Journal Description
Fractal and Fractional
Fractal and Fractional
is an international, scientific, peer-reviewed, open access journal of fractals and fractional calculus and their applications in different fields of science and engineering published monthly online by MDPI.
- Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
- High Visibility: indexed within Scopus, SCIE (Web of Science), Inspec, and other databases.
- Journal Rank: JCR - Q1 (Mathematics, Interdisciplinary Applications) / CiteScore - Q1 (Analysis)
- Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 18.9 days after submission; acceptance to publication is undertaken in 3.5 days (median values for papers published in this journal in the second half of 2023).
- Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Impact Factor:
5.4 (2022);
5-Year Impact Factor:
4.7 (2022)
Latest Articles
Further Hermite–Hadamard-Type Inequalities for Fractional Integrals with Exponential Kernels
Fractal Fract. 2024, 8(6), 345; https://doi.org/10.3390/fractalfract8060345 - 7 Jun 2024
Abstract
This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example
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This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example with graphical representations and provides some practical applications, highlighting their relevance to special means. This study presents novel results, offering new insights into classical integrals as the fractional order approaches 1, in addition to the fractional integrals we examined.
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(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)
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Fractal Tent Map with Application to Surrogate Testing
by
Ekaterina Kopets, Vyacheslav Rybin, Oleg Vasilchenko, Denis Butusov, Petr Fedoseev and Artur Karimov
Fractal Fract. 2024, 8(6), 344; https://doi.org/10.3390/fractalfract8060344 - 7 Jun 2024
Abstract
Discrete chaotic maps are a mathematical basis for many useful applications. One of the most common is chaos-based pseudorandom number generators (PRNGs), which should be computationally cheap and controllable and possess necessary statistical properties, such as mixing and diffusion. However, chaotic PRNGs have
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Discrete chaotic maps are a mathematical basis for many useful applications. One of the most common is chaos-based pseudorandom number generators (PRNGs), which should be computationally cheap and controllable and possess necessary statistical properties, such as mixing and diffusion. However, chaotic PRNGs have several known shortcomings, e.g., being prone to chaos degeneration, falling in short periods, and having a relatively narrow parameter range. Therefore, it is reasonable to design novel simple chaotic maps to overcome these drawbacks. In this study, we propose a novel fractal chaotic tent map, which is a generalization of the well-known tent map with a fractal function introduced into the right-hand side. We construct and investigate a PRNG based on the proposed map, showing its high level of randomness by applying the NIST statistical test suite. The application of the proposed PRNG to the task of generating surrogate data and a surrogate testing procedure is shown. The experimental results demonstrate that our approach possesses superior accuracy in surrogate testing across three distinct signal types—linear, chaotic, and biological signals—compared to the MATLAB built-in randn() function and PRNGs based on the logistic map and the conventional tent map. Along with surrogate testing, the proposed fractal tent map can be efficiently used in chaos-based communications and data encryption tasks.
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(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)
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Quantized Nonfragile State Estimation of Memristor-Based Fractional-Order Neural Networks with Hybrid Time Delays Subject to Sensor Saturations
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Xiaoguang Shao, Yanjuan Lu, Jie Zhang, Ming Lyu and Yu Yang
Fractal Fract. 2024, 8(6), 343; https://doi.org/10.3390/fractalfract8060343 - 6 Jun 2024
Abstract
This study addresses the issue of nonfragile state estimation for memristor-based fractional-order neural networks with hybrid randomly occurring delays. Considering the finite bandwidth of the signal transmission channel, quantitative processing is introduced to reduce network burden and prevent signal blocking and packet loss.
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This study addresses the issue of nonfragile state estimation for memristor-based fractional-order neural networks with hybrid randomly occurring delays. Considering the finite bandwidth of the signal transmission channel, quantitative processing is introduced to reduce network burden and prevent signal blocking and packet loss. In a real-world setting, the designed estimator may experience potential gain variations. To address this issue, a fractional-order nonfragile estimator is developed by incorporating a logarithmic quantizer, which ultimately improves the reliability of the state estimator. In addition, by combining the generalized fractional-order Lyapunov direct method with novel Caputo–Wirtinger integral inequalities, a lower conservative criterion is derived to guarantee the asymptotic stability of the augmented system. At last, the accuracy and practicality of the desired estimation scheme are demonstrated through two simulation examples.
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(This article belongs to the Special Issue Fractional Order Systems with Time Delay: Theory, Stability Analysis and Applications)
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Well-Posedness and Hyers–Ulam Stability of Fractional Stochastic Delay Systems Governed by the Rosenblatt Process
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Ghada AlNemer, Mohamed Hosny, Ramalingam Udhayakumar and Ahmed M. Elshenhab
Fractal Fract. 2024, 8(6), 342; https://doi.org/10.3390/fractalfract8060342 - 6 Jun 2024
Abstract
Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall’s inequality,
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Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall’s inequality, sufficient criteria for Hyers–Ulam stability are established. Ultimately, an example is presented to demonstrate the effectiveness of the obtained findings.
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(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
Open AccessArticle
A Dynamical Analysis and New Traveling Wave Solution of the Fractional Coupled Konopelchenko–Dubrovsky Model
by
Jin Wang and Zhao Li
Fractal Fract. 2024, 8(6), 341; https://doi.org/10.3390/fractalfract8060341 - 6 Jun 2024
Abstract
The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using
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The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the trigonometric function solutions, rational function solutions, solitary wave solutions and the elliptic function solutions of the fractional coupled Konopelchenko–Dubrovsky model are derived by means of the polynomial complete discriminant system method. Moreover, a two-dimensional phase portrait is drawn. Finally, a 3D-diagram and a 2D-diagram of the fractional coupled Konopelchenko–Dubrovsky model are plotted in Maple 2022 software.
Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
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Day of the Week Effect on the World Exchange Rates through Fractal Analysis
by
Werner Kristjanpoller and Benjamin Miranda Tabak
Fractal Fract. 2024, 8(6), 340; https://doi.org/10.3390/fractalfract8060340 - 6 Jun 2024
Abstract
The foreign exchange rate market is one of the most liquid and efficient. In this study, we address the efficient analysis of this market by verifying the day-of-the-week effect with fractal analysis. The presence of fractality was evident in the return series of
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The foreign exchange rate market is one of the most liquid and efficient. In this study, we address the efficient analysis of this market by verifying the day-of-the-week effect with fractal analysis. The presence of fractality was evident in the return series of each day and when analyzing an upward trend and a downward trend. The econometric models showed that the day-of-the-week effect in the studied currencies did not align with previous studies. However, analyzing the Hurst exponent of each day revealed that there a weekday effect in the fractal dimension. Thirty main world currencies from all continents were analyzed, showing weekday effects according to their fractal behavior. These results show a form of market inefficiency, as the returns or price variations of each day for the analyzed currencies should have behaved similarly and tended towards random walks. This fractal day-of-the-week effect in world currencies allows us to generate investment strategies and to better complement or support buying and selling decisions on certain days.
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(This article belongs to the Special Issue Fractal and Multifractal Analyses in Financial Markets and Economics, 2nd Edition)
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Command Filter-Based Adaptive Neural Control for Nonstrict-Feedback Nonlinear Systems with Prescribed Performance
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Xiaoli Yang, Jing Li, Shuzhi (Sam) Ge, Xiaoling Liang and Tao Han
Fractal Fract. 2024, 8(6), 339; https://doi.org/10.3390/fractalfract8060339 - 5 Jun 2024
Abstract
In this paper, a new command filter-based adaptive NN control strategy is developed to address the prescribed tracking performance issue for a class of nonstrict-feedback nonlinear systems. Compared with the existing performance functions, a new performance function, the fixed-time performance function, which does
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In this paper, a new command filter-based adaptive NN control strategy is developed to address the prescribed tracking performance issue for a class of nonstrict-feedback nonlinear systems. Compared with the existing performance functions, a new performance function, the fixed-time performance function, which does not depend on the accurate initial value of the error signal and has the ability of fixed-time convergence, is proposed for the first time. A radial basis function neural network is introduced to identify unknown nonlinear functions, and the characteristic of Gaussian basis functions is utilized to overcome the difficulties of the nonstrict-feedback structure. Moreover, in contrast to the traditional Backstepping technique, a command filter-based adaptive control algorithm is constructed, which solves the “explosion of complexity” problem and relaxes the assumption on the reference signal. Additionally, it is guaranteed that the tracking error falls within a prescribed small neighborhood by the designed performance functions in fixed time, and the closed-loop system is semi-globally uniformly ultimately bounded (SGUUB). The effectiveness of the proposed control scheme is verified by numerical simulation.
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(This article belongs to the Special Issue Fractional- and Integer-Order System: Control Theory and Applications, 2nd Edition)
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Enhancing the Pitch-Rate Control Performance of an F-16 Aircraft Using Fractional-Order Direct-MRAC Adaptive Control
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Gustavo E. Ceballos Benavides, Manuel A. Duarte-Mermoud, Marcos E. Orchard and Alfonso Ehijo
Fractal Fract. 2024, 8(6), 338; https://doi.org/10.3390/fractalfract8060338 - 5 Jun 2024
Abstract
This study presents a comparative analysis of classical model reference adaptive control (IO-DMRAC) and its fractional-order counterpart (FO-DMRAC), which are applied to the pitch-rate control of an F-16 aircraft longitudinal model. The research demonstrates a significant enhancement in control performance with fractional-order adaptive
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This study presents a comparative analysis of classical model reference adaptive control (IO-DMRAC) and its fractional-order counterpart (FO-DMRAC), which are applied to the pitch-rate control of an F-16 aircraft longitudinal model. The research demonstrates a significant enhancement in control performance with fractional-order adaptive control. Notably, the FO-DMRAC achieves lower performance indices such as the Integral Square-Error criterion (ISE) and Integral Square-Input criterion (ISU) and eliminates system output oscillations during transient periods. This study marks the pioneering application of FO-DMRAC in aircraft pitch-rate control within the literature. Through simulations on an F-16 short-period model with a relative degree of 1, the FO-DMRAC design is assessed under specific flight conditions and compared with its IO-DMRAC counterpart. Furthermore, the study ensures the boundedness of all signals, including internal ones such as ω(t).
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(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control, 2nd Edition)
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Nonlocal Changing-Sign Perturbation Tempered Fractional Sub-Diffusion Model with Weak Singularity
by
Xinguang Zhang, Jingsong Chen, Peng Chen, Lishuang Li and Yonghong Wu
Fractal Fract. 2024, 8(6), 337; https://doi.org/10.3390/fractalfract8060337 - 5 Jun 2024
Abstract
In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to
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In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder’s fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables.
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(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
Open AccessArticle
Fractional Lévy Stable Motion from a Segmentation Perspective
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Aleksander A. Stanislavsky and Aleksander Weron
Fractal Fract. 2024, 8(6), 336; https://doi.org/10.3390/fractalfract8060336 - 4 Jun 2024
Abstract
The segmentation analysis of the Golding–Cox mRNA dataset clarifies the description of these trajectories as a Fractional Lévy Stable Motion (FLSM). The FLSM method has several important advantages. Using only a few parameters, it allows for the detection of jumps in segmented trajectories
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The segmentation analysis of the Golding–Cox mRNA dataset clarifies the description of these trajectories as a Fractional Lévy Stable Motion (FLSM). The FLSM method has several important advantages. Using only a few parameters, it allows for the detection of jumps in segmented trajectories with non-Gaussian confined parts. The value of each parameter indicates the contribution of confined segments. Non-Gaussian features in mRNA trajectories are attributed to trajectory segmentation. Each segment can be in one of the following diffusion modes: free diffusion, confined motion, and immobility. When free diffusion segments alternate with confined or immobile segments, the mean square displacement of the segmented trajectory resembles subdiffusion. Confined segments have both Gaussian (normal) and non-Gaussian statistics. If random trajectories are estimated as FLSM, they can exhibit either subdiffusion or Lévy diffusion. This approach can be useful for analyzing empirical data with non-Gaussian behavior, and statistical classification of diffusion trajectories helps reveal anomalous dynamics.
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(This article belongs to the Special Issue Modern Methods for Fractal and Multifractal Analysis of Time Series: Theoretical Frameworks and Practical Applications)
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Quantifying the Pore Heterogeneity of Alkaline Lake Shale during Hydrous Pyrolysis by Using the Multifractal Method
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Yanxin Liu, Hong Zhang, Zhengchen Zhang, Luda Jing and Kouqi Liu
Fractal Fract. 2024, 8(6), 335; https://doi.org/10.3390/fractalfract8060335 - 4 Jun 2024
Abstract
Distinguishing itself from marine shale formations, alkaline lake shale, as a significant hydrocarbon source rock and petroleum reservoir, exhibits distinct multifractal characteristics and evolutionary patterns. This study employs a combination of hydrous pyrolysis experimentation, nitrogen adsorption analysis, and multifractal theory to investigate the
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Distinguishing itself from marine shale formations, alkaline lake shale, as a significant hydrocarbon source rock and petroleum reservoir, exhibits distinct multifractal characteristics and evolutionary patterns. This study employs a combination of hydrous pyrolysis experimentation, nitrogen adsorption analysis, and multifractal theory to investigate the factors influencing pore heterogeneity and multifractal dimension during the maturation process of shale with abundant rich alkaline minerals. Utilizing partial least squares (PLS) analysis, a comparative examination is conducted, elucidating the disparate influence of mineralogical composition on their respective multifractal dimensions. The findings reveal a dynamic evolution of pore characteristics throughout the maturation process of alkaline lake shale, delineated into three distinct stages. Initially, in Stage 1 (200 °C to 300 °C), both ΔD and H demonstrate an incremental trend, rising from 1.2699 to 1.3 and from 0.8615 to 0.8636, respectively. Subsequently, in Stages 2 and 3, fluctuations are observed in the values of ΔD and D, while the H value undergoes a pronounced decline to 0.85. Additionally, the parameter D1 exhibits a diminishing trajectory across all stages, decreasing from 0.859 to 0.829, indicative of evolving pore structure characteristics throughout the maturation process. The distinct alkaline environment and mineral composition of alkaline lake shale engender disparate diagenetic effects during its maturation process compared with other shale varieties. Consequently, this disparity results in contrasting evolutionary trajectories in pore heterogeneity and multifractal characteristics. Specifically, multifractal characteristics of alkaline lake shale are primarily influenced by quartz, potassium feldspar, clay minerals, and alkaline minerals.
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(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)
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An Enhanced Multiple Unmanned Aerial Vehicle Swarm Formation Control Using a Novel Fractional Swarming Strategy Approach
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Abdul Wadood, Al-Fahad Yousaf and Aadel Mohammed Alatwi
Fractal Fract. 2024, 8(6), 334; https://doi.org/10.3390/fractalfract8060334 - 3 Jun 2024
Abstract
This paper addresses the enhancement of multiple Unmanned Aerial Vehicle (UAV) swarm formation control in challenging terrains through the novel fractional memetic computing approach known as fractional-order velocity-pausing particle swarm optimization (FO-VPPSO). Existing particle swarm optimization (PSO) algorithms often suffer from premature convergence
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This paper addresses the enhancement of multiple Unmanned Aerial Vehicle (UAV) swarm formation control in challenging terrains through the novel fractional memetic computing approach known as fractional-order velocity-pausing particle swarm optimization (FO-VPPSO). Existing particle swarm optimization (PSO) algorithms often suffer from premature convergence and an imbalanced exploration–exploitation trade-off, which limits their effectiveness in complex optimization problems such as UAV swarm control in rugged terrains. To overcome these limitations, FO-VPPSO introduces an adaptive fractional order β and a velocity pausing mechanism, which collectively enhance the algorithm’s adaptability and robustness. This study leverages the advantages of a meta-heuristic computing approach; specifically, fractional-order velocity-pausing particle swarm optimization is utilized to optimize the flying path length, mitigate the mountain terrain costs, and prevent collisions within the UAV swarm. Leveraging fractional-order dynamics, the proposed hybrid algorithm exhibits accelerated convergence rates and improved solution optimality compared to traditional PSO methods. The methodology involves integrating terrain considerations and diverse UAV control parameters. Simulations under varying conditions, including complex terrains and dynamic threats, substantiate the effectiveness of the approach, resulting in superior fitness functions for multi-UAV swarms. To validate the performance and efficiency of the proposed optimizer, it was also applied to 13 benchmark functions, including uni- and multimodal functions in terms of the mean average fitness value over 100 independent trials, and furthermore, an improvement at percentages of 29.05% and 2.26% is also obtained against PSO and VPPSO in the case of the minimum flight length, as well as 16.46% and 1.60% in mountain terrain costs and 55.88% and 31.63% in collision avoidance. This study contributes valuable insights to the optimization challenges in UAV swarm-formation control, particularly in demanding terrains. The FO-VPPSO algorithm showcases potential advancements in swarm intelligence for real-world applications.
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(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)
Open AccessArticle
Dynamics of a Delayed Fractional-Order Predator–Prey Model with Cannibalism and Disease in Prey
by
Hui Zhang and Ahmadjan Muhammadhaji
Fractal Fract. 2024, 8(6), 333; https://doi.org/10.3390/fractalfract8060333 - 3 Jun 2024
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In this study, a class of delayed fractional-order predation models with disease and cannibalism in the prey was studied. In addition, we considered the prey stage structure and the refuge effect. A Holling type-II functional response function was used to describe predator–prey interactions.
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In this study, a class of delayed fractional-order predation models with disease and cannibalism in the prey was studied. In addition, we considered the prey stage structure and the refuge effect. A Holling type-II functional response function was used to describe predator–prey interactions. First, the existence and uniform boundedness of the solutions of the systems without delay were proven. The local stability of the equilibrium point was also analyzed. Second, we used the digestion delay of predators as a bifurcation parameter to determine the conditions under which Hopf bifurcation occurs. Finally, a numerical simulation was performed to validate the obtained results. Numerical simulations have shown that cannibalism contributes to the elimination of disease in diseased prey populations. In addition, the size of the bifurcation point decreased with an increase in the fractional order, and this had a significant effect on the stability of the system.
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A High-Performance Fractional Order Controller Based on Chaotic Manta-Ray Foraging and Artificial Ecosystem-Based Optimization Algorithms Applied to Dual Active Bridge Converter
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Felipe Ruiz, Eduardo Pichardo, Mokhtar Aly, Eduardo Vazquez, Juan G. Avalos and Giovanny Sánchez
Fractal Fract. 2024, 8(6), 332; https://doi.org/10.3390/fractalfract8060332 - 31 May 2024
Abstract
Over the last decade, dual active bridge (DAB) converters have become critical components in high-frequency power conversion systems. Recently, intensive efforts have been directed at optimizing DAB converter design and control. In particular, several strategies have been proposed to improve the performance of
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Over the last decade, dual active bridge (DAB) converters have become critical components in high-frequency power conversion systems. Recently, intensive efforts have been directed at optimizing DAB converter design and control. In particular, several strategies have been proposed to improve the performance of DAB control systems. For example, fractional-order (FO) control methods have proven potential in several applications since they offer improved controllability, flexibility, and robustness. However, the FO controller design process is critical for industrializing their use. Conventional FO control design methods use frequency domain-based design schemes, which result in complex and impractical designs. In addition, several nonlinear equations need to be solved to determine the optimum parameters. Currently, metaheuristic algorithms are used to design FO controllers due to their effectiveness in improving system performance and their ability to simultaneously tune possible design parameters. Moreover, metaheuristic algorithms do not require precise and detailed knowledge of the controlled system model. In this paper, a hybrid algorithm based on the chaotic artificial ecosystem-based optimization (AEO) and manta-ray foraging optimization (MRFO) algorithms is proposed with the aim of combining the best features of each. Unlike the conventional MRFO method, the newly proposed hybrid AEO-CMRFO algorithm enables the use of chaotic maps and weighting factors. Moreover, the AEO and CMRFO hybridization process enables better convergence performance and the avoidance of local optima. Therefore, superior FO controller performance was achieved compared to traditional control design methods and other studied metaheuristic algorithms. An exhaustive study is provided, and the proposed control method was compared with traditional control methods to verify its advantages and superiority.
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(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)
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Reduced Order Modeling of System by Dynamic Modal Decom-Position with Fractal Dimension Feature Embedding
by
Mingming Zhang, Simeng Bai, Aiguo Xia, Wei Tuo and Yongzhao Lv
Fractal Fract. 2024, 8(6), 331; https://doi.org/10.3390/fractalfract8060331 - 31 May 2024
Abstract
The balance between accuracy and computational complexity is currently a focal point of research in dynamical system modeling. From the perspective of model reduction, this paper addresses the mode selection strategy in Dynamic Mode Decomposition (DMD) by integrating an embedded fractal theory based
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The balance between accuracy and computational complexity is currently a focal point of research in dynamical system modeling. From the perspective of model reduction, this paper addresses the mode selection strategy in Dynamic Mode Decomposition (DMD) by integrating an embedded fractal theory based on fractal dimension (FD). The existing model selection methods lack interpretability and exhibit arbitrariness in choosing mode dimension truncation levels. To address these issues, this paper analyzes the geometric features of modes for the dimensional characteristics of dynamical systems. By calculating the box counting dimension (BCD) of modes and the correlation dimension (CD) and embedding dimension (ED) of the original dynamical system, it achieves guidance on the importance ranking of modes and the truncation order of modes in DMD. To validate the practicality of this method, it is applied to the reduction applications on the reconstruction of the velocity field of cylinder wake flow and the force field of compressor blades. Theoretical results demonstrate that the proposed selection technique can effectively characterize the primary dynamic features of the original dynamical systems. By employing a loss function to measure the accuracy of the reconstruction models, the computed results show that the overall errors of the reconstruction models are below 5%. These results indicate that this method, based on fractal theory, ensures the model’s accuracy and significantly reduces the complexity of subsequent computations, exhibiting strong interpretability and practicality.
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(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)
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A Fractional Heston-Type Model as a Singular Stochastic Equation Driven by Fractional Brownian Motion
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Marc Mukendi Mpanda
Fractal Fract. 2024, 8(6), 330; https://doi.org/10.3390/fractalfract8060330 - 30 May 2024
Abstract
This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven
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This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven by a fractional Brownian motion with a Hurst parameter . We establish the Malliavin differentiability of the fHt model and derive an expression for the expected payoff function, revealing potential discontinuities. Simulation experiments are conducted to illustrate the dynamics of the stock price process and option prices.
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(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
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Existence of Weak Solutions for the Class of Singular Two-Phase Problems with a ψ-Hilfer Fractional Operator and Variable Exponents
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Tahar Bouali, Rafik Guefaifia, Rashid Jan, Salah Boulaaras and Taha Radwan
Fractal Fract. 2024, 8(6), 329; https://doi.org/10.3390/fractalfract8060329 - 30 May 2024
Abstract
In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a -Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here,
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In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a -Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results.
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Matrix-Wigner Distribution
by
Long Wang, Manjun Cui, Ze Qin, Zhichao Zhang and Jianwei Zhang
Fractal Fract. 2024, 8(6), 328; https://doi.org/10.3390/fractalfract8060328 - 30 May 2024
Abstract
In order to achieve time–frequency superresolution in comparison to the conventional Wigner distribution (WD), this study generalizes the well-known -Wigner distribution ( -WD) with only one parameter to the multiple-parameter matrix-Wigner distribution (M-WD) with the parameter matrix . According
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In order to achieve time–frequency superresolution in comparison to the conventional Wigner distribution (WD), this study generalizes the well-known -Wigner distribution ( -WD) with only one parameter to the multiple-parameter matrix-Wigner distribution (M-WD) with the parameter matrix . According to operator theory, we construct Heisenberg’s inequalities on the uncertainty product in M-WD domains and formulate two kinds of attainable lower bounds dependent on . We solve the problem of lower bound minimization and obtain the optimality condition of , under which the M-WD achieves superior time–frequency resolution. It turns out that the M-WD breaks through the limitation of the -WD and gives birth to some novel distributions other than the WD that could generate the highest time–frequency resolution. As an example, the two-dimensional linear frequency-modulated signal is carried out to demonstrate the time–frequency concentration superiority of the M-WD over the short-time Fourier transform and wavelet transform.
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Abundant Closed-Form Soliton Solutions to the Fractional Stochastic Kraenkel–Manna–Merle System with Bifurcation, Chaotic, Sensitivity, and Modulation Instability Analysis
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J. R. M. Borhan, M. Mamun Miah, Faisal Alsharif and Mohammad Kanan
Fractal Fract. 2024, 8(6), 327; https://doi.org/10.3390/fractalfract8060327 - 29 May 2024
Abstract
An essential mathematical structure that demonstrates the nonlinear short-wave movement across the ferromagnetic materials having zero conductivity in an exterior region is known as the fractional stochastic Kraenkel–Manna–Merle system. In this article, we extract abundant wave structure closed-form soliton solutions to the fractional
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An essential mathematical structure that demonstrates the nonlinear short-wave movement across the ferromagnetic materials having zero conductivity in an exterior region is known as the fractional stochastic Kraenkel–Manna–Merle system. In this article, we extract abundant wave structure closed-form soliton solutions to the fractional stochastic Kraenkel–Manna–Merle system with some important analyses, such as bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability. This fractional system renders a substantial impact on signal transmission, information systems, control theory, condensed matter physics, dynamics of chemical reactions, optical fiber communication, electromagnetism, image analysis, species coexistence, speech recognition, financial market behavior, etc. The Sardar sub-equation approach was implemented to generate several genuine innovative closed-form soliton solutions. Additionally, phase portraiture of bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability were employed to monitor the qualitative characteristics of the dynamical system. A certain number of the accumulated outcomes were graphed, including singular shape, kink-shaped, soliton-shaped, and dark kink-shaped soliton in terms of 3D and contour plots to better understand the physical mechanisms of fractional system. The results show that the proposed methodology with analysis in comparison with the other methods is very structured, simple, and extremely successful in analyzing the behavior of nonlinear evolution equations in the field of fractional PDEs. Assessments from this study can be utilized to provide theoretical advice for improving the fidelity and efficiency of soliton dissemination.
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(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)
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Darbo’s Fixed-Point Theorem: Establishing Existence and Uniqueness Results for Hybrid Caputo–Hadamard Fractional Sequential Differential Equations
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Muhammad Yaseen, Sadia Mumtaz, Reny George, Azhar Hussain and Hossam A. Nabwey
Fractal Fract. 2024, 8(6), 326; https://doi.org/10.3390/fractalfract8060326 - 29 May 2024
Abstract
This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo’s fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The
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This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo’s fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The hybrid approach considered in this work combines the advantages of both the Caputo and Hadamard fractional derivatives, leading to a more comprehensive and versatile model for describing sequential processes. To address the problem of the existence and uniqueness of solutions for such hybrid fractional sequential differential equations, we turn to Darbo’s fixed-point theorem, a powerful mathematical tool that establishes the existence of fixed points for certain types of mappings. By appropriately transforming the differential equation into an equivalent fixed-point formulation, we can exploit the properties of Darbo’s theorem to analyze the solutions’ existence and uniqueness. The outcomes of this research expand the understanding of HCHFSDEs and contribute to the growing body of knowledge in fractional calculus and fixed-point theory. These findings are expected to have significant implications in various scientific and engineering applications, where sequential processes are prevalent, such as in physics, biology, finance, and control theory.
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(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)
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