977","DOIUrl":"https://doi.org/10.1142/s0218348x23401977","url":null,"abstract":"This study employs three advanced computational and numerical techniques to solve the nonlinear fractional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV–ZK) equation in magnetized plasma. The reductive perturbation approach is utilized to investigate the dynamics of various components, namely isothermal species, immobile background species, and warm adiabatic fluid, in magnetized plasma. Emphasis is placed on unraveling the asymmetrical propagation characteristics of nonlinear electrostatic waves. The model’s solutions encompass diverse types of solitons, including ion-acoustic, dust acoustic, and electron acoustic solitons. Analytical solutions are obtained using a variety of mathematical functions, such as exponents, trigonometry, and hyperbolas. Two- and three-dimensional density graphs illustrate the practical behavior of a single soliton. The primary objective of employing numerical schemes is to assess the accuracy of the derived solutions, and the outcomes demonstrate the efficacy of the analytical method in solving nonlinear mathematical and physical problems. Several techniques are employed to validate the consistency between calculated and estimated results, ensuring the study’s accuracy and reliability. Overall, this investigation underscores the effectiveness of numerical and analytical techniques in tackling complex mathematical models, offering a promising avenue for future research in the field. The findings carry significant implications for comprehending nonlinear phenomena in magnetized plasma and contribute to advancing the field.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135667698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-19DOI: 10.1142/s0218348x23401989
Zareen A. Khan, Humaira Kalsoom
The main goals of this paper are to provide an introduction to the idea of interval-valued [Formula: see text]-polynomial [Formula: see text]-type convex functions and to investigate the algebraic properties of this type of function. This new generalization aims to show the existence of new Hermite–Hadamard inequalities for the recently presented class of interval-valued [Formula: see text]-polynomials of [Formula: see text]-type convex describing the [Formula: see text]-fractional integral operator. In the classical sense, some special cases are figured out, and the two examples are also given. There are some recently discovered inequalities for interval-valued functions that are regulated by fractional calculus applicable to interval-valued [Formula: see text]-polynomial [Formula: see text]-type convexity. The results obtained show that future research will be simple to implement, highly efficient, feasible, and extremely precise in its investigation. It could also help solve modeling problems, optimization problems, and fuzzy interval-valued functions that involve both discrete and continuous variables.
{"title":"Some new inequalities for <i>n</i>-polynomial s-type convexity pertaining to inter-valued functions governed by fractional calculus","authors":"Zareen A. Khan, Humaira Kalsoom","doi":"10.1142/s0218348x23401989","DOIUrl":"https://doi.org/10.1142/s0218348x23401989","url":null,"abstract":"The main goals of this paper are to provide an introduction to the idea of interval-valued [Formula: see text]-polynomial [Formula: see text]-type convex functions and to investigate the algebraic properties of this type of function. This new generalization aims to show the existence of new Hermite–Hadamard inequalities for the recently presented class of interval-valued [Formula: see text]-polynomials of [Formula: see text]-type convex describing the [Formula: see text]-fractional integral operator. In the classical sense, some special cases are figured out, and the two examples are also given. There are some recently discovered inequalities for interval-valued functions that are regulated by fractional calculus applicable to interval-valued [Formula: see text]-polynomial [Formula: see text]-type convexity. The results obtained show that future research will be simple to implement, highly efficient, feasible, and extremely precise in its investigation. It could also help solve modeling problems, optimization problems, and fuzzy interval-valued functions that involve both discrete and continuous variables.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135667696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-18DOI: 10.1142/s0218348x23401965
Esra Karatas Akgul, Wasim Jamshed, Sherzod Shukhratovich Abdullaev, Fethi Bin Muhammed Belgacem, Sayed M. El Din
In this research, we study the Caputo fractional and constant proportional derivative numerical approximation of electrical symmetric circuits. It has been assumed that the derivative is in the order [Formula: see text]. For the fractional electrical symmetric circuits, the RC, LC, and RLC solutions are obtained by using the Sumudu transformation. We also compare the numerical simulation of each equation to its classical equivalent. We use a highly efficient integral transform to examine the impact of the power-law kernel. In our upcoming works, we will apply this to electrical circuits that are more intricate.
{"title":"Computational Solutions of Fractional Electric Symmetric Circuits by Sumudu Transformation","authors":"Esra Karatas Akgul, Wasim Jamshed, Sherzod Shukhratovich Abdullaev, Fethi Bin Muhammed Belgacem, Sayed M. El Din","doi":"10.1142/s0218348x23401965","DOIUrl":"https://doi.org/10.1142/s0218348x23401965","url":null,"abstract":"In this research, we study the Caputo fractional and constant proportional derivative numerical approximation of electrical symmetric circuits. It has been assumed that the derivative is in the order [Formula: see text]. For the fractional electrical symmetric circuits, the RC, LC, and RLC solutions are obtained by using the Sumudu transformation. We also compare the numerical simulation of each equation to its classical equivalent. We use a highly efficient integral transform to examine the impact of the power-law kernel. In our upcoming works, we will apply this to electrical circuits that are more intricate.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":"408 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-18DOI: 10.1142/s0218348x23501190
Kang-Le Wang
In this work, the coupled fractional nonlinear Hirota equation is defined by using a powerful fractional derivative sense, which is M-truncate derivative. We explore the fractional functional method and fractional simple equation method to investigate the structure of the solutions of the coupled fractional nonlinear Hirota equations, and some new periodic solutions and solitary wave solutions are successfully acquired. The two proposed approaches are simple, effective and direct. Moreover, some 3D and 2D graphs are sketched to elaborate the behavior of these solutions. These obtained solitary wave and periodic solutions are helpful to improve the understanding of the physical behavior of the corresponding mathematical model.
{"title":"New Analysis Methods for the Coupled Fractional Nonlinear Hirota Equation","authors":"Kang-Le Wang","doi":"10.1142/s0218348x23501190","DOIUrl":"https://doi.org/10.1142/s0218348x23501190","url":null,"abstract":"In this work, the coupled fractional nonlinear Hirota equation is defined by using a powerful fractional derivative sense, which is M-truncate derivative. We explore the fractional functional method and fractional simple equation method to investigate the structure of the solutions of the coupled fractional nonlinear Hirota equations, and some new periodic solutions and solitary wave solutions are successfully acquired. The two proposed approaches are simple, effective and direct. Moreover, some 3D and 2D graphs are sketched to elaborate the behavior of these solutions. These obtained solitary wave and periodic solutions are helpful to improve the understanding of the physical behavior of the corresponding mathematical model.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-17DOI: 10.1142/s0218348x23402028
Raghda A. M. Attia, Suleman H. Alfalqi, Jameel F. Alzaidi, Mostafa M. A. Khater
This study investigates novel solitary wave solutions of the Gilson–Pickering ([Formula: see text]) equation, which is a model that describes the motion of a fluid in a porous medium. An analytical scheme is applied to construct these solutions, utilizing the extended Khater method in conjunction with the homogenous balance technique. The derived expressions for the solitary wave solutions are exact and are presented in terms of hyperbolic functions. The [Formula: see text] equation is valuable for a wide range of applications, including oil and gas reservoir engineering, groundwater flow, and flow in biological tissues. Additionally, this model is employed to describe the behavior of waves in various physical systems such as fluids and plasmas. Specifically, it models the propagation of dispersive waves in a media that exhibits both dispersion and dissipation. To ensure the accuracy of the constructed solutions, a numerical scheme is employed. The properties of the solitary wave solutions are analyzed, and their physical implications are explored. The results of this investigation reveal a rich variety of solitary wave solutions that exhibit interesting behaviors, including oscillatory and non-oscillatory behavior, which are elucidated through various types of distinct graphs. Consequently, this study provides significant insights into the behavior of fluid flow in porous media and its applications in various fields, including oil and gas reservoir engineering and groundwater flow modeling. The analytical and numerical methods employed in this investigation demonstrate their potential for studying nonlinear evolution equations and their applications in the physical sciences.
{"title":"Unraveling the Complex Dynamics of Fluid Flow in Porous Media: Effects of Viscosity, Porosity, and Inertia on the Motion of Fluids","authors":"Raghda A. M. Attia, Suleman H. Alfalqi, Jameel F. Alzaidi, Mostafa M. A. Khater","doi":"10.1142/s0218348x23402028","DOIUrl":"https://doi.org/10.1142/s0218348x23402028","url":null,"abstract":"This study investigates novel solitary wave solutions of the Gilson–Pickering ([Formula: see text]) equation, which is a model that describes the motion of a fluid in a porous medium. An analytical scheme is applied to construct these solutions, utilizing the extended Khater method in conjunction with the homogenous balance technique. The derived expressions for the solitary wave solutions are exact and are presented in terms of hyperbolic functions. The [Formula: see text] equation is valuable for a wide range of applications, including oil and gas reservoir engineering, groundwater flow, and flow in biological tissues. Additionally, this model is employed to describe the behavior of waves in various physical systems such as fluids and plasmas. Specifically, it models the propagation of dispersive waves in a media that exhibits both dispersion and dissipation. To ensure the accuracy of the constructed solutions, a numerical scheme is employed. The properties of the solitary wave solutions are analyzed, and their physical implications are explored. The results of this investigation reveal a rich variety of solitary wave solutions that exhibit interesting behaviors, including oscillatory and non-oscillatory behavior, which are elucidated through various types of distinct graphs. Consequently, this study provides significant insights into the behavior of fluid flow in porous media and its applications in various fields, including oil and gas reservoir engineering and groundwater flow modeling. The analytical and numerical methods employed in this investigation demonstrate their potential for studying nonlinear evolution equations and their applications in the physical sciences.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-17DOI: 10.1142/s0218348x23501116
Kang-Jia Wang
This study proposes a new fractal modified equal width-Burgers equation (MEWBE) with the local fractional derivative (LFD) for the first time. By defining the Mittag-Leffler function (MLF) on the Cantor set (CS), two special functions, namely, the [Formula: see text] and [Formula: see text] functions, are derived for constructing the auxiliary function to seek the non-differentiable (ND) exact solutions. And 16 groups of the ND exact solutions are successfully established. The solutions on the CS are depicted graphically to interpret the nonlinear dynamic behaviors. Furthermore, the comparative results of the fractal MEWBE and the classical MEWBE are also discussed. The obtained results confirm that the proposed method is effective and powerful, and can provide a promising way to find the ND exact solutions of the local fractional PDEs.
{"title":"New exact solutions of the local fractional modified equal width-Burgers equation on the Cantor sets","authors":"Kang-Jia Wang","doi":"10.1142/s0218348x23501116","DOIUrl":"https://doi.org/10.1142/s0218348x23501116","url":null,"abstract":"This study proposes a new fractal modified equal width-Burgers equation (MEWBE) with the local fractional derivative (LFD) for the first time. By defining the Mittag-Leffler function (MLF) on the Cantor set (CS), two special functions, namely, the [Formula: see text] and [Formula: see text] functions, are derived for constructing the auxiliary function to seek the non-differentiable (ND) exact solutions. And 16 groups of the ND exact solutions are successfully established. The solutions on the CS are depicted graphically to interpret the nonlinear dynamic behaviors. Furthermore, the comparative results of the fractal MEWBE and the classical MEWBE are also discussed. The obtained results confirm that the proposed method is effective and powerful, and can provide a promising way to find the ND exact solutions of the local fractional PDEs.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-17DOI: 10.1142/s0218348x23501153
Wenjia Ma, Keqin Cui, Lifeng Xi
In this paper, we investigate the Zagreb eccentricity index of the level-[Formula: see text] Sierpinski gasket [Formula: see text]. Based on the self-similarity and finite pattern, we compute the Zagreb eccentricity index of [Formula: see text] which is the integral of square of eccentricity in terms of the self-similar singular measure.
{"title":"Zagreb Eccentricity Index of Level-N Sierpinski Gasket","authors":"Wenjia Ma, Keqin Cui, Lifeng Xi","doi":"10.1142/s0218348x23501153","DOIUrl":"https://doi.org/10.1142/s0218348x23501153","url":null,"abstract":"In this paper, we investigate the Zagreb eccentricity index of the level-[Formula: see text] Sierpinski gasket [Formula: see text]. Based on the self-similarity and finite pattern, we compute the Zagreb eccentricity index of [Formula: see text] which is the integral of square of eccentricity in terms of the self-similar singular measure.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":"126 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-17DOI: 10.1142/s0218348x23501219
LEONARDO H. S. FERNANDES, FERNANDO H. A. DE ARAUJO, JOSÉ W. L. SILVA, MARCO C. M. FILHO, BENJAMIN M. TABAK
We explore the synergic interplay between entropy (disorder), predictability, and informational efficiency of the daily closing price time series of 13 sectoral economics components of the Shanghai index letter considering three non-overlapping periods (before and during COVID-19 and Russia–Ukraine war). Our findings reveal that the telecom services, financials, and consumer discretionary sectors are marked by higher informational efficiency. Otherwise, the industrials, utilities, and transportation sectors exhibit lower informational efficiency. These insights are relevant for financial agents to make informed decisions, manage risk, and seek opportunities in an ever-changing market environment.
{"title":"THE SYNERGIC INTERPLAY BETWEEN ENTROPY, PREDICTABILITY, AND INFORMATIONAL EFFICIENCY OF THE SHANGHAI SECTORAL INDEX","authors":"LEONARDO H. S. FERNANDES, FERNANDO H. A. DE ARAUJO, JOSÉ W. L. SILVA, MARCO C. M. FILHO, BENJAMIN M. TABAK","doi":"10.1142/s0218348x23501219","DOIUrl":"https://doi.org/10.1142/s0218348x23501219","url":null,"abstract":"We explore the synergic interplay between entropy (disorder), predictability, and informational efficiency of the daily closing price time series of 13 sectoral economics components of the Shanghai index letter considering three non-overlapping periods (before and during COVID-19 and Russia–Ukraine war). Our findings reveal that the telecom services, financials, and consumer discretionary sectors are marked by higher informational efficiency. Otherwise, the industrials, utilities, and transportation sectors exhibit lower informational efficiency. These insights are relevant for financial agents to make informed decisions, manage risk, and seek opportunities in an ever-changing market environment.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":"174 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136037953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-17DOI: 10.1142/s0218348x23501207
Kang-Jia Wang
The fractal calculus has gained more widespread attention in the last years. The fractal variational principle plays a major role in the fractal travelling wave theory of the fractal PDEs. This paper develops the fractal generalized variational principle (GVP) of the fractal Gardner equation by virtue of the Semi-inverse method (SIM) for the first time. On the other hand, we also discuss and verify the fractal GVP via the fractal two-scale transform (FTST) from another dimension field. The extracted fractal GVP shows the conservation laws through the energy form in the fractal space, and can be manipulated to explore the fractal solitary wave properties.
{"title":"On the generalized variational principle of the fractal Gardner equation","authors":"Kang-Jia Wang","doi":"10.1142/s0218348x23501207","DOIUrl":"https://doi.org/10.1142/s0218348x23501207","url":null,"abstract":"The fractal calculus has gained more widespread attention in the last years. The fractal variational principle plays a major role in the fractal travelling wave theory of the fractal PDEs. This paper develops the fractal generalized variational principle (GVP) of the fractal Gardner equation by virtue of the Semi-inverse method (SIM) for the first time. On the other hand, we also discuss and verify the fractal GVP via the fractal two-scale transform (FTST) from another dimension field. The extracted fractal GVP shows the conservation laws through the energy form in the fractal space, and can be manipulated to explore the fractal solitary wave properties.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-14DOI: 10.1142/s0218348x23501256
MUHAMMAD MARWAN, MAOAN HAN, MAWIA OSMAN
In this work, we have adopted an advanced Julia-based function that helps not only in sorting out hidden wings in the fractals of chaotic systems, but can also generate an extra wing in chaotic systems based on a single wing. For verification, two examples Lorenz and modified stretch-twist-fold (STF) systems based on more than one wing, whereas chemical reaction-based chaotic system with a unique wing is considered. The existence of another wing in chaotic systems based on a single wing was a big question mark on the creation of multi-wings in the theory of fractals, but with the aid of advanced Julia functions, we have elaborated in detail that the existence of a second wing in such systems is also possible. The stretching and squeezing in the trajectories of fractals are also integral parts of our findings. Moreover, our study has solved another problem related to fractals. In the past, authors have shown multi-wings in chaotic systems with empty space inside all the time. In this study, we have shown for the first time that these inner spaces have special meaning due to the existence of hidden wings. Furthermore, we have shown that fractals can be divided into outer and inner wings, where the inner wings reflect the answer to the question about the empty space in between the outer wings of chaotic systems. For convenience, an extra file named as MiddleSpace.pdf is attached as a supplementary file to better understand the concept of covering empty space inside fractals.
{"title":"HIDDEN COVERS (WINGS) IN THE FRACTALS OF CHAOTIC SYSTEMS USING ADVANCED JULIA FUNCTION","authors":"MUHAMMAD MARWAN, MAOAN HAN, MAWIA OSMAN","doi":"10.1142/s0218348x23501256","DOIUrl":"https://doi.org/10.1142/s0218348x23501256","url":null,"abstract":"In this work, we have adopted an advanced Julia-based function that helps not only in sorting out hidden wings in the fractals of chaotic systems, but can also generate an extra wing in chaotic systems based on a single wing. For verification, two examples Lorenz and modified stretch-twist-fold (STF) systems based on more than one wing, whereas chemical reaction-based chaotic system with a unique wing is considered. The existence of another wing in chaotic systems based on a single wing was a big question mark on the creation of multi-wings in the theory of fractals, but with the aid of advanced Julia functions, we have elaborated in detail that the existence of a second wing in such systems is also possible. The stretching and squeezing in the trajectories of fractals are also integral parts of our findings. Moreover, our study has solved another problem related to fractals. In the past, authors have shown multi-wings in chaotic systems with empty space inside all the time. In this study, we have shown for the first time that these inner spaces have special meaning due to the existence of hidden wings. Furthermore, we have shown that fractals can be divided into outer and inner wings, where the inner wings reflect the answer to the question about the empty space in between the outer wings of chaotic systems. For convenience, an extra file named as MiddleSpace.pdf is attached as a supplementary file to better understand the concept of covering empty space inside fractals.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135803610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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