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Fractal Fract., Volume 8, Issue 6 (June 2024) – 59 articles

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20 pages, 3893 KiB  
Article
Stability Analysis of a Fractional-Order Time-Delayed Solow Growth Model with Environmental Pollution
by Yajuan Gu and Hu Wang
Fractal Fract. 2024, 8(6), 361; https://doi.org/10.3390/fractalfract8060361 (registering DOI) - 18 Jun 2024
Viewed by 122
Abstract
Economic growth is resulting in serious environmental problems. Effectively establishing an economic growth model that considers environmental pollution is an important topic. To analyze the interplay between economic growth and environmental pollution, a fractional-order time-delayed economic growth model with environmental purification is proposed [...] Read more.
Economic growth is resulting in serious environmental problems. Effectively establishing an economic growth model that considers environmental pollution is an important topic. To analyze the interplay between economic growth and environmental pollution, a fractional-order time-delayed economic growth model with environmental purification is proposed in this paper. The established model considers not only the environment and economic production but also the labor force population and total factor productivity. Furthermore, the asymptotic stability conditions and parameter stability interval are provided. Finally, in numerical experiments, the correctness of the theory is verified. Full article
(This article belongs to the Section General Mathematics, Analysis)
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20 pages, 6111 KiB  
Article
Fractal and Multifractal Analysis of Microscopic Pore Structure of UHPC Matrix Modified with Nano Silica
by Dian Guan, Tinghong Pan, Rongxin Guo, Ya Wei, Rongqing Qi, Chaoshu Fu, Ziqi Zhang and Yukai Zhu
Fractal Fract. 2024, 8(6), 360; https://doi.org/10.3390/fractalfract8060360 - 17 Jun 2024
Viewed by 197
Abstract
Nano silica (NS) has been found to have a positive impact on enhancing the microporous structure of Ultra-High-Performance Concrete (UHPC). However, there is a lack of effective methods to accurately characterize the regulatory improvement mechanism of NS on the pore structure of UHPC. [...] Read more.
Nano silica (NS) has been found to have a positive impact on enhancing the microporous structure of Ultra-High-Performance Concrete (UHPC). However, there is a lack of effective methods to accurately characterize the regulatory improvement mechanism of NS on the pore structure of UHPC. In this study, our objective is to investigate the influence of NS on various characteristic parameters of the pore structure in UHPC, including porosity, average pore size, box fractal dimension, and multifractal spectral parameters. To analyze these effects, we employ a combination of X- CT image processing techniques and fractal theory. Furthermore, we conducted regression analysis using linear functions to explore the correlation between these parameters and the 28d compressive strength of UHPC. The experimental results demonstrate that NS promotes the refinement of matrix pore size, leading to a denser microstructure of the matrix. Fractal analysis revealed that the pore structure of NS-modified UHPC exhibited favorable fractal characteristics. The fractal dimension and multiple fractal parameters provided complementary insights into the pore structure of NS-modified UHPC from different perspectives. The fractal dimension described the global information, indicating that NS improved matrix defects and reduced the complexity of the pore structure. On the other hand, the multiple fractal parameters supplemented local information, highlighting how the increase in micropores contributed to the heterogeneity of the pore structure. The results of the correlation analysis indicate that the developed mathematical model has a good fit with the 28d compressive strength of UHPC. Full article
(This article belongs to the Special Issue Fractal Mechanics of Engineering Materials)
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14 pages, 4470 KiB  
Article
Unveiling Temporal Cyclicities in Seismic b-Values and Major Earthquake Events in Japan by Local Singularity Analysis and Wavelet Methods
by Siyuan Li, Yuanzhi Zhou and Qiuming Cheng
Fractal Fract. 2024, 8(6), 359; https://doi.org/10.3390/fractalfract8060359 - 17 Jun 2024
Viewed by 228
Abstract
Studying the temporal characteristics of earthquake activity contributes to enhancing earthquake prediction capabilities. The seismic b-value is a key indicator describing the relationship between seismic frequency and magnitude. This study investigates the correlation between the occurrence of major earthquakes and seismic b-values using [...] Read more.
Studying the temporal characteristics of earthquake activity contributes to enhancing earthquake prediction capabilities. The seismic b-value is a key indicator describing the relationship between seismic frequency and magnitude. This study investigates the correlation between the occurrence of major earthquakes and seismic b-values using earthquake activity records in Japan from 1990 to 2023. Local singularity analysis and wavelet analysis of earthquake frequency and b-value time series reveal significant 5-year periodic features in seismic activity in Japan. Furthermore, our research identifies that this periodicity is also prominent in major earthquakes with magnitudes of 7 and above. Additionally, through a detailed analysis of the cross-correlation between seismic b-values and the occurrence time of major earthquakes, we uncover a notable pattern: major earthquakes often occur approximately two years after the peak of seismic b-values. This discovery offers a new perspective on earthquake prediction and may play a crucial role in future earthquake early warning systems. Full article
(This article belongs to the Section Engineering)
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14 pages, 1513 KiB  
Article
Anisotropy of Magnetohydrodynamic and Kinetic Scale Fluctuations through Correlation Tensor in Solar Wind at 0.8 au
by Mirko Stumpo, Simone Benella, Pier Paolo Di Bartolomeo, Luca Sorriso-Valvo and Tommaso Alberti
Fractal Fract. 2024, 8(6), 358; https://doi.org/10.3390/fractalfract8060358 - 14 Jun 2024
Viewed by 188
Abstract
Space plasma turbulence is inherently characterized by anisotropic fluctuations. The generalized k-th order correlation tensor of magnetic field increments allow us to separate the mixed isotropic and anisotropic structure functions from the purely anisotropic ones. In this work, we quantified the relative [...] Read more.
Space plasma turbulence is inherently characterized by anisotropic fluctuations. The generalized k-th order correlation tensor of magnetic field increments allow us to separate the mixed isotropic and anisotropic structure functions from the purely anisotropic ones. In this work, we quantified the relative importance of anisotropic fluctuations in solar wind turbulence using two Alfvénic data samples gathered by the Solar Orbiter at 0.8 astronomical units. The results based on the joined statistics suggest that the anisotropic fluctuations are ubiquitous in solar wind turbulence and persist at kinetic scales. Using the RTN coordinate system, we show that their presence depends on the anisotropic sector under consideration, e.g., the RN and RT sectors exhibit enhanced anisotropy toward kinetic scales, in contrast with the TN. We then study magnetic field fluctuations parallel and perpendicular to the local mean magnetic field separately. We find that perpendicular fluctuations are representative of the global statistics, resembling the typical picture of magnetohydrodynamic turbulence, whereas parallel fluctuations exhibit a scaling law with slope ∼1 for all the joined isotropic and anisotropic components. These results are in agreement with predictions based on the critical balance phenomenology. This topic is potentially of interest for future space missions measuring kinetic and MHD scales simultaneously in a multi-spacecraft configuration. Full article
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41 pages, 5315 KiB  
Article
Estimation of Fractal Dimension and Segmentation of Brain Tumor with Parallel Features Aggregation Network
by Haseeb Sultan, Nadeem Ullah, Jin Seong Hong, Seung Gu Kim, Dong Chan Lee, Seung Yong Jung and Kang Ryoung Park
Fractal Fract. 2024, 8(6), 357; https://doi.org/10.3390/fractalfract8060357 - 14 Jun 2024
Viewed by 279
Abstract
The accurate recognition of a brain tumor (BT) is crucial for accurate diagnosis, intervention planning, and the evaluation of post-intervention outcomes. Conventional methods of manually identifying and delineating BTs are inefficient, prone to error, and time-consuming. Subjective methods for BT recognition are biased [...] Read more.
The accurate recognition of a brain tumor (BT) is crucial for accurate diagnosis, intervention planning, and the evaluation of post-intervention outcomes. Conventional methods of manually identifying and delineating BTs are inefficient, prone to error, and time-consuming. Subjective methods for BT recognition are biased because of the diffuse and irregular nature of BTs, along with varying enhancement patterns and the coexistence of different tumor components. Hence, the development of an automated diagnostic system for BTs is vital for mitigating subjective bias and achieving speedy and effective BT segmentation. Recently developed deep learning (DL)-based methods have replaced subjective methods; however, these DL-based methods still have a low performance, showing room for improvement, and are limited to heterogeneous dataset analysis. Herein, we propose a DL-based parallel features aggregation network (PFA-Net) for the robust segmentation of three different regions in a BT scan, and we perform a heterogeneous dataset analysis to validate its generality. The parallel features aggregation (PFA) module exploits the local radiomic contextual spatial features of BTs at low, intermediate, and high levels for different types of tumors and aggregates them in a parallel fashion. To enhance the diagnostic capabilities of the proposed segmentation framework, we introduced the fractal dimension estimation into our system, seamlessly combined as an end-to-end task to gain insights into the complexity and irregularity of structures, thereby characterizing the intricate morphology of BTs. The proposed PFA-Net achieves the Dice scores (DSs) of 87.54%, 93.42%, and 91.02%, for the enhancing tumor region, whole tumor region, and tumor core region, respectively, with the multimodal brain tumor segmentation (BraTS)-2020 open database, surpassing the performance of existing state-of-the-art methods. Additionally, PFA-Net is validated with another open database of brain tumor progression and achieves a DS of 64.58% for heterogeneous dataset analysis, surpassing the performance of existing state-of-the-art methods. Full article
19 pages, 7496 KiB  
Article
Fractal Characterization of the Pore-Throat Structure in Tight Sandstone Based on Low-Temperature Nitrogen Gas Adsorption and High-Pressure Mercury Injection
by Taping He, Yaoqi Zhou, Zhaobing Chen, Zhenwei Zhang, Huanyu Xie, Yuehan Shang and Gaixia Cui
Fractal Fract. 2024, 8(6), 356; https://doi.org/10.3390/fractalfract8060356 - 14 Jun 2024
Viewed by 308
Abstract
The pore-throat structure is a critical factor in the study of unconventional oil and gas reservoirs, drawing particular attention from petroleum geologists, and it is of paramount significance to analyze to enhance oil and gas production. In tight sandstone, which serves as a [...] Read more.
The pore-throat structure is a critical factor in the study of unconventional oil and gas reservoirs, drawing particular attention from petroleum geologists, and it is of paramount significance to analyze to enhance oil and gas production. In tight sandstone, which serves as a significant hydrocarbon reservoir, the internal pore-throat structure plays a decisive role in the storage and migration of fluids such as water, gases, and hydrocarbons. This paper employs casting thin section (CTS), field emission scanning electron microscope (FE-SEM), high-pressure mercury injection (HPMI), and low-temperature nitrogen gas adsorption (LT−N2−GA) experimental tests to qualitatively and quantitatively analyze the characteristics of the pore-throat structure in tight sandstone. The results indicate that the pore types in tight sandstone include intergranular residual pores, dissolution pores, intercrystalline pores, and microfractures, while the throat types encompass sheet-shaped, curved-sheet-shaped, and tubular throats. Analysis of the physical and structural parameters from 13 HPMI and 5 LT−N2−GA samples reveals a bimodal distribution of pore-throat radii. The complexity of the pore-throat structure is identified as a primary controlling factor for reservoir permeability. The fractal dimension (D) exhibits an average value of 2.45, displaying a negative correlation with porosity (R2 = 0.22), permeability (R2 = 0.65), the pore-throat diameter (R2 = 0.58), and maximum mercury saturation (R2 = 0.86) and a positive correlation with threshold pressure (R2 = 0.56), median saturation pressure (R2 = 0.49), BET specific surface area (R2 = 0.51), and BJH total pore volume (R2 = 0.14). As D increases, reservoir pores tend to decrease in size, leading to reduced flow and deteriorated physical properties, indicative of a more complex pore-throat structure. Full article
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20 pages, 6930 KiB  
Article
Riemann–Hilbert Method Equipped with Mixed Spectrum for N-Soliton Solutions of New Three-Component Coupled Time-Varying Coefficient Complex mKdV Equations
by Sheng Zhang, Xianghui Wang and Bo Xu
Fractal Fract. 2024, 8(6), 355; https://doi.org/10.3390/fractalfract8060355 - 14 Jun 2024
Viewed by 278
Abstract
This article extends the celebrated Riemann–Hilbert (RH) method equipped with mixed spectrum to a new integrable system of three-component coupled time-varying coefficient complex mKdV equations (ccmKdVEs for short) generated by the mixed spectral equations (msEs). Firstly, the ccmKdVEs and the msEs for generating [...] Read more.
This article extends the celebrated Riemann–Hilbert (RH) method equipped with mixed spectrum to a new integrable system of three-component coupled time-varying coefficient complex mKdV equations (ccmKdVEs for short) generated by the mixed spectral equations (msEs). Firstly, the ccmKdVEs and the msEs for generating the ccmKdVEs are proposed. Then, based on the msEs, a solvable RH problem related to the ccmKdVEs is constructed. By using the constructed RH problem with mixed spectrum, scattering data for the recovery of potential formulae are further determined. In the case of reflectionless coefficients, explicit N-soliton solutions of the ccmKdVEs are ultimately obtained. Taking N equal to 1 and 2 as examples, this paper reveals that the spatiotemporal solution structures with time-varying nonlinear dynamic characteristics localized in the ccmKdVEs is attributed to the multiple selectivity of mixed spectrum and time-varying coefficients. In addition, to further highlight the application of our work in fractional calculus, by appropriately selecting these time-varying coefficients, the ccmKdVEs are transformed into a conformable time-fractional order system of three-component coupled complex mKdV equations. Based on the obtained one-soliton solutions, a set of initial values are assigned to the transformed fractional order system, and the N-th iteration formulae of approximate solutions for this fractional order system are derived through the variational iteration method (VIM). Full article
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22 pages, 28940 KiB  
Article
Fractional Active Disturbance Rejection Positioning and Docking Control of Remotely Operated Vehicles: Analysis and Experimental Validation
by Weidong Liu, Liwei Guo, Le Li, Jingming Xu and Guanghao Yang
Fractal Fract. 2024, 8(6), 354; https://doi.org/10.3390/fractalfract8060354 - 14 Jun 2024
Viewed by 222
Abstract
In this paper, a fractional active disturbance rejection control (FADRC) scheme is proposed for remotely operated vehicles (ROVs) to enhance high-precision positioning and docking control in the presence of ocean current disturbances and model uncertainties. The scheme comprises a double closed-loop fractional-order [...] Read more.
In this paper, a fractional active disturbance rejection control (FADRC) scheme is proposed for remotely operated vehicles (ROVs) to enhance high-precision positioning and docking control in the presence of ocean current disturbances and model uncertainties. The scheme comprises a double closed-loop fractional-order PIλDμ controller (DFOPID) and a model-assisted finite-time sliding-mode extended state observer (MFSESO). Among them, DFOPID effectively compensates for non-matching disturbances, while its fractional-order term enhances the dynamic performance and steady-state accuracy of the system. MFSESO contributes to enhancing the estimation accuracy through the integration of sliding-mode technology and model information, ensuring the finite-time convergence of observation errors. Numerical simulations and pool experiments have shown that the proposed control scheme can effectively resist disturbances and successfully complete high-precision tasks in the absence of an accurate model. This underscores the independence of this control scheme on accurate model data of an operational ROV. Meanwhile, it also has the advantages of a simple structure and easy parameter tuning. The FADRC scheme presented in this paper holds practical significance and can serve as a valuable reference for applications involving ROVs. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Calculus in Robotics)
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20 pages, 412 KiB  
Article
Fractional Operators and Fractionally Integrated Random Fields on Zν
by Vytautė Pilipauskaitė and Donatas Surgailis
Fractal Fract. 2024, 8(6), 353; https://doi.org/10.3390/fractalfract8060353 - 13 Jun 2024
Viewed by 200
Abstract
We consider fractional integral operators (IT)d,d(1,1) acting on functions g:ZνR,ν1, where T is the transition operator of a random [...] Read more.
We consider fractional integral operators (IT)d,d(1,1) acting on functions g:ZνR,ν1, where T is the transition operator of a random walk on Zν. We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ(s;d),sZν of (IT)d. The asymptotic behavior of τ(s;d) as |s| is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Zν solving the difference equation (IT)dX=ε with white noise on the right-hand side is discussed and their scaling limits. Several examples, including fractional lattice Laplace and heat operators, are studied in detail. Full article
19 pages, 5495 KiB  
Article
Exploring Novel Soliton Solutions to the Time-Fractional Coupled Drinfel’d–Sokolov–Wilson Equation in Industrial Engineering Using Two Efficient Techniques
by Md Nur Hossain, M. Mamun Miah, Moataz Alosaimi, Faisal Alsharif and Mohammad Kanan
Fractal Fract. 2024, 8(6), 352; https://doi.org/10.3390/fractalfract8060352 - 13 Jun 2024
Viewed by 699
Abstract
The time-fractional coupled Drinfel’d–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma [...] Read more.
The time-fractional coupled Drinfel’d–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma physics. In this study, we present novel soliton solutions for the DSW equation, which significantly enhance the accuracy of describing soliton phenomena. To achieve these results, we employed two distinct methods to derive the solutions: the Sardar subequation method, which works with one variable, and the ΩΩ, 1Ω method which utilizes two variables. These approaches supply significant improvements in efficiency, accuracy, and the ability to explore a broader spectrum of soliton solutions compared to traditional computational methods. By using these techniques, we construct a wide range of wave structures, including rational, trigonometric, and hyperbolic functions. Rigorous validation with Mathematica software 13.1 ensures precision, while dynamic visual representations illustrate soliton solutions with diverse patterns such as dark solitons, multiple dark solitons, singular solitons, multiple singular solitons, kink solitons, bright solitons, and bell-shaped patterns. These findings highlight the effectiveness of these methods in discovering new soliton solutions and supplying deeper insights into the DSW model’s behavior. The novel soliton solutions obtained in this study significantly enhance our understanding of the DSW equation’s underlying dynamics and offer potential applications across various scientific fields. Full article
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16 pages, 3554 KiB  
Article
A Novel Contact Stiffness Model for Grinding Joint Surface Based on the Generalized Ubiquitiformal Sierpinski Carpet Theory
by Qi An, Yue Liu, Min Huang and Shuangfu Suo
Fractal Fract. 2024, 8(6), 351; https://doi.org/10.3390/fractalfract8060351 - 12 Jun 2024
Viewed by 298
Abstract
A novel analytical model based on the generalized ubiquitiformal Sierpinski carpet is proposed which can more accurately obtain the normal contact stiffness of the grinding joint surface. Firstly, the profile and the distribution of asperities on the grinding surface are characterized. Then, based [...] Read more.
A novel analytical model based on the generalized ubiquitiformal Sierpinski carpet is proposed which can more accurately obtain the normal contact stiffness of the grinding joint surface. Firstly, the profile and the distribution of asperities on the grinding surface are characterized. Then, based on the generalized ubiquitiformal Sierpinski carpet, the contact characterization of the grinding joint surface is realized. Secondly, a contact mechanics analysis of the asperities on the grinding surface is carried out. The analytical expressions for contact stiffness in various deformation stages are derived, culminating in the establishment of a comprehensive analytical model for the grinding joint surface. Subsequently, a comparative analysis is conducted between the outcomes of the presented model, the KE model, and experimental data. The findings reveal that, under identical contact pressure conditions, the results obtained from the presented model exhibit a closer alignment with experimental observations compared to the KE model. With an increase in contact pressure, the relative error of the presented model shows a trend of first increasing and then decreasing, while the KE model has a trend of increasing. For the relative error values of the four surfaces under different contact pressures, the maximum relative error of the presented model is 5.44%, while the KE model is 22.99%. The presented model can lay a solid theoretical foundation for the optimization design of high-precision machine tools and provide a scientific theoretical basis for the performance analysis of machine tool systems. Full article
(This article belongs to the Special Issue Fractal Analysis and Fractal Dimension in Materials Chemistry)
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30 pages, 4990 KiB  
Article
Optimizing Economic Dispatch with Renewable Energy and Natural Gas Using Fractional-Order Fish Migration Algorithm
by Abdallah Aldosary
Fractal Fract. 2024, 8(6), 350; https://doi.org/10.3390/fractalfract8060350 - 12 Jun 2024
Viewed by 358
Abstract
This work presents a model for solving the Economic-Environmental Dispatch (EED) challenge, which addresses the integration of thermal, renewable energy schemes, and natural gas (NG) units, that consider both toxin emission and fuel costs as its primary objectives. Three cases are examined using [...] Read more.
This work presents a model for solving the Economic-Environmental Dispatch (EED) challenge, which addresses the integration of thermal, renewable energy schemes, and natural gas (NG) units, that consider both toxin emission and fuel costs as its primary objectives. Three cases are examined using the IEEE 30-bus system, where thermal units (TUs) are replaced with NGs to minimize toxin emissions and fuel costs. The system constraints include equality and inequality conditions. A detailed modeling of NGs is performed, which also incorporates the pressure pipelines and the flow velocity of gas as procedure limitations. To obtain Pareto optimal solutions for fuel costs and emissions, three optimization algorithms, namely Fractional-Order Fish Migration Optimization (FOFMO), Coati Optimization Algorithm (COA), and Non-Dominated Sorting Genetic Algorithm (NSGA-II) are employed. Three cases are investigated to validate the effectiveness of the proposed model when applied to the IEEE 30-bus system with the integration of renewable energy sources (RESs) and natural gas units. The results from Case III, where NGs are installed in place of two thermal units (TUs), demonstrate that the economic dispatching approach presented in this study significantly reduces emission levels to 0.4232 t/h and achieves a lower fuel cost of 796.478 USD/MWh. Furthermore, the findings indicate that FOFMO outperforms COA and NSGA-II in effectively addressing the EED problem. Full article
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22 pages, 1814 KiB  
Article
A Novel Stochastic Model for Human Norovirus Dynamics: Vaccination Impact with Lévy Noise
by Yuqin Song, Peijiang Liu and Anwarud Din
Fractal Fract. 2024, 8(6), 349; https://doi.org/10.3390/fractalfract8060349 - 12 Jun 2024
Viewed by 255
Abstract
The epidemic norovirus causes vomiting and diarrhea and is a highly contagious infection. The disease is affecting human lives in terms of deaths and medical expenses. This study examines the governing dynamics of norovirus by incorporating Lévy noise into a stochastic [...] Read more.
The epidemic norovirus causes vomiting and diarrhea and is a highly contagious infection. The disease is affecting human lives in terms of deaths and medical expenses. This study examines the governing dynamics of norovirus by incorporating Lévy noise into a stochastic SIRWF (susceptible, infected, recovered, water contamination, and food contamination) model. The existence of a non-negative solution and its uniqueness are proved after model formulation. Subsequently, the threshold parameter is calculated, and this number is used to explore the conditions under which disease tends to exist in the population. Likewise, additional conditions are derived that ensure the elimination of the disease from the community. It is proved that the norovirus is extinct whenever the threshold parameter is less than one and it persists for Rs>1. The work assumes two working examples to numerically explain the theoretical findings. Simulations of the study are visually presented, and comparisons are made. The results of this study suggest a robust approach for handling complex biological and epidemic phenomena. Full article
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8 pages, 4284 KiB  
Article
Dynamic Behavior and Optical Soliton for the M-Truncated Fractional Paraxial Wave Equation Arising in a Liquid Crystal Model
by Jie Luo and Zhao Li
Fractal Fract. 2024, 8(6), 348; https://doi.org/10.3390/fractalfract8060348 - 12 Jun 2024
Viewed by 282
Abstract
The main purpose of this article is to investigate the dynamic behavior and optical soliton for the M-truncated fractional paraxial wave equation arising in a liquid crystal model, which is usually used to design camera lenses for high-quality photography. The traveling wave transformation [...] Read more.
The main purpose of this article is to investigate the dynamic behavior and optical soliton for the M-truncated fractional paraxial wave equation arising in a liquid crystal model, which is usually used to design camera lenses for high-quality photography. The traveling wave transformation is applied to the M-truncated fractional paraxial wave equation. Moreover, a two-dimensional dynamical system and its disturbance system are obtained. The phase portraits of the two-dimensional dynamic system and Poincaré sections and a bifurcation portrait of its perturbation system are drawn. The obtained three-dimensional graphs of soliton solutions, two-dimensional graphs of soliton solutions, and contour graphs of the M-truncated fractional paraxial wave equation arising in a liquid crystal model are drawn. Full article
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21 pages, 1994 KiB  
Article
Iterative Learning Formation Control via Input Sharing for Fractional-Order Singular Multi-Agent Systems with Local Lipschitz Nonlinearity
by Guangxu Wang, Rui Wang, Danhu Yi, Xingyu Zhou and Shuyu Zhang
Fractal Fract. 2024, 8(6), 347; https://doi.org/10.3390/fractalfract8060347 - 11 Jun 2024
Viewed by 360
Abstract
For a class of fractional-order singular multi-agent systems (FOSMASs) with local Lipschitz nonlinearity, this paper proposes a closed-loop Dα-type iterative learning formation control law via input sharing to achieve the stable formation of FOSMASs in a finite time. Firstly, the formation [...] Read more.
For a class of fractional-order singular multi-agent systems (FOSMASs) with local Lipschitz nonlinearity, this paper proposes a closed-loop Dα-type iterative learning formation control law via input sharing to achieve the stable formation of FOSMASs in a finite time. Firstly, the formation control issue of FOSMASs with local Lipschitz nonlinearity under the fixed communication topology (FCT) is transformed into the consensus tracking control scenario. Secondly, by virtue of utilizing the characteristics of fractional calculus and the generalized Gronwall inequality, sufficient conditions for the convergence of formation error are given. Then, drawing upon the FCT, the iteration-varying switching communication topology is considered and examined. Ultimately, the validity of the Dα-type learning method is showcased through two numerical cases. Full article
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22 pages, 1056 KiB  
Article
Common Attractors for Generalized F-Iterated Function Systems in G-Metric Spaces
by Talat Nazir and Sergei Silvestrov
Fractal Fract. 2024, 8(6), 346; https://doi.org/10.3390/fractalfract8060346 - 10 Jun 2024
Viewed by 278
Abstract
In this paper, we study the generalized F-iterated function system in G-metric space. Several results of common attractors of generalized iterated function systems obtained by using generalized F-Hutchinson operators are also established. We prove that the triplet of F-Hutchinson [...] Read more.
In this paper, we study the generalized F-iterated function system in G-metric space. Several results of common attractors of generalized iterated function systems obtained by using generalized F-Hutchinson operators are also established. We prove that the triplet of F-Hutchinson operators defined for a finite number of general contractive mappings on a complete G-metric space is itself a generalized F-contraction mapping on a space of compact sets. We also present several examples in 2-D and 3-D for our results. Full article
(This article belongs to the Section General Mathematics, Analysis)
24 pages, 376 KiB  
Article
Further Hermite–Hadamard-Type Inequalities for Fractional Integrals with Exponential Kernels
by Hong Li, Badreddine Meftah, Wedad Saleh, Hongyan Xu, Adem Kiliçman and Abdelghani Lakhdari
Fractal Fract. 2024, 8(6), 345; https://doi.org/10.3390/fractalfract8060345 - 7 Jun 2024
Viewed by 658
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 [...] Read more.
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 β approaches 1, in addition to the fractional integrals we examined. Full article
(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)
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17 pages, 53744 KiB  
Article
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
Viewed by 361
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 [...] Read more.
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. Full article
(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)
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19 pages, 2368 KiB  
Article
Quantized Nonfragile State Estimation of Memristor-Based Fractional-Order Neural Networks with Hybrid Time Delays Subject to Sensor Saturations
by 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
Viewed by 336
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. [...] Read more.
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. Full article
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15 pages, 315 KiB  
Article
Well-Posedness and Hyers–Ulam Stability of Fractional Stochastic Delay Systems Governed by the Rosenblatt Process
by 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
Viewed by 330
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, [...] Read more.
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. Full article
10 pages, 919 KiB  
Article
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
Viewed by 367
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 [...] Read more.
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
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26 pages, 9798 KiB  
Article
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
Viewed by 382
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 [...] Read more.
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. Full article
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17 pages, 581 KiB  
Article
Command Filter-Based Adaptive Neural Control for Nonstrict-Feedback Nonlinear Systems with Prescribed Performance
by 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
Viewed by 392
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 [...] Read more.
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. Full article
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13 pages, 3488 KiB  
Article
Enhancing the Pitch-Rate Control Performance of an F-16 Aircraft Using Fractional-Order Direct-MRAC Adaptive Control
by 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
Viewed by 350
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 [...] Read more.
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). Full article
(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control, 2nd Edition)
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15 pages, 323 KiB  
Article
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
Viewed by 318
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
11 pages, 624 KiB  
Article
Fractional Lévy Stable Motion from a Segmentation Perspective
by Aleksander A. Stanislavsky and Aleksander Weron
Fractal Fract. 2024, 8(6), 336; https://doi.org/10.3390/fractalfract8060336 - 4 Jun 2024
Viewed by 330
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 [...] Read more.
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. Full article
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21 pages, 5168 KiB  
Article
Quantifying the Pore Heterogeneity of Alkaline Lake Shale during Hydrous Pyrolysis by Using the Multifractal Method
by 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
Viewed by 331
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)
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30 pages, 1860 KiB  
Article
An Enhanced Multiple Unmanned Aerial Vehicle Swarm Formation Control Using a Novel Fractional Swarming Strategy Approach
by Abdul Wadood, Al-Fahad Yousaf and Aadel Mohammed Alatwi
Fractal Fract. 2024, 8(6), 334; https://doi.org/10.3390/fractalfract8060334 - 3 Jun 2024
Viewed by 160
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 [...] Read more.
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. Full article
24 pages, 2964 KiB  
Article
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
Viewed by 170
Abstract
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. [...] Read more.
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 τ0 decreased with an increase in the fractional order, and this had a significant effect on the stability of the system. Full article
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19 pages, 1714 KiB  
Article
A High-Performance Fractional Order Controller Based on Chaotic Manta-Ray Foraging and Artificial Ecosystem-Based Optimization Algorithms Applied to Dual Active Bridge Converter
by 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
Viewed by 287
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 [...] Read more.
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. Full article
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