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Fractal Fract., Volume 9, Issue 1 (January 2025) – 25 articles

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23 pages, 358 KiB  
Article
New Approaches to Fractal–Fractional Bullen’s Inequalities Through Generalized Convexity
by Wedad Saleh, Hamid Boulares, Abdelkader Moumen, Hussien Albala and Badreddine Meftah
Fractal Fract. 2025, 9(1), 25; https://doi.org/10.3390/fractalfract9010025 - 3 Jan 2025
Abstract
This paper introduces a new identity involving fractal–fractional integrals, which allow us to derive several new Bullen-type inequalities via generalized convexity. This study provides a significant advancement in the area of fractal–fractional inequalities, presenting a range of results not only for fractional integrals [...] Read more.
This paper introduces a new identity involving fractal–fractional integrals, which allow us to derive several new Bullen-type inequalities via generalized convexity. This study provides a significant advancement in the area of fractal–fractional inequalities, presenting a range of results not only for fractional integrals and fractal calculus, but also offering a refinement of the well-known Bullen-type inequality. We further explore the connections between generalized convexity and fractal–fractional integrals, showing how the concept of generalized convexity enables the establishment of error bounds for fractal–fractional integrals involving lower-order derivatives, with an emphasis on their applications in various fields. The findings expand the current understanding of fractal–fractional inequalities and offer new insights into the use of local fractional derivatives for analyzing functions with fractional-order properties. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
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<p>A comparison of Corollary 3 and Theorem 1.</p>
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16 pages, 293 KiB  
Article
Modeling Anomalous Transport of Cosmic Rays in the Heliosphere Using a Fractional Fokker–Planck Equation
by José Luis Díaz Palencia
Fractal Fract. 2025, 9(1), 24; https://doi.org/10.3390/fractalfract9010024 - 2 Jan 2025
Viewed by 230
Abstract
Cosmic rays exhibit anomalous diffusion behaviors in the heliospheric environment that cannot be adequately described by classical diffusion models. In this paper, we develop a theoretical framework employing a fractional Fokker–Planck equation to model the anomalous transport of cosmic rays. This approach accounts [...] Read more.
Cosmic rays exhibit anomalous diffusion behaviors in the heliospheric environment that cannot be adequately described by classical diffusion models. In this paper, we develop a theoretical framework employing a fractional Fokker–Planck equation to model the anomalous transport of cosmic rays. This approach accounts for the observed non-Gaussian distributions, long-range correlations and memory effects in cosmic ray fluxes. We derive analytical solutions using the Adomian Decomposition Method and express them in terms of Mittag-Leffler functions and Lévy stable distributions. The model parameters, including the fractional orders α and μ and the entropic index q, are estimated by a short comparison between theoretical predictions and observational data from cosmic ray experiments. Our findings suggest that the integration of fractional calculus and non-extensive statistics can be employed for describing the cosmic ray propagation and the anomalous diffusion observed in the heliosphere. Full article
21 pages, 5722 KiB  
Article
Analytical Solutions of Time-Fractional Navier–Stokes Equations Employing Homotopy Perturbation–Laplace Transform Method
by Awatif Muflih Alqahtani, Hamza Mihoubi, Yacine Arioua and Brahim Bouderah
Fractal Fract. 2025, 9(1), 23; https://doi.org/10.3390/fractalfract9010023 - 31 Dec 2024
Viewed by 249
Abstract
The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional [...] Read more.
The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional derivative (CHFD). The effectiveness of this method is demonstrated and validated through two test problems. The results show that the proposed method is robust, efficient, and easy to implement for both linear and nonlinear problems in science and engineering. Additionally, its computational efficiency requires less computation compared to other schemes. Full article
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Figure 1

Figure 1
<p>Plots solution application 1 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>y</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>0.9</mn> <mo>,</mo> <mo> </mo> <mn>0.6</mn> <mo>,</mo> <mo> </mo> <mn>0.3</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 1 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>0.9</mn> <mo>,</mo> <mo> </mo> <mn>0.6</mn> <mo>,</mo> <mo> </mo> <mn>0.3</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 1 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mo> </mo> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 1 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mo> </mo> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 4 Cont.
<p>Plots solution application 1 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mo> </mo> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 1 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mo> </mo> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 1 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mo> </mo> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 2 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>y</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>0.9</mn> <mo>,</mo> <mo> </mo> <mn>0.6</mn> <mo>,</mo> <mo> </mo> <mn>0.3</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 1 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>0.9</mn> <mo>,</mo> <mo> </mo> <mn>0.6</mn> <mo>,</mo> <mo> </mo> <mn>0.3</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 2 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 2 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 10
<p>Plots solution application 2 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <mi>α</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 2 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>α</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>.</p>
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<p>Plots solution application 2 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>α</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 12 Cont.
<p>Plots solution application 2 with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> <mo>,</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>α</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p>
Full article ">
18 pages, 10069 KiB  
Article
Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects
by José Luis Echenausía-Monroy, Luis Alberto Quezada-Tellez, Hector Eduardo Gilardi-Velázquez, Omar Fernando Ruíz-Martínez, María del Carmen Heras-Sánchez, Jose E. Lozano-Rizk, José Ricardo Cuesta-García, Luis Alejandro Márquez-Martínez, Raúl Rivera-Rodríguez, Jonatan Pena Ramirez and Joaquín Álvarez
Fractal Fract. 2025, 9(1), 22; https://doi.org/10.3390/fractalfract9010022 - 31 Dec 2024
Viewed by 234
Abstract
Fractional calculus (or arbitrary order calculus) refers to the integration and derivative operators of an order different than one and was developed in 1695. They have been widely used to study dynamical systems, especially chaotic systems, as the use of arbitrary-order operators broke [...] Read more.
Fractional calculus (or arbitrary order calculus) refers to the integration and derivative operators of an order different than one and was developed in 1695. They have been widely used to study dynamical systems, especially chaotic systems, as the use of arbitrary-order operators broke the milestone of restricting autonomous continuous systems of order three to obtain chaotic behavior and triggered the study of fractional chaotic systems. In this paper, we study the chaotic behavior in fractional systems in more detail and characterize the geometric variations that the dynamics of the system undergo when using arbitrary-order operators by asking the following question: is the Lyapunov exponent sufficient to describe the dynamical variations in a chaotic system of fractional order? By quantifying the convex envelope generated by the 2D projection of the system into all its phase portraits, the changes in the area of the system, as well as the volume of the attractor, are characterized. The results are compared with standard metrics for the study of chaotic systems, such as the Kaplan–Yorke dimension and the fractal dimension, and we also evaluate the frequency fluctuations in the dynamical response. It is found that our methodology can better describe the changes occurring in the systems, while the traditional dimensions are limited to confirming chaotic behaviors; meanwhile, the frequency spectrum hardly changes. The results deepen the study of fractional-order chaotic systems, contribute to understanding the implications and effects observed in the dynamics of the systems, and provide a reference framework for decision-making when using arbitrary-order operators to model dynamical systems. Full article
23 pages, 4954 KiB  
Article
Automatic Voltage Regulator Betterment Based on a New Fuzzy FOPI+FOPD Tuned by TLBO
by Mokhtar Shouran and Mohammed Alenezi
Fractal Fract. 2025, 9(1), 21; https://doi.org/10.3390/fractalfract9010021 - 31 Dec 2024
Viewed by 250
Abstract
This paper presents a novel Fuzzy Logic Controller (FLC) framework aimed at enhancing the performance and stability of Automatic Voltage Regulators (AVRs) in power systems. The proposed system combines fuzzy control theory with the Fractional Order Proportional Integral Derivative (FOPID) technique and employs [...] Read more.
This paper presents a novel Fuzzy Logic Controller (FLC) framework aimed at enhancing the performance and stability of Automatic Voltage Regulators (AVRs) in power systems. The proposed system combines fuzzy control theory with the Fractional Order Proportional Integral Derivative (FOPID) technique and employs cascading control theory to significantly improve reliability and robustness. The unique control architecture, termed Fuzzy Fractional Order Proportional Integral (PI) plus Fractional Order Proportional Derivative (PD) plus Integral (Fuzzy FOPI+FOPD+I), integrates advanced control methodologies to achieve superior performance. To optimize the controller parameters, the Teaching–Learning-Based Optimization (TLBO) algorithm is utilized in conjunction with the Integral Time Absolute Error (ITAE) objective function, ensuring precise tuning for optimal control behavior. The methodology is validated through comparative analyses with controllers reported in prior studies, highlighting substantial improvements in performance metrics. Key findings demonstrate significant reductions in peak overshoot, peak undershoot, and settling time, emphasizing the proposed controller’s effectiveness. Additionally, the robustness of the controller is extensively evaluated under challenging scenarios, including parameter uncertainties and load disturbances. Results confirm its ability to maintain stability and performance across a wide range of conditions, outperforming existing methods. This study presents a notable contribution by introducing an innovative control structure that addresses critical challenges in AVR systems, paving the way for more resilient and efficient power system operations. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)
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<p>Schematic representation of generalized AVR components.</p>
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<p>The traditional AVR model without a controller.</p>
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<p>Step response of AVR system without controller.</p>
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<p>A root locus diagram of the AVR system without a controller.</p>
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<p>The proposed Fuzzy FOPI+FOPD+I AVR system.</p>
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<p>The membership functions of the fuzzy controller.</p>
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<p>The TLBO-tuned Fuzzy FOPI+FOPD+I for AVR.</p>
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<p>The flowchart of the TLBO algorithm.</p>
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<p>The convergence curve of TLBO.</p>
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<p>The dynamic response of the AVR system based on different controllers.</p>
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<p>Settling and rise times of different controllers.</p>
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<p>ITAE of different controllers.</p>
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<p>Peak overshoot and undershoot of different controllers.</p>
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<p>Step responses of AVR systems without controller under different parametric uncertainty conditions.</p>
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<p>Step responses of AVR systems when system is subjected to parametric uncertainties.</p>
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<p>Step responses of AVR systems when system is subjected to parametric uncertainties with load disturbance.</p>
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23 pages, 444 KiB  
Article
A Study on the Existence, Uniqueness, and Stability of Fractional Neutral Volterra-Fredholm Integro-Differential Equations with State-Dependent Delay
by Prabakaran Raghavendran, Tharmalingam Gunasekar, Junaid Ahmad and Walid Emam
Fractal Fract. 2025, 9(1), 20; https://doi.org/10.3390/fractalfract9010020 (registering DOI) - 31 Dec 2024
Viewed by 235
Abstract
This paper presents an analysis of the existence, uniqueness, and stability of solutions to fractional neutral Volterra-Fredholm integro-differential equations, incorporating Caputo fractional derivatives and semigroup operators with state-dependent delays. By employing Krasnoselskii’s fixed point theorem, conditions under which solutions exist are established. To [...] Read more.
This paper presents an analysis of the existence, uniqueness, and stability of solutions to fractional neutral Volterra-Fredholm integro-differential equations, incorporating Caputo fractional derivatives and semigroup operators with state-dependent delays. By employing Krasnoselskii’s fixed point theorem, conditions under which solutions exist are established. To ensure uniqueness, the Banach Contraction Principle is applied, and the contraction condition is verified. Stability is analyzed using Ulam’s stability concept, emphasizing the resilience of solutions to perturbations and providing insights into their long-term behavior. An example is included, accompanied by graphical analysis that visualizes the solutions and their dynamic properties. Full article
(This article belongs to the Section General Mathematics, Analysis)
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Figure 1

Figure 1
<p>2D Contour Plot of <math display="inline"><semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>ϑ</mi> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> over <math display="inline"><semantics> <mrow> <mi>ϑ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mrow> </semantics></math>, showing the regions where the solution maintains constant values. The color gradient represents the amplitude of the solution, highlighting the oscillatory behavior induced by the sine and cosine components. This plot visually demonstrates how the solution evolves in time <math display="inline"><semantics> <mi>ϑ</mi> </semantics></math> and space <span class="html-italic">h</span>, offering insight into the periodic nature of the system and its interaction with the SDD.</p>
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<p>3D Surface Plot of <math display="inline"><semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>ϑ</mi> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> over <math display="inline"><semantics> <mrow> <mi>ϑ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mrow> </semantics></math>, providing a three-dimensional perspective of the solution’s variation. The plot reveals the amplitude and frequency characteristics of the solution, showcasing the maxima and minima corresponding to the oscillatory behavior. The surface visualization offers a clearer understanding of the solution’s dependence on both time and space, with particular emphasis on how SDD affect the system’s dynamics.</p>
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<p>2D Line Plot of <math display="inline"><semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>ϑ</mi> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> over <math display="inline"><semantics> <mrow> <mi>ϑ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math> for a fixed value of <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>, showing the oscillatory nature of the solution over time. This plot captures the periodic fluctuations of the solution, emphasizing the temporal behavior while maintaining the spatial influence at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>.</p>
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36 pages, 10932 KiB  
Review
Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity Effects
by Danil Makarov, Roman Parovik and Zafar Rakhmonov
Fractal Fract. 2025, 9(1), 19; https://doi.org/10.3390/fractalfract9010019 - 31 Dec 2024
Viewed by 302
Abstract
The article is devoted to the study of economic cycles and crises, which are studied within the framework of the theory of N.D. Kondratiev long waves (K-waves). The object of the study is the fractional mathematical models of S. V. Dubovsky, consisting of [...] Read more.
The article is devoted to the study of economic cycles and crises, which are studied within the framework of the theory of N.D. Kondratiev long waves (K-waves). The object of the study is the fractional mathematical models of S. V. Dubovsky, consisting of two nonlinear differential equations of fractional order and describing the dynamics of the efficiency of new technologies and the efficiency of capital productivity, taking into account constant and variable heredity. Fractional mathematical models also take into account the dependence of the rate of accumulation on capital productivity and the influx of external investment and new technological solutions. The effects of heredity lead to a delayed effect of the response of the system in question to the impact. The property of heredity in mathematical models is taken into account using fractional derivatives of constant and variable orders in the sense of Gerasimov–Caputo. The fractional mathematical models of S. V. Dubovsky are further studied numerically using the Adams–Bashforth–Moulton algorithm. Using a numerical algorithm, oscillograms and phase trajectories were constructed for various values and model parameters. It is shown that the fractional mathematical models of S. V. Dubovsky may have limit cycles, which are not always stable. Full article
(This article belongs to the Section Mathematical Physics)
Show Figures

Figure 1

Figure 1
<p>Example of a simply connected region for <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>0.5</mn> <mo>,</mo> <mn>1.3</mn> </mfenced> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>0</mn> <mo>,</mo> <mn>0.775</mn> </mfenced> </mrow> </semantics></math>.</p>
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<p>Oscillograms and phase trajectories, constructed for various values of <math display="inline"><semantics> <mi>λ</mi> </semantics></math> and <span class="html-italic">a</span>.</p>
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<p>Examples of a simply connected region: (<b>a</b>) for dependence (<a href="#FD13-fractalfract-09-00019" class="html-disp-formula">13</a>); (<b>b</b>) for dependence (<a href="#FD14-fractalfract-09-00019" class="html-disp-formula">14</a>).</p>
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<p>Examples of oscillograms and phase trajectories for dependence (<a href="#FD13-fractalfract-09-00019" class="html-disp-formula">13</a>). The points on the oscillogram are the coordinates between the two maximum peaks.</p>
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<p>Examples of oscillograms and phase trajectories for dependence (<a href="#FD14-fractalfract-09-00019" class="html-disp-formula">14</a>). The points on the oscillogram are the coordinates between the two maximum peaks.</p>
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<p>Oscillograms and phase trajectory of system (<a href="#FD16-fractalfract-09-00019" class="html-disp-formula">16</a>) for dependence (<a href="#FD13-fractalfract-09-00019" class="html-disp-formula">13</a>) taking into account the following parameter values: <math display="inline"><semantics> <mrow> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
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<p>Oscillograms and phase trajectory of system (<a href="#FD16-fractalfract-09-00019" class="html-disp-formula">16</a>) for dependence (<a href="#FD13-fractalfract-09-00019" class="html-disp-formula">13</a>) taking into account the following parameter values: <math display="inline"><semantics> <mrow> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mo> </mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
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<p>Oscillograms and phase trajectory of system (<a href="#FD16-fractalfract-09-00019" class="html-disp-formula">16</a>) for dependence (<a href="#FD14-fractalfract-09-00019" class="html-disp-formula">14</a>) taking into account the values of the following parameters: <math display="inline"><semantics> <mrow> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>.</p>
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<p>Oscillograms and phase trajectory of system (<a href="#FD16-fractalfract-09-00019" class="html-disp-formula">16</a>) for dependence (<a href="#FD14-fractalfract-09-00019" class="html-disp-formula">14</a>) taking into account the values of the following parameters: <math display="inline"><semantics> <mrow> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>15</mn> </mrow> </semantics></math>.</p>
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<p>Oscillograms and phase trajectory of system (<a href="#FD16-fractalfract-09-00019" class="html-disp-formula">16</a>) for dependence (<a href="#FD14-fractalfract-09-00019" class="html-disp-formula">14</a>) taking into account the values of the following parameters: <math display="inline"><semantics> <mrow> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo> </mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
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<p>Oscillograms and phase trajectories, constructed for various values of <math display="inline"><semantics> <mi>λ</mi> </semantics></math> and <span class="html-italic">a</span>.</p>
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<p>Oscillograms and phase trajectories, constructed for various values <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>=</mo> <mfenced separators="" open="[" close="]"> <mn>0.8</mn> <mo>,</mo> <mn>0.9</mn> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </semantics></math>.</p>
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<p>Oscillograms and phase trajectories, constructed for various values <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>=</mo> <mfenced separators="" open="[" close="]"> <mn>0.8</mn> <mo>,</mo> <mn>0.9</mn> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </semantics></math>.</p>
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<p>Oscillogram for <math display="inline"><semantics> <mrow> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>.</p>
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<p>Oscillograms and phase trajectories constructed for different values of the initial conditions <math display="inline"><semantics> <mfenced separators="" open="(" close=")"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mfenced> </semantics></math>.</p>
Full article ">Figure 16
<p>Oscillograms and phase trajectories plotted for different values of <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics></math>.</p>
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<p>Oscillograms and phase trajectories plotted for different values of <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics></math>.</p>
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<p>Phase trajectories obtained for different values of the parameters <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with fixed values of <math display="inline"><semantics> <msub> <mi>l</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>l</mi> <mn>2</mn> </msub> </semantics></math>.</p>
Full article ">Figure 19
<p>Phase trajectories obtained for different values of the parameters <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with fixed values <math display="inline"><semantics> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 20
<p>Phase trajectories obtained for different values of the parameters <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with fixed values of <math display="inline"><semantics> <msub> <mi>l</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>l</mi> <mn>2</mn> </msub> </semantics></math> for dependency (<a href="#FD14-fractalfract-09-00019" class="html-disp-formula">14</a>).</p>
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<p>Simulation results in Example 9 for different values of <math display="inline"><semantics> <msub> <mi>α</mi> <mn>10</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>α</mi> <mn>20</mn> </msub> </semantics></math>.</p>
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<p>Simulation results in Example 9 for different values of <math display="inline"><semantics> <msub> <mi>M</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>M</mi> <mn>2</mn> </msub> </semantics></math>.</p>
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<p>Simulation results in Example 9 for various values of <math display="inline"><semantics> <msub> <mi>ω</mi> <mn>2</mn> </msub> </semantics></math> and <math display="inline"><semantics> <mi>λ</mi> </semantics></math>.</p>
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<p>Simulation results in Example 9 for various values of <math display="inline"><semantics> <msub> <mi>ω</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>δ</mi> <mn>1</mn> </msub> </semantics></math>.</p>
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<p>Simulation results in Example 9 for various values of <span class="html-italic">a</span> and <span class="html-italic">b</span>.</p>
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<p>Simulation results in Example 10 for various values of <math display="inline"><semantics> <msub> <mi>M</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>M</mi> <mn>2</mn> </msub> </semantics></math>.</p>
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<p>Simulation results in Example 10 for various values of <math display="inline"><semantics> <msub> <mi>l</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>l</mi> <mn>2</mn> </msub> </semantics></math>.</p>
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<p>Simulation results in Example 10 for various values of <span class="html-italic">a</span> and <span class="html-italic">b</span>.</p>
Full article ">
29 pages, 7819 KiB  
Article
Dynamic Behavior and Fixed-Time Synchronization Control of Incommensurate Fractional-Order Chaotic System
by Xianchen Wang, Zhen Wang and Shihong Dang
Fractal Fract. 2025, 9(1), 18; https://doi.org/10.3390/fractalfract9010018 - 30 Dec 2024
Viewed by 350
Abstract
In this paper, an incommensurate fractional-order chaotic system is established based on Chua’s system. Combining fractional-order calculus theory and the Adomian algorithm, the dynamic phenomena of the incommensurate system caused by different fractional orders are studied. Meanwhile, the incommensurate system parameters and initial [...] Read more.
In this paper, an incommensurate fractional-order chaotic system is established based on Chua’s system. Combining fractional-order calculus theory and the Adomian algorithm, the dynamic phenomena of the incommensurate system caused by different fractional orders are studied. Meanwhile, the incommensurate system parameters and initial values are used as variables to study the dynamic characteristics of the incommensurate system. It is found that there are rich coexistence bifurcation diagrams and coexistence Lyapunov exponent spectra which are further verified with the phase diagrams. Moreover, a special dynamic phenomenon, such as chaotic degenerate dynamic behavior, is found in the incommensurate system. Secondly, for the feasibility of practical application, the equivalent analog circuit of incommensurate system is realized according to fractional-order time–frequency frequency domain algorithm. Finally, in order to overcome the limitation that the convergence time of the finite-time synchronization control scheme depends on the initial value, a fixed-time synchronization control scheme is proposed in the selection of synchronization control scheme. The rationality of this scheme is proved by theoretical analysis and numerical simulation. Full article
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Figure 1

Figure 1
<p>(<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> phase plane. (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> phase plane. (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>4</mn> </msub> </mrow> </semantics></math> phase plane. (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> phase plane. (<b>e</b>) <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> phase plane. (<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>4</mn> </msub> </mrow> </semantics></math> phase plane. Phase diagrams: (<b>a</b>–<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>q</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>0.85</mn> </mrow> </semantics></math>; (<b>d</b>–<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.70</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>0.90</mn> </mrow> </semantics></math>.</p>
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<p>Poincaré section of incommensurate system at <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. (<b>a</b>) Double-scroll Poincaré section. (<b>b</b>) Single-scroll Poincaré section.</p>
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<p>Bifurcation diagram and LEs of incommensurate system when <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>i</mi> </msub> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0.8</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mfenced separators="" open="(" close=")"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mfenced> </mrow> </semantics></math>. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>∈</mo> <mfenced separators="" open="(" close=")"> <mn>0.8</mn> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </semantics></math>. (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>∈</mo> <mfenced separators="" open="(" close=")"> <mn>0.8</mn> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </semantics></math>. (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>∈</mo> <mfenced separators="" open="(" close=")"> <mn>0.8</mn> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </semantics></math>. (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>4</mn> </msub> <mo>∈</mo> <mfenced separators="" open="(" close=")"> <mn>0.8</mn> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </semantics></math>.</p>
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<p>Coexistence dynamics phenomenon of parameter <span class="html-italic">a</span> change. (<b>a</b>) Bifurcation of coexistence of initial values. (<b>b</b>) LEs of coexistence of initial values.</p>
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<p>Coexistence attractor types of incommensurate system with different parameter <span class="html-italic">a</span> in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> plane. (<b>a</b>) Symmetric periodic-1 coexistence attractors. (<b>b</b>) Symmetric periodic-2 coexistence attractors. (<b>c</b>) Symmetric periodic-4 coexistence attractor. (<b>d</b>) Symmetric single-scroll coexistence attractor. (<b>e</b>) Symmetric single-helix coexistence attractor. (<b>f</b>) Symmetric double-scroll coexistence attractor.</p>
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<p>Coexistence dynamics phenomenon of parameter <span class="html-italic">b</span> change. (<b>a</b>) Bifurcation of coexistence of initial values. (<b>b</b>) LEs of coexistence of initial values.</p>
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<p>Coexistence attractor types of incommensurate system with different parameter <span class="html-italic">b</span> in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> plane. (<b>a</b>) Symmetric periodic-1 coexistence attractors. (<b>b</b>) Symmetric periodic-2 coexistence attractors. (<b>c</b>) Symmetric periodic-4 coexistence attractors. (<b>d</b>) Symmetric single-scroll coexistence attractors. (<b>e</b>) Symmetric double-vortex period-1 attractors. (<b>f</b>) Symmetric double-vortex period-2 attractors.</p>
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<p>Coexistence dynamics phenomenon of parameter <span class="html-italic">d</span> change. (<b>a</b>) Bifurcation of coexistence of initial values. (<b>b</b>) LEs of coexistence of initial values.</p>
Full article ">Figure 9
<p>Coexistence attractor types of incommensurate system with different parameter <span class="html-italic">d</span> in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> plane. (<b>a</b>) Single-scroll coexistence attractors. (<b>b</b>) Periodic- and single-scroll coexisting attractors. (<b>c</b>) Single-vortex and periodic coexistence attractors. (<b>d</b>) Coexistence attractors of period-2 and period-4. (<b>e</b>) Symmetric period-2 attractors. (<b>f</b>) Symmetric period-1 attractors.</p>
Full article ">Figure 10
<p>Coexistence dynamics phenomenon of parameter <span class="html-italic">f</span> change. (<b>a</b>) Bifurcation of coexistence of initial values. (<b>b</b>) LEs of coexistence of initial values.</p>
Full article ">Figure 11
<p>Coexistence attractor types of incommensurate system with different parameter <span class="html-italic">f</span> in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> plane. (<b>a</b>) Symmetric periodic-1 coexistence attractors. (<b>b</b>) Symmetric periodic-2 coexistence attractors. (<b>c</b>) Symmetric periodic-4 coexistence attractors. (<b>d</b>) Coexistence attractors of single-scroll and limited cycles. (<b>e</b>) Single-scroll coexistence attractors. (<b>f</b>) Double-vortex period-1 coexistence attractors.</p>
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<p>Chaotic degeneration behavior of incommensurate system. (<b>a</b>) Time–domain waveform of state variable <math display="inline"><semantics> <msub> <mi>x</mi> <mn>4</mn> </msub> </semantics></math> in time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> </mrow> </semantics></math> (0–40 s). (<b>b</b>) Time–domain waveform of state variable <math display="inline"><semantics> <msub> <mi>x</mi> <mn>4</mn> </msub> </semantics></math> in time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> </mrow> </semantics></math> (30–40 s). (<b>c</b>) The phase diagram of transient double-scroll chaotic attractor in time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> </mrow> </semantics></math> (0–20 s). (<b>d</b>) Phase diagram of stable periodic limit cycles in time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> </mrow> </semantics></math> (20–40 s).</p>
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<p>Circuit schematic diagram of incommensurate system (<a href="#FD13-fractalfract-09-00018" class="html-disp-formula">13</a>).</p>
Full article ">Figure 14
<p>Coexistence attractor types of incommensurate system with different parameter <span class="html-italic">f</span> in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> plane. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> plane phase diagram. (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> plane phase diagram. (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>4</mn> </msub> </mrow> </semantics></math> plane phase diagram.</p>
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<p>Synchronization curve of state variables for incommensurate system. (<b>a</b>) Synchronization curve of state variables <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> </semantics></math>. (<b>b</b>) Synchronization curve of state variables <math display="inline"><semantics> <msub> <mi>x</mi> <mn>2</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>s</mi> <mn>2</mn> </mrow> </msub> </semantics></math>. (<b>c</b>) Synchronization curve of state variables <math display="inline"><semantics> <msub> <mi>x</mi> <mn>3</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>s</mi> <mn>3</mn> </mrow> </msub> </semantics></math>. (<b>d</b>) Synchronization curve of state variables <math display="inline"><semantics> <msub> <mi>x</mi> <mn>4</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>s</mi> <mn>4</mn> </mrow> </msub> </semantics></math>.</p>
Full article ">Figure 16
<p>Control input curve of incommensurate system. (<b>a</b>) Control input <math display="inline"><semantics> <msub> <mi>u</mi> <mn>1</mn> </msub> </semantics></math> curve. (<b>b</b>) Control input <math display="inline"><semantics> <msub> <mi>u</mi> <mn>2</mn> </msub> </semantics></math> curve. (<b>c</b>) Control input <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>3</mn> </msub> </mrow> </semantics></math> curve. (<b>d</b>) Control input <math display="inline"><semantics> <msub> <mi>u</mi> <mn>4</mn> </msub> </semantics></math> curve.</p>
Full article ">Figure 17
<p>Incommensurate system sliding surface motion curve. (<b>a</b>) <math display="inline"><semantics> <msub> <mi>s</mi> <mn>1</mn> </msub> </semantics></math> sliding surface curve. (<b>b</b>) <math display="inline"><semantics> <msub> <mi>s</mi> <mn>2</mn> </msub> </semantics></math> sliding surface curve. (<b>c</b>) <math display="inline"><semantics> <msub> <mi>s</mi> <mn>3</mn> </msub> </semantics></math> sliding surface curve. (<b>d</b>) <math display="inline"><semantics> <msub> <mi>s</mi> <mn>4</mn> </msub> </semantics></math> sliding surface curve.</p>
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12 pages, 2318 KiB  
Article
An Adsorption Model Considering Fictitious Stress
by Xiaohua Tan, Xinjian Ma, Xiaoping Li and Yilong Li
Fractal Fract. 2025, 9(1), 17; https://doi.org/10.3390/fractalfract9010017 - 30 Dec 2024
Viewed by 287
Abstract
The adsorption of coalbed methane alters the pore structure of reservoirs, subsequently affecting the coal seam’s gas adsorption capacity. However, traditional gas adsorption models often neglect this crucial aspect. In this article, we introduce a fractal capillary bundle model that accounts for the [...] Read more.
The adsorption of coalbed methane alters the pore structure of reservoirs, subsequently affecting the coal seam’s gas adsorption capacity. However, traditional gas adsorption models often neglect this crucial aspect. In this article, we introduce a fractal capillary bundle model that accounts for the expansion of coal seam adsorption. We utilize curvature fractal dimension and capillary fractal dimension to characterize the complexity of the coal seam’s pore structure. By incorporating the concept of fictitious stress, we have described the relationship between gas adsorption, matrix porosity, and permeability changes. We have developed a model that describes the changes in matrix porosity and permeability during the gas adsorption process. After fitting this model to experimental data, it demonstrated high accuracy in predictions. Furthermore, our investigation into how factors such as curvature fractal dimension, capillary fractal dimension, and fictitious stress influence gas adsorption capacity reveals several key findings. Firstly, the specific surface area within the pore structure of coal seams is the primary factor controlling gas adsorption capacity. Secondly, the virtual stress generated during the gas adsorption process alters the coal seam’s maximum gas adsorption capacity, a factor that cannot be overlooked. Lastly, we found that gas adsorption primarily affects the gas migration process, while under high-pressure conditions, gas desorption does not cause significant changes in the matrix porosity and permeability. Full article
(This article belongs to the Section Engineering)
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Figure 1

Figure 1
<p>The capillary bundle model.</p>
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<p>Adsorption model fitting results. (<b>a</b>) Fit data 1; (<b>b</b>) Fit data 2; (<b>c</b>) Fit data 3; (<b>d</b>) Fit data 4; (<b>e</b>) Fit data 5; (<b>f</b>) Fit data 6.</p>
Full article ">Figure 2 Cont.
<p>Adsorption model fitting results. (<b>a</b>) Fit data 1; (<b>b</b>) Fit data 2; (<b>c</b>) Fit data 3; (<b>d</b>) Fit data 4; (<b>e</b>) Fit data 5; (<b>f</b>) Fit data 6.</p>
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<p>Curve of methane adsorption under the influence of curvature and capillary fractal dimension.</p>
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<p>The trend of maximum methane adsorption under different fractal dimensions.</p>
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<p>Curve of adsorption capacity variation amplitude with pressure variation.</p>
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<p>Gas adsorption capacity and fictitious stress variation curve with pressure.</p>
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<p>Adsorption model fitting results.</p>
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<p>Variation curve of porosity with pressure under gas adsorption.</p>
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<p>Pressure-dependent permeability curve under gas adsorption.</p>
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18 pages, 3859 KiB  
Article
The Use of Artificial Intelligence in Data Analysis with Error Recognitions in Liver Transplantation in HIV-AIDS Patients Using Modified ABC Fractional Order Operators
by Hasib Khan, Jehad Alzabut, D. K. Almutairi and Wafa Khalaf Alqurashi
Fractal Fract. 2025, 9(1), 16; https://doi.org/10.3390/fractalfract9010016 - 30 Dec 2024
Viewed by 264
Abstract
In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments and control. The model is structured with eight classes and a modified Atangana–Baleanu [...] Read more.
In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments and control. The model is structured with eight classes and a modified Atangana–Baleanu derivative in Caputo’s sense. The model has several interlinking parameters which show the rates of transmission between classes. We assumed natural death and death on the disease severity in patients. The model was analyzed mathematically as well as computationally. In the mathematical aspects, R0 was plotted for different cases which play a vital role in the infection spread in the population. The model was passed through qualitative analysis for the existence of solutions and stability results. A computational scheme is developed for the model and is applied for the numerical results to analyze the intricate dynamics of the infection. It has been observed that there is a good resemblance in the results for the correlation between the hospitalization, vaccination and recovery rate of the patients. These are reaffirmed with the neural networking tools for the regression, probability, clustering, mean square error and fitting data. Full article
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<p>HIV-AIDS transmission and treatment with liver transplant and positive recovery.</p>
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<p>Flowchart for liver transplantation in HIV-AIDS infected individuals of (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>).</p>
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<p>Different graphs of <math display="inline"><semantics> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </semantics></math> for variant parametric values. (<b>a</b>) Graphical representation of <math display="inline"><semantics> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </semantics></math> for the variant <math display="inline"><semantics> <mi>β</mi> </semantics></math> and <math display="inline"><semantics> <mi>γ</mi> </semantics></math>; (<b>b</b>) Graphical representation of <math display="inline"><semantics> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </semantics></math> for the variant <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>μ</mi> </semantics></math>; (<b>c</b>) Graphical representation of <math display="inline"><semantics> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </semantics></math> for the variant <math display="inline"><semantics> <mi>β</mi> </semantics></math> and <math display="inline"><semantics> <mi>σ</mi> </semantics></math>; (<b>d</b>) Graphical representation of <math display="inline"><semantics> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </semantics></math> for the variant <math display="inline"><semantics> <mi>β</mi> </semantics></math> and <math display="inline"><semantics> <mi>μ</mi> </semantics></math>; (<b>e</b>) Graphical representation of <math display="inline"><semantics> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </semantics></math> versus <math display="inline"><semantics> <mi mathvariant="sans-serif">Λ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>f</b>) Graphical representation of <math display="inline"><semantics> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </semantics></math> versus <math display="inline"><semantics> <mi>μ</mi> </semantics></math> and <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math>.</p>
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<p>HIV-AIDS model (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>) for fractional order 0.98. (<b>a</b>) The time-varying number of susceptible entities <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">S</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>η</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>b</b>) The time-varying quantity of exposed population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">E</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>η</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>c</b>) The time-varying number of infected individuals <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">I</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>η</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>d</b>) The time-varying number of hospitalized entities <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">H</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>η</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>e</b>) The time-varying number of vaccinated people <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">V</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>f</b>) The time-varying number of treated entities <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">T</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>g</b>) The time-varying number of susceptible entities <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">V</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>h</b>) The time-varying number of susceptible entities <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">R</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>.</p>
Full article ">Figure 4 Cont.
<p>HIV-AIDS model (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>) for fractional order 0.98. (<b>a</b>) The time-varying number of susceptible entities <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">S</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>η</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>b</b>) The time-varying quantity of exposed population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">E</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>η</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>c</b>) The time-varying number of infected individuals <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">I</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>η</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>d</b>) The time-varying number of hospitalized entities <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">H</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>η</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>e</b>) The time-varying number of vaccinated people <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">V</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>f</b>) The time-varying number of treated entities <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">T</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>g</b>) The time-varying number of susceptible entities <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">V</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>h</b>) The time-varying number of susceptible entities <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">R</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>.</p>
Full article ">Figure 5
<p>Impacts of the parameters over different classes of Model (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>) for fractional order 0.98. (<b>a</b>) Susceptible population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">S</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>θ</mi> </semantics></math>; (<b>b</b>) The impact of <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> over the <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">S</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> with respect to <span class="html-italic">t</span>; (<b>c</b>) Illustration of the impact of <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> over the exposed population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">E</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> with respect to <span class="html-italic">t</span>; (<b>d</b>) Impact of the <math display="inline"><semantics> <mi>θ</mi> </semantics></math> over the <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">E</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> population; (<b>e</b>) HIV infected population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">I</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>θ</mi> </semantics></math>; (<b>f</b>) The timely variation in <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">C</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>g</b>) Hospitalized population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">H</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>θ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>h</b>) Recovered population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">R</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>η</mi> </semantics></math>.</p>
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<p>Impacts of the parameters over different classes of Model (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>) for fractional order 0.98. (<b>a</b>) Susceptible population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">S</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>θ</mi> </semantics></math>; (<b>b</b>) The impact of <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> over the <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">S</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> with respect to <span class="html-italic">t</span>; (<b>c</b>) Illustration of the impact of <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> over the exposed population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">E</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> with respect to <span class="html-italic">t</span>; (<b>d</b>) Impact of the <math display="inline"><semantics> <mi>θ</mi> </semantics></math> over the <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">E</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> population; (<b>e</b>) HIV infected population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">I</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>θ</mi> </semantics></math>; (<b>f</b>) The timely variation in <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">C</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>g</b>) Hospitalized population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">H</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>θ</mi> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>; (<b>h</b>) Recovered population <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">R</mi> <mo>⊛</mo> </msub> <mo>)</mo> </mrow> </semantics></math> under the influence of <math display="inline"><semantics> <mi>η</mi> </semantics></math>.</p>
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<p>Neural networking for the liver transplantation in the HIV-AIDS dynamical system for the modified ABC sense of derivative (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>). (<b>a</b>) Training of data for the model (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>); (<b>b</b>) Mean square error analysis with best validation performance <math display="inline"><semantics> <mrow> <mn>9.2151</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </mrow> </semantics></math>; (<b>c</b>) Clustering of cirrhosis and hospitalized patients; (<b>d</b>) Error analysis of the data.</p>
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<p>Neural networking for the liver transplantation in the HIV-AIDS patients of model (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>). (<b>a</b>) Error analysis in the hospitalized population by histograms for 20 bins; (<b>b</b>) Correlation between the cirrhosis and hospitalized patients by means of histogram; (<b>c</b>) Neutron index for the input weights visualization; (<b>d</b>) Regression of the HIV-AIDS data for the model (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>).</p>
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<p>Neural networking for the liver transplantation in the HIV-AIDS patients of model (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>). (<b>a</b>) Error analysis in the hospitalized population by histograms for 20 bins; (<b>b</b>) Correlation between the cirrhosis and hospitalized patients by means of histogram; (<b>c</b>) Neutron index for the input weights visualization; (<b>d</b>) Regression of the HIV-AIDS data for the model (<a href="#FD1-fractalfract-09-00016" class="html-disp-formula">1</a>).</p>
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18 pages, 1416 KiB  
Article
Fractional-Order Sliding Mode Terrain-Tracking Control of Autonomous Underwater Vehicle with Sparse Identification
by Zheping Yan, Lichao Hao, Qiqi Pi and Tao Chen
Fractal Fract. 2025, 9(1), 15; https://doi.org/10.3390/fractalfract9010015 - 30 Dec 2024
Viewed by 223
Abstract
This paper has addressed the terrain-following problem of an autonomous underwater vehicle for widely used ocean survey missions. Considering the terrain feature description with limited sensing ability in underwater scenarios, a vertically installed multi-beam sonar and a downward single-beam echo sounder are equipped [...] Read more.
This paper has addressed the terrain-following problem of an autonomous underwater vehicle for widely used ocean survey missions. Considering the terrain feature description with limited sensing ability in underwater scenarios, a vertically installed multi-beam sonar and a downward single-beam echo sounder are equipped to obtain seafloor detecting data online, and a local polynomial fitting algorithm is carried out with a receding horizon strategy in order to generate a proper tracking path to keep the desired height above the sea bottom. With the construction of the autonomous underwater vehicle dynamic model in the North East Down frame regarding the vertical plane, an online sparse identification algorithm is implemented to obtain the model parameters during the diving process. Then, a fractional-order sliding mode controller is proposed to enable accurate tracking of the path planned and Lyapunov-based theory is utilized to prove the stability of the control algorithm. With the simulation results, the tracking effectiveness of the fractional-order sliding mode controller with in situ identification is verified. Full article
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<p>Control purpose of terrain tracking.</p>
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<p>Sonar configuration.</p>
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<p>Coordinate frames of AUV.</p>
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<p>Receding path planning example.</p>
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<p>Sparse identification process for controller design.</p>
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<p>State value comparisons in surge.</p>
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<p>Elevator input for identification.</p>
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<p>State value comparisons in pitch.</p>
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<p>AUV trajectory during surge dynamic idenfitication.</p>
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<p>AUV trajectory during pitch dynamic idenfitication.</p>
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<p>AUV tracking path.</p>
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<p>AUV tracking error.</p>
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<p>Receding planned tracking path.</p>
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<p>Tracking error with sliding mode control.</p>
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24 pages, 4168 KiB  
Article
Multifractal Characteristics and Information Flow Analysis of Stock Markets Based on Multifractal Detrended Cross-Correlation Analysis and Transfer Entropy
by Wenjuan Zhou, Jingjing Huang and Maofa Wang
Fractal Fract. 2025, 9(1), 14; https://doi.org/10.3390/fractalfract9010014 - 30 Dec 2024
Viewed by 262
Abstract
Understanding cross-correlation and information flow between stocks is crucial for stock market analysis. However, traditional methods often struggle to capture financial markets’ complex and multifaceted dynamics. This paper presents a robust combination of techniques, integrating three advanced methods: Multifractal Detrended Cross-Correlation Analysis (MFDCCA), [...] Read more.
Understanding cross-correlation and information flow between stocks is crucial for stock market analysis. However, traditional methods often struggle to capture financial markets’ complex and multifaceted dynamics. This paper presents a robust combination of techniques, integrating three advanced methods: Multifractal Detrended Cross-Correlation Analysis (MFDCCA), transfer entropy (TE), and complex networks. To address inherent non-stationarity and noise in financial data, we employ Ensemble Empirical Mode Decomposition (EEMD) for preprocessing, which helps reduce noise and handle non-stationary effects. The application and effectiveness of this combination of techniques are demonstrated through examples, uncovering significant multifractal properties and long-range cross correlations among the stocks studied. This combination of techniques also captures the magnitude and direction of information flow between stocks. This holistic analysis provides valuable insights for investors and policymakers, enhancing their understanding of stock market behavior and supporting better-informed portfolio decisions and risk management strategies. Full article
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<p>The flowchart of the combination of techniques.</p>
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<p>EEMD decomposition results of AAPL.</p>
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<p>The significance testing of IMF based on lnE and lnT.</p>
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<p>Normalized original time series 3D chart of eight stocks.</p>
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<p>Normalized reconstructed time series 3D chart of eight stocks.</p>
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<p>DCCA coefficients of JPM versus the other seven stocks.</p>
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<p>The log–log plot of fluctuation function <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mi>q</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> vs. time series scale s.</p>
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<p>The cross-correlation Hurst exponents <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math> vs. <span class="html-italic">q</span>.</p>
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<p>Singularity strength <math display="inline"><semantics> <mi>α</mi> </semantics></math> vs. multifractal spectrum <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </semantics></math>.</p>
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<p>The histogram line plot of <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Δ</mi> <mi>H</mi> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Δ</mi> <mi>α</mi> </mrow> </semantics></math>.</p>
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<p>Transfer entropy heatmap between the eight stocks.</p>
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<p>Directed weighted network diagram with transfer entropy as weights.</p>
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19 pages, 12105 KiB  
Article
Particle Flow Simulation of the Mechanical Properties and Fracture Behavior of Multi-Mineral Rock Models with Different Fractal Dimensions
by Run Shi and Huaiguang Xiao
Fractal Fract. 2025, 9(1), 13; https://doi.org/10.3390/fractalfract9010013 - 29 Dec 2024
Viewed by 349
Abstract
To study the effects of rock models with different fractal dimensions on their mechanical properties and fracture behavior, three representative numerical rock models, including the digital texture model, the Voronoi polygon model, and the Weibull distribution model, are established in this paper. These [...] Read more.
To study the effects of rock models with different fractal dimensions on their mechanical properties and fracture behavior, three representative numerical rock models, including the digital texture model, the Voronoi polygon model, and the Weibull distribution model, are established in this paper. These models are used to simulate the structure of multi-mineral rocks and to investigate the influence of fractal dimensions on the mechanical properties and fracture behavior of rocks. Uniaxial compression numerical tests are carried out on 2D and 3D intact rocks under different fractal dimensions using the particle flow simulation method. The relationship between fractal dimensions and uniaxial compression strength and fracture behavior was analyzed. The results show that the fractal dimension of the Weibull distribution model is the largest, followed by the digital texture model, and the fractal dimension of the Voronoi polygon model is the smallest. With the increase in fractal dimension, the uniaxial compressive strength of intact rocks increases significantly, and their relationship is approximately linear. The influence of fractal dimension on rock strength shows a similar trend in both the 2D and 3D models. This study provides a new perspective for the application of fractal dimensions in multi-mineral rock models. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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<p>Rock samples: (<b>a</b>) cylinder sample; (<b>b</b>) digital rock model; (<b>c</b>) mineral compositions of rock samples.</p>
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<p>2D numerical model of PFC for multi-mineral rocks: (<b>a</b>) texture model, (<b>b</b>) Voronoi model, (<b>c</b>) Weibull distribution model.</p>
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<p>3D numerical model of PFC for multi-mineral rocks: (<b>a</b>) texture model, (<b>b</b>) Voronoi model, (<b>c</b>) Weibull distribution model.</p>
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<p>Bonding model between balls in PFC (adapted from [<a href="#B20-fractalfract-09-00013" class="html-bibr">20</a>,<a href="#B21-fractalfract-09-00013" class="html-bibr">21</a>]): (<b>a</b>) particle movements after breakage of parallel bond (<b>b</b>) linear parallel bond model.</p>
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<p>Calibration of deformation and failure for experiment and numerical results.</p>
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<p>Calibration of failure modes: (<b>a</b>) experiment result, (<b>b</b>) numerical result.</p>
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<p>Fractal dimensions of minerals in the 2D texture model, Voronoi model, and Weibull model.</p>
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<p>Fractal dimensions of minerals in the 3D texture model, Voronoi model, and Weibull model.</p>
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<p>Stress–strain of 2D models with different fractal dimensions.</p>
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<p>Failure modes of 2D models with different fractal dimensions.</p>
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<p>The crack number in 2D models with different fractal dimensions.</p>
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<p>Stress–strain of 3D models with different fractal dimensions.</p>
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<p>Failure modes of 3D textured rock models.</p>
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<p>Failure modes of 3D Voronoi rock models.</p>
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<p>Failure modes of 3D Weibull rock models.</p>
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<p>Relationship between fractal dimensions and uniaxial compression strength.</p>
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20 pages, 11238 KiB  
Article
Analysis of Nanostructures and Wettability of Marine Shale in Southern China, Based on Different Fractal Models
by Yang Wang, Baoyuan Zhong, Yunsheng Zhang, Yanming Zhu and Meng Wang
Fractal Fract. 2025, 9(1), 12; https://doi.org/10.3390/fractalfract9010012 - 29 Dec 2024
Viewed by 402
Abstract
The wetting behavior of shale oil and gas on shale surfaces is determined by the interplay of organic matter (OM), mineral composition, and the intricate pore network structure of the shale. In this paper, the sensitivity responses of the Frenkel–Halsey–Hill (FHH), Neimark (NM), [...] Read more.
The wetting behavior of shale oil and gas on shale surfaces is determined by the interplay of organic matter (OM), mineral composition, and the intricate pore network structure of the shale. In this paper, the sensitivity responses of the Frenkel–Halsey–Hill (FHH), Neimark (NM), and Wang–Li (WL) fractal models to marine shale with varying material components are analyzed, based on liquid nitrogen adsorption experiments and fractal theory. The wettability evolution model of shale with different maturity stages is established to reveal the heterogeneity characteristics of wettability in shale with complex pore structures. Results show that the NM and WL models offer distinct advantages in evaluating the reservoir structure of shale oil and gas resources. The existence of large-diameter pores is conducive to the homogeneous development of the pore structure. The coupling relationship between pore volume, pore size and pore specific surface affects the fractal characteristics of the pore structure. For highly overmature shale, with an increase in fractal dimension, the wettability of shale changes from neutral-wet to water-wet. For ultramature shale, the higher heterogeneity of the pore structure leads to larger contact angles, causing the wettability to transition gradually from water-wet to oil-wet. In addition, the sensitivity analysis of wettability to fractal structure parameters is examined from the perspective of OM maturation and evolution. Full article
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<p>Permian strata release and sampling point map.</p>
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<p>Stacking diagram of mineral component contents.</p>
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<p>FE-SEM images of different types of pores in the LMS-5 and LMS-8 shale samples. (<b>a</b>) Organic matter (OM). (<b>b</b>) Convoluted OM pores in an organic area with clay. (<b>c</b>) Intragranular pores of strawberry-like pyrite after shedding. (<b>d</b>) Spongy OM pores in an organic-rich area. (<b>e</b>) Intraparticle pores in clay. (<b>f</b>) Interparticle pores in clay. (<b>g</b>) Organic pores with different particle sizes. (<b>h</b>) Intraparticle pores in quartz.</p>
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<p>Joint PSD based on LP-N<sub>2</sub> GA and MIP.</p>
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<p>Hysteresis-loop image based on the nitrogen adsorption–desorption isotherm. (<b>a</b>) Typical samples of <span class="html-italic">H<sub>3</sub></span> type (IUPAC) hysteresis loops are shown. (<b>b</b>) Typical samples of <span class="html-italic">H<sub>2</sub></span> type hysteresis loops are shown.</p>
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<p>Illustrative diagrams of liquid nitrogen adsorption isotherms for shale samples. (<b>a</b>) Fractal dimensions (<span class="html-italic">D<sub>f</sub></span>, <span class="html-italic">D<sub>f1</sub></span>, <span class="html-italic">D<sub>f2</sub></span>, <span class="html-italic">D<sub>f3</sub></span>) calculated across different relative pressure ranges based on the FHH model. (<b>b</b>) Power-law fitting curves for calculating <span class="html-italic">m</span> and <span class="html-italic">b</span> values in the NM and WL models. (<b>c</b>) Fractal dimension (<span class="html-italic">D<sub>n</sub></span>) calculated using the NM model. (<b>d</b>) Slope fitting for fractal dimension determination in the WL model.</p>
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<p>Differences in fractal dimension values calculated using various fractal models.</p>
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<p>CAs on argon-ion-polished surfaces of marine shale samples. (<b>a</b>) Distribution of CAs across nine shale samples. (<b>b</b>) The CA of a highly hydrophilic shale sample, represented by LMS<span class="html-italic">-</span>2. (<b>c</b>) The CA of an oil-wet shale sample, represented by LMS<span class="html-italic">-</span>9.</p>
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<p>The connections between fractal dimensions derived from various fractal models and the material constituents. (<b>a</b>) The connection between TOC and fractal dimensions from the FHH, NM, and WL models. (<b>b</b>) The connection between quartz content and fractal dimensions from the FHH, NM, and WL models. (<b>c</b>) The connection between clay mineral content and fractal dimensions from the FHH, NM, and WL models.</p>
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<p>Relationships between fractal dimensions calculated by different models and pore structure parameters obtained from liquid nitrogen adsorption experiments. (<b>a</b>) Response of different fractal dimensions to total pore volume. (<b>b</b>) Response of different fractal dimensions to average pore size. (<b>c</b>) Response of different fractal dimensions to specific surface area (BET).</p>
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<p>Relationship between shale CA and fractal dimensions. (<b>a</b>) Correlation between CA and fractal characteristics for highly overmature shale. (<b>b</b>) Correlation between contact angle and fractal characteristics for ultramature shale.</p>
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<p>Evolutionary model of wettability characteristics and pore fractal features in overmature shale.</p>
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25 pages, 7489 KiB  
Article
Pore Structure and Fractal Characteristics of Inter-Layer Sandstone in Marine–Continental Transitional Shale: A Case Study of the Upper Permian Longtan Formation in Southern Sichuan Basin, South China
by Jianguo Wang, Jizhen Zhang, Xiao Xiao, Ya’na Chen and Denglin Han
Fractal Fract. 2025, 9(1), 11; https://doi.org/10.3390/fractalfract9010011 - 29 Dec 2024
Viewed by 375
Abstract
With the evolution of unconventional oil and gas exploration concepts from source rocks and reservoirs to carrier beds, the inter-layer sandstone carrier bed within marine–continental transitional shale strata has emerged as a significant target for oil and gas exploration. The inter-layer sandstone is [...] Read more.
With the evolution of unconventional oil and gas exploration concepts from source rocks and reservoirs to carrier beds, the inter-layer sandstone carrier bed within marine–continental transitional shale strata has emerged as a significant target for oil and gas exploration. The inter-layer sandstone is closely associated with the source rock and differs from conventional tight sandstone in terms of sedimentary environment, matrix composition, and the characteristics of reservoir microscopic pore development. Preliminary exploration achievements display that the inter-layer sandstone is plentiful in gas content and holds promising prospects for exploration and development. Consequently, it is essential to investigate the gas-rich accumulation theory specific to the inter-layer sandstone reservoir in transitional facies. Pore development characteristics and heterogeneity are crucial aspects of oil and gas accumulation research, as they influence reservoir seepage performance and capacity. This paper employs total organic carbon analysis, X-ray diffraction, rock thin section examination, field emission scanning electron microscopy, physical analysis, high-pressure mercury intrusion analysis, gas adsorption experiments, and fractal theory to explore the reservoir development characteristics of the sandstone samples from the Upper Permian marine–continental transitional facies Longtan Formation in the southern Sichuan Basin. It also attempts to combine high-pressure mercury intrusion analysis and gas adsorption experiments to describe the structural and fractal characteristics of pores at different scales in a segmented manner. The findings reveal that the sandstone type of the Longtan Formation is mainly lithic sandstone. The pore size distribution of the sandstone primarily falls below 30 nm and above 1000 nm, with the main pore types being inter-granular pores and micro-fractures in clay minerals. The pore volume and specific surface area are largely attributed to the micropores and mesopores of clay minerals. The pore morphology is complex, exhibiting strong heterogeneity, predominantly characterized by slit-like and ink bottle-like features. Notably, there are discernible differences in reservoir structural characteristics and homogeneity between muddy sandstone and non-muddy sandstone. The pore morphology is complex, exhibiting strong heterogeneity, predominantly characterized by slit-like and ink bottle-like features. Notably, there are discernible differences in reservoir structural characteristics and homogeneity between muddy sandstone and non-muddy sandstone. Full article
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<p>(<b>a</b>) Structural background map of Sichuan Basin [<a href="#B2-fractalfract-09-00011" class="html-bibr">2</a>]; (<b>b</b>) paleogeographic map and Sampling Well Locations of the Longtan Formation in the Study Area [<a href="#B47-fractalfract-09-00011" class="html-bibr">47</a>]; and (<b>c</b>) stratigraphic Column of Longtan Formation [<a href="#B47-fractalfract-09-00011" class="html-bibr">47</a>].</p>
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<p>Comparison diagram of sandstone component content of Longtan Formation.</p>
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<p>(<b>a</b>) XRD pattern; and (<b>b</b>) sandstone type triangle diagram of Longtan Formation [<a href="#B18-fractalfract-09-00011" class="html-bibr">18</a>,<a href="#B27-fractalfract-09-00011" class="html-bibr">27</a>,<a href="#B41-fractalfract-09-00011" class="html-bibr">41</a>,<a href="#B55-fractalfract-09-00011" class="html-bibr">55</a>].</p>
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<p>(<b>a</b>–<b>f</b>) are sandstone thin section identification of Longtan Formation: (<b>a</b>) the linear contact relationship of minerals and brittle minerals micro-fracture, sample ZG1-8, orthogonal light; (<b>b</b>) directional arrangement of minerals and cementation of carbonate minerals, sample ZG1-6, orthogonal light; (<b>c</b>) muddy components and feldspar dissolution, sample N242-3; (<b>d</b>) authigenic quartz, sample ZG1-1, orthogonal light; (<b>e</b>) siderite enrichment, sample ZG1-3, orthogonal light; and (<b>f</b>) carbonaceous fragment filled with minerals components, sample N242-3. (<b>g</b>–<b>o</b>) are pore types of sandstone reservoir in Longtan Formation: (<b>g</b>) inter-particle pores and inter-crystalline pores of clay minerals, sample GS133-1; (<b>h</b>) clay minerals micro-fracture, sample ZG1-1; (<b>i</b>) brittle minerals intra-particle dissolution pores, sample ZG1-2; (<b>j</b>) quartz distributed inside clay minerals, sample N242-3; (<b>k</b>) inter-crystalline pores in brittle mineral aggregates, sample YJ1-6; (<b>l</b>) brittle minerals inter-particle fracture, sample GS133-1; (<b>m</b>) brittle minerals micro-fracture, sample ZG1-1; and (<b>n</b>,<b>o</b>), carbonaceous fragment and wood fiber filled in it, sample ZG1-1 and ZG1-8.</p>
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<p>Comparison diagram of pore structure parameters of sandstone in Longtan Formation.</p>
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<p>Mercury saturation curves of different types of Longtan formation sandstone.</p>
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<p>N<sub>2</sub> adsorption–desorption isotherms of Longtan formation sandstone.</p>
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<p>CO<sub>2</sub> adsorption–desorption isotherms of Longtan formation sandstone.</p>
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<p>The pore size to PV joint characterization diagram of different grain size sandstone in Longtan Formation.</p>
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<p>Relationship diagram between lg (<span class="html-italic">S<sub>Hg</sub></span>) and lg (<span class="html-italic">P<sub>c</sub></span>) based on HPMI data.</p>
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<p>Relationship curve of ln(<span class="html-italic">V</span>) to ln (ln(<span class="html-italic">P</span><sub>0</sub>/<span class="html-italic">P</span>)) based on N<sub>2</sub> adsorption experimental data.</p>
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<p>(<b>a</b>) Correlation heat map between different parameters of Longtan Formation sandstone; and (<b>b</b>) interrelation between the PV and SSA and TOC.</p>
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<p>Relationship between pore structure parameters and porosity (<b>a</b>,<b>b</b>) and permeability (<b>c</b>,<b>d</b>) in different sandstone types.</p>
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<p>Correlation between pore fractal dimensions of different scale pores and grain size (<b>a</b>) and pore structure parameters (<b>b</b>–<b>d</b>).</p>
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<p>Correlation between pore fractal dimensions of different scale pores and porosity (<b>a</b>) and permeability (<b>b</b>–<b>d</b>).</p>
Full article ">Figure 16
<p>Interrelation between pore fractal dimensions of different scale pores and clay minerals, quartz, carbonate minerals, and TOC (<b>a</b>–<b>c</b>,<b>e</b>); and correlation between TOC and fractal dimensions of different scale pores in muddy and non-muddy sandstone (<b>d</b>,<b>f</b>).</p>
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16 pages, 6041 KiB  
Article
Relating the Morphology of Bipolar Neurons to Fractal Dimension
by Bret Brouse, Jr., Conor Rowland and Richard P. Taylor
Fractal Fract. 2025, 9(1), 9; https://doi.org/10.3390/fractalfract9010009 - 28 Dec 2024
Viewed by 220
Abstract
By analyzing reconstructed three-dimensional images of retinal bipolar neurons, we show that their dendritic arbors weave through space in a manner that generates fractal-like behavior quantified by an ‘effective’ fractal dimension. Examining this fractal weave along with traditional morphological parameters reveals a dependence [...] Read more.
By analyzing reconstructed three-dimensional images of retinal bipolar neurons, we show that their dendritic arbors weave through space in a manner that generates fractal-like behavior quantified by an ‘effective’ fractal dimension. Examining this fractal weave along with traditional morphological parameters reveals a dependence of arbor fractal dimension on the summation of the lengths of the arbor’s dendrites. We discuss the implications of this behavior for healthy neurons and also for the morphological deterioration of unhealthy neurons in response to diseases. Full article
(This article belongs to the Special Issue Fractal Analysis in Biology and Medicine)
Show Figures

Figure 1

Figure 1
<p>A schematic illustration of the cross section of the human eye. The layer of photoreceptors (shaded dark blue) is positioned behind the layer of retinal neurons (shaded red). A retinal implant (shaded cyan) is positioned to replace diseased photoreceptors (<b>A</b>). A zoom-in schematic illustration of the retina. The bipolar neurons are shaded yellow to distinguish them from other retinal neurons (shaded red) and the photoreceptors (shaded blue) (<b>B</b>). A side view image of an individual retinal bipolar cell extracted from a rodent retina. A spherical shape has been added to the image to represent the neuron’s soma. One dendrite extends from the soma and branches out to form the neuron arbor (<b>C</b>). A rotated view of the bipolar cell is shown in (<b>D</b>). Scale bars in panels (<b>C</b>,<b>D</b>) represent approximately 10 µm.</p>
Full article ">Figure 2
<p>A mathematical H-Tree fractal with <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>F</mi> </mrow> </msub> </mrow> </semantics></math> = 1.4 (<b>A</b>) and a typical dendritic arbor of a retinal bipolar neuron shown using the orientation used in <a href="#fractalfract-09-00009-f001" class="html-fig">Figure 1</a>C (<b>B</b>). The branch level of both fractal structures is colored such that a darker shade of blue corresponds to a higher branch level. Histograms for an H-Tree (<b>C</b>) and neuron (<b>D</b>) plotting number of branches <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>F</mi> </mrow> </msub> </mrow> </semantics></math> with a given value of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>F</mi> </mrow> </msub> <mo>/</mo> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>. Equivalent plots of panels (<b>C</b>,<b>D</b>) in log–log space are shown in (<b>E</b>,<b>F</b>) with uncertainty bars shown for the neuron in panel (<b>F</b>). The red lines in panels (<b>C</b>–<b>F</b>) correspond to the power law behaviors described in the main text. The distribution of weaving angles for each fractal structure is shown in (<b>G</b>,<b>H</b>). A schematic of how the weave angle, <span class="html-italic">θ</span>, and forking length, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>F</mi> </mrow> </msub> <mo>,</mo> </mrow> </semantics></math> are determined is shown in the inset of panel (<b>G</b>). Accounting for angles and lengths using a box counting algorithm of each is shown in (<b>I</b>,<b>J</b>). The slope of the fit in (<b>J</b>) is approximately −1.47. Note that <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math> in panel (<b>D</b>) corresponds to the largest forking length whereas in panel (<b>J</b>), <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math> corresponds to the largest box size in the box counting algorithm.</p>
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<p>A visual comparison of a natural bipolar neuron (<b>left</b>) and a modified version in which the branches have been straightened (<b>right</b>). The box length corresponds to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>F</mi> </mrow> </msub> </mrow> </semantics></math>/<math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math> = 0.05 in both cases.</p>
Full article ">Figure 4
<p>Three reconstructed bipolar neurons with increasing fractal dimension (<b>top</b> row, left to right). A histogram of the normalized count <span class="html-italic">n</span> for a given fractal dimension <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math> for the total dataset (<b>bottom</b> row).</p>
Full article ">Figure 5
<p>Three reconstructed bipolar neurons with increasing arbor radius, <span class="html-italic">R</span> (<b>top</b> row, left to right). A histogram of the normalized count <span class="html-italic">n</span> for a given arbor radius <span class="html-italic">R</span> for the total dataset (<b>bottom</b> row). For the same dataset, convex hull volume given arbor radius is plotted with a fitting curve of approximately degree 3 (<b>bottom</b> row histogram, inset). We note that the branch width is constant across all neurons—the “thicker” look for some neurons is due to zooming in for neurons with smaller arbor radii than their counterparts.</p>
Full article ">Figure 6
<p>Fractal dimension <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math> plotted against arbor radius <span class="html-italic">R</span> for the total dataset, grouped according to three size regimes: 0–10 µm (violet), 10–20 µm (blue), and &gt;20 µm (cyan). These groups contain 16 (small), 342 (medium), and 95 (large) neurons. The dashed lines represent the mean <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math> values for each group. The inset plots the total dendritic arbor length <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> </mrow> </semantics></math> against arbor radius <span class="html-italic">R</span>, with a linear fit showing a mild correlation.</p>
Full article ">Figure 7
<p>Fractal dimension <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math> plotted against total dendritic arbor length <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> </mrow> </semantics></math> for the entire dataset color coded by arbor radius <span class="html-italic">R</span> in the same categories as <a href="#fractalfract-09-00009-f006" class="html-fig">Figure 6</a>. Linear fit curves are shown for each group, showing correlations between <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math> and <span class="html-italic">R</span> within each group. For the small <span class="html-italic">R</span> group, a steep line is observed, whereas the slopes of the other two groups are lower. Grid lines show that all <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math> values fall between the range of 1.25 and 1.75.</p>
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<p>Graph of the fractal dimension, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math>, versus fork length, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>F</mi> </mrow> </msub> </mrow> </semantics></math>, and weave angle, <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math>. Here we plot the values of the <span class="html-italic">average</span> fractal dimension, <math display="inline"><semantics> <mrow> <mo>⟨</mo> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>⟩</mo> </mrow> </semantics></math>, <span class="html-italic">average</span> fork length, <math display="inline"><semantics> <mrow> <mo>⟨</mo> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>F</mi> </mrow> </msub> <mo>⟩</mo> </mrow> </semantics></math>, and <span class="html-italic">average</span> weave angle, <math display="inline"><semantics> <mrow> <mo>⟨</mo> <mi>θ</mi> <mo>⟩</mo> </mrow> </semantics></math>, for the small (violet), medium (blue), and large (cyan) group sizes as determined by their <math display="inline"><semantics> <mrow> <mi>R</mi> </mrow> </semantics></math> values. The fractal dimension decreases as the average fork length increases, but the same relationship is not observed when considering the average weave angle.</p>
Full article ">Figure 9
<p>Reconstructed bipolar neurons are arranged in a grid that increases vertically with respect to arbor radius <span class="html-italic">R</span> and horizontally with respect to total dendritic arbor length, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> </mrow> </semantics></math>. The <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math> value of each neuron is labelled, indicating the rise in <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math> with R and the decrease in <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> </mrow> </semantics></math> (consistent with the trends of <a href="#fractalfract-09-00009-f007" class="html-fig">Figure 7</a>).</p>
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31 pages, 4644 KiB  
Article
New Solitary Wave Solutions of the Lakshamanan–Porsezian–Daniel Model with the Application of the Φ6 Method in Fractional Sense
by Hicham Saber, Hussien Albala, Khaled Aldwoah, Amer Alsulami, Khidir Shaib Mohamed, Mohammed Hassan and Abdelkader Moumen
Fractal Fract. 2025, 9(1), 10; https://doi.org/10.3390/fractalfract9010010 - 28 Dec 2024
Viewed by 349
Abstract
This paper explores a significant fractional model, which is the fractional Lakshamanan–Porsezian–Daniel (FLPD) model, widely used in various fields like nonlinear optics and plasma physics. An advanced analytical solution for it is attained by the Φ6 technique. According to this [...] Read more.
This paper explores a significant fractional model, which is the fractional Lakshamanan–Porsezian–Daniel (FLPD) model, widely used in various fields like nonlinear optics and plasma physics. An advanced analytical solution for it is attained by the Φ6 technique. According to this methodology, effective and accurate solutions for wave structures within various types can be produced in the FLPD model framework. Solutions such as dark, bright, singular, periodic, and plane waves are studied in detail to identify their stability and behavior. Validations are also brought forward to assess the precision and flexibility of the Φ6 technique in modeling fractional models. Therefore, it is established in this study that the Φ6 technique represents a powerful tool for examining wave patterns in differential fractional order models. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
Show Figures

Figure 1

Figure 1
<p>The visualization of Equation (<a href="#FD24-fractalfract-09-00010" class="html-disp-formula">24</a>), under the parametric values <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9999</mn> <mo>,</mo> <mi>η</mi> <mo>=</mo> <mo>−</mo> <mn>1.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>a</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 2
<p>The visualization of Equation (<a href="#FD28-fractalfract-09-00010" class="html-disp-formula">28</a>), under the parametric values <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9999</mn> <mo>,</mo> <mi>η</mi> <mo>=</mo> <mo>−</mo> <mn>1.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>a</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>The visualization of Equation (<a href="#FD36-fractalfract-09-00010" class="html-disp-formula">36</a>), under the parametric values <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9999</mn> <mo>,</mo> <mi>η</mi> <mo>=</mo> <mo>−</mo> <mn>1.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>a</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>The visualization of Equation (<a href="#FD40-fractalfract-09-00010" class="html-disp-formula">40</a>), under the parametric values <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9999</mn> <mo>,</mo> <mi>η</mi> <mo>=</mo> <mo>−</mo> <mn>1.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>a</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>The visualization of Equation (<a href="#FD48-fractalfract-09-00010" class="html-disp-formula">48</a>), under the parametric values <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9999</mn> <mo>,</mo> <mi>η</mi> <mo>=</mo> <mo>−</mo> <mn>1.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>a</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>The visualization of Equation (<a href="#FD70-fractalfract-09-00010" class="html-disp-formula">70</a>), under the parametric values <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9999</mn> <mo>,</mo> <mi>η</mi> <mo>=</mo> <mo>−</mo> <mn>1.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>a</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>The visualization of Equation (<a href="#FD160-fractalfract-09-00010" class="html-disp-formula">160</a>), under the parametric values <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9999</mn> <mo>,</mo> <mi>η</mi> <mo>=</mo> <mo>−</mo> <mn>1.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>a</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>The visualization of Equation (<a href="#FD171-fractalfract-09-00010" class="html-disp-formula">171</a>), under the parametric values <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9999</mn> <mo>,</mo> <mi>η</mi> <mo>=</mo> <mo>−</mo> <mn>1.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>a</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p>
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12 pages, 2292 KiB  
Article
Chladni and Fractal Dynamics: Dual Mode Marker to Map Cancer Cell Nucleus Disintegration Phases
by Parama Dey, Anup Singhania, Ajaikumar B. Kunnumakkara, Subrata Ghosh and Anirban Bandyopadhyay
Fractal Fract. 2025, 9(1), 8; https://doi.org/10.3390/fractalfract9010008 - 27 Dec 2024
Viewed by 238
Abstract
Conventional cancer drugs are small molecules that target specific pathways. We introduced PCMS, a 26 kDa supramolecule combining sensors (S), molecular motors (M), and switching molecules (C), integrated within a fourth-generation PAMAM structure (P). PCMS identifies and deactivates cancer cell nucleus dynamics. A [...] Read more.
Conventional cancer drugs are small molecules that target specific pathways. We introduced PCMS, a 26 kDa supramolecule combining sensors (S), molecular motors (M), and switching molecules (C), integrated within a fourth-generation PAMAM structure (P). PCMS identifies and deactivates cancer cell nucleus dynamics. A decade ago, we demonstrated programmable, clock-like interactions among the S-C-M components. In this study, we captured images of fractal patterns formed by chromosomal compartments and developed a theoretical model of their fractal dynamics. We showed that the nucleus behaves like a cavity, producing resonance effects similar to Chladni patterns. When the external agent, PCMS, interacts with this cavity, it generates a fractal pattern. We identified and mapped five key phase transitions that ultimately lead to the breakdown of cancer cell nuclei. Full article
(This article belongs to the Special Issue Fractals in Biophysics and Their Applications)
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<p>PCMS and five phases in lung cancer cells: (<b>a</b>) The PCMS molecular structure, with components labeled as P = PAMAM, C = Controller, M = Molecular Motor, and S = Sensor. (<b>b</b>) White and yellow arrows highlight chromosomal compartments within a lung cancer cell and modifications in compartments, respectively. (<b>c</b>) Temporal variation of the cancer cell nucleus is shown, with the green arrow denoting the administration of PCMS and the red plot indicating instances where the nucleus reverts to its original shape, suggesting potential resistance to destruction. (<b>d</b>) Five phases of chromosome compartmentalization within the cancer cell nucleus are presented, with the top row showing schematics and the bottom row displaying corresponding fluorescence images, scale bar is 2 μm. Phases 4 and 5 represent points of no return.</p>
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<p>Snapshot of Chladni pattern analysis live feed: Chladni pattern analysis of seventeen lung cancer cells observed before the addition of PCMS. Panel (<b>a</b>) shows the number of loops or strings per unit area relative to the total area, with a maximum value of 1, represented by a color scale. In panel (<b>b</b>), the distribution of filament length is depicted by the number of pixels along their length, color-coded to show average filament length or inter-compartmental distances (ICDs). Panel (<b>c</b>) highlights large loop structures in comparison to smaller loop structures, while panel (<b>d</b>) displays regions with intertwined loops and filaments, with a color scale indicating high and low intertwinement. Panel (<b>e</b>) assesses the completion percentage of filamentary loops, with bright colors marking more circular structures. Finally, panel (<b>f</b>) presents the degree of symmetry, with red regions indicating asymmetry and blue regions representing symmetry.</p>
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<p>Chladni pattern analysis on seventeen lung cancer cells: Seventeen lung cancer cells are zoomed in just before PCMS addition, followed by Chladni pattern analysis on each cell. The 50 min duration is divided into 22 intervals, capturing 22 snapshots at equal time points. As illustrated in <a href="#fractalfract-09-00008-f002" class="html-fig">Figure 2</a>, six snapshots show different Chladni patterns, with four selected here to plot (<b>a</b>) variations in density; (<b>b</b>) complexity; (<b>c</b>) number of patterns; (<b>d</b>) average area.</p>
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<p>Segmenting and labeling self-similar objects: Seventeen lung cancer cells are zoomed in just before PCMS addition, and Chladni pattern analysis is performed on each cell. The image is converted to grayscale for analysis (<b>left</b>). Each “compartment” represents a closed area; a total of one hundred and sixty objects are detected (<b>right</b>). Based on compartment areas, colors are assigned, revealing an initial distribution of compartmental symmetries within the cancer cell cluster.</p>
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<p>Temporal evolution of fractal pattern: Seventeen lung cancer cells are zoomed in just before PCMS addition, and Fractal pattern analysis is performed on each cell. (<b>a</b>) After PCMS is introduced, the compartmental structure of each cell changes over time to estimate the scale free features. The total duration of 50 min is divided into 22 equal intervals, producing 22 snapshots. Upon PCMS addition, the symmetry distribution of each cell transforms. The changes in compartmental shapes and symmetry across cells are significant, with the number of compartments varying non-linearly. (<b>b</b>) Fractal dimension varies as a function of time, indicating structural complexity changes. (<b>c</b>) Lacunarity peaks just before nuclear disintegration, displayed by ripples in the graph.</p>
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11 pages, 272 KiB  
Article
Some Existence, Uniqueness, and Stability Results for a Class of ϑ-Fractional Stochastic Integral Equations
by Fahad Alsharari, Raouf Fakhfakh, Omar Kahouli and Abdellatif Ben Makhlouf
Fractal Fract. 2025, 9(1), 7; https://doi.org/10.3390/fractalfract9010007 - 27 Dec 2024
Viewed by 316
Abstract
This paper focuses on the existence and uniqueness of solutions for ϑ-fractional stochastic integral equations (ϑ-FSIEs) using the Banach fixed point theorem (BFPT). We explore the Ulam–Hyers stability (UHS) of ϑ-FSIEs through traditional methods of stochastic calculus and the [...] Read more.
This paper focuses on the existence and uniqueness of solutions for ϑ-fractional stochastic integral equations (ϑ-FSIEs) using the Banach fixed point theorem (BFPT). We explore the Ulam–Hyers stability (UHS) of ϑ-FSIEs through traditional methods of stochastic calculus and the BFPT. Moreover, the continuous dependence of solutions on initial conditions is proven. Additionally, we provide three examples to demonstrate our findings. Full article
12 pages, 289 KiB  
Article
Forced-Perturbed Fractional Differential Equations of Higher Order: Asymptotic Properties of Non-Oscillatory Solutions
by Said R. Grace, Gokula N. Chhatria, S. Kaleeswari, Yousef Alnafisah and Osama Moaaz
Fractal Fract. 2025, 9(1), 6; https://doi.org/10.3390/fractalfract9010006 - 27 Dec 2024
Viewed by 329
Abstract
This study investigates the asymptotic behavior of non-oscillatory solutions to forced-perturbed fractional differential equations with the Caputo fractional derivative. The main aim is to unify the Beta Integral Lemma (Lemma 2) and the Gamma Integral Lemma (Lemma 3) into a single framework. By [...] Read more.
This study investigates the asymptotic behavior of non-oscillatory solutions to forced-perturbed fractional differential equations with the Caputo fractional derivative. The main aim is to unify the Beta Integral Lemma (Lemma 2) and the Gamma Integral Lemma (Lemma 3) into a single framework. By combining these two powerful tools, we propose new criteria that effectively characterize the asymptotic behavior of non-oscillatory solutions to the given equations. The analysis of such solutions has significant implications in the fields of oscillation and stability theory. Notably, our findings extend prior work by exploring a wider range of equations with more general functions and coefficients, thereby broadening the applicability and deepening the understanding of both asymptotic and oscillatory behaviors. Moreover, the criteria we introduce offer improvements over previous approaches, as demonstrated by the example provided, which highlights the advantages of our results in comparison to earlier methods. Full article
27 pages, 4051 KiB  
Article
Fractal-Based Robotic Trading Strategies Using Detrended Fluctuation Analysis and Fractional Derivatives: A Case Study in the Energy Market
by Ekaterina Popovska and Galya Georgieva-Tsaneva
Fractal Fract. 2025, 9(1), 5; https://doi.org/10.3390/fractalfract9010005 - 26 Dec 2024
Viewed by 394
Abstract
This paper presents an integrated robotic trading strategy developed for the day-ahead energy market that includes different methods for time series analysis and forecasting, such as Detrended Fluctuation Analysis (DFA), Rescaled Range Analysis (R/S analysis), fractional derivatives, Long Short-Term Memory (LSTM) Networks, and [...] Read more.
This paper presents an integrated robotic trading strategy developed for the day-ahead energy market that includes different methods for time series analysis and forecasting, such as Detrended Fluctuation Analysis (DFA), Rescaled Range Analysis (R/S analysis), fractional derivatives, Long Short-Term Memory (LSTM) Networks, and Seasonal Autoregressive Integrated Moving Average (SARIMA) models. DFA and R/S analysis may capture the long-range dependencies and fractal features inherited by the nature of the electricity price time series and give information about persistence and variability in their behavior. Given this, fractional derivatives can be used to analyze price movements concerning the minor changes in price and time acceleration for that change, which makes the proposed framework more flexible for quickly changing market conditions. LSTM, from their perspective, may capture complex and non-linear dependencies, while SARIMA models may help handle seasonal trends. This integrated approach improves market signal interpretation and optimizes the market risk through adjustable stop-loss and take-profit levels which could lead to better portfolio performance. The proposed integrated strategy is based on actual data from the Bulgarian electricity market for the years 2017–2024. Findings from this research show how the combination of fractals with statistical and machine learning models can improve complex trading strategies implementation for the energy markets. Full article
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<p>Analysis and Forecasting Strategy Workflow.</p>
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<p>Hourly Day-Ahead prices dataset.</p>
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<p>α parameter of Bulgarian hourly electricity price market (2019–2024).</p>
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<p>Annual Electricity Prices (2019–2024) and their First and Second Derivatives.</p>
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<p>Annual Electricity Prices (2019–2024) and their First and Second Derivatives.</p>
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<p>Fractional derivatives of price time series using the Caputo method with different alpha values.</p>
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<p>Actual vs. Predicted Prices and Forecasted Prices for the Next 60 Days Using LSTM Model.</p>
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<p>Graphical analysis of SARIMA model results.</p>
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16 pages, 352 KiB  
Article
Sandwich-Type Results and Existence Results of Analytic Functions Associated with the Fractional q-Calculus Operator
by Sudhansu Palei, Madan Mohan Soren, Daniel Breaz and Luminiţa-Ioana Cotîrlǎ
Fractal Fract. 2025, 9(1), 4; https://doi.org/10.3390/fractalfract9010004 - 25 Dec 2024
Viewed by 295
Abstract
In the present investigation, we present certain subordination and superordination results for the q-integral operator of a fractional order associated with analytic functions in the open unit disk U. Using this q-integral operator, we obtain sandwich-type results. Furthermore, we employ [...] Read more.
In the present investigation, we present certain subordination and superordination results for the q-integral operator of a fractional order associated with analytic functions in the open unit disk U. Using this q-integral operator, we obtain sandwich-type results. Furthermore, we employ the existence of univalent solutions to a q-differential equation connected with a fractional q-integral operator of fractional order. We use these results to demonstrate the significance of our findings for particular functions. We also derive some examples and corollaries that are pertinent to our results. Full article
27 pages, 11214 KiB  
Article
Fractal Characteristics of the Spatial Distribution of Mine Earthquake Sources in the Vicinity of a Fault: A Case Study in the Ashele Copper Mine
by Congcong Zhao, Shigen Fu and Yinghua Huang
Fractal Fract. 2025, 9(1), 3; https://doi.org/10.3390/fractalfract9010003 - 24 Dec 2024
Viewed by 274
Abstract
Potential faults are common sensitive geological bodies that affect the safe mining of underground mines, often leading to major accidents such as rock instability and rockburst during mining. The failure mechanism of faults has been widely studied. However, due to the spatiotemporal specificity [...] Read more.
Potential faults are common sensitive geological bodies that affect the safe mining of underground mines, often leading to major accidents such as rock instability and rockburst during mining. The failure mechanism of faults has been widely studied. However, due to the spatiotemporal specificity of fault occurrence, there are few theoretical and mathematical methods suitable for effective analysis in mine safety risk management. This study aims to introduce fractal theory to characterize the spatiotemporal activity fractal characteristics of induced faults intersecting the mining site and roadway during the mining process of the Ashele copper mine in China. Using microseismic systems and fractal theory, a spatiotemporal fractal model of the fault slip process is constructed, and a fractal analysis method is proposed. The fractal dimension value is calculated based on the spatiotemporal parameters of different segments and stages. The fractal dimension is used to characterize and analyze the evolution of the fault. The physical formation process of potential faults and the relationship between fractal dimension values and multiple parameters, including spatial clustering, regional distribution characteristics, and energy-release characteristics, were analyzed based on the division of events into different time stages. Discovering fractal dimension’s temporal and spatial–temporal characteristics can provide technical references for mine disaster prevention. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)
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<p>The mine location, landform, microseismic monitoring topology system, and underground system. (<b>a</b>) geographical location; (<b>b</b>) topographic features; (<b>c</b>) development Engineering and Microseismic Monitoring; (<b>d</b>) topological structure.</p>
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<p>The layout of the target area and monitoring network: (<b>a</b>) stereo diagram; (<b>b</b>) top plan view.</p>
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<p>The relationships between the positions of the microseismic monitoring sensors and the middle levels in the study area. (<b>a</b>) +450m middle and its microseismic events.; (<b>b</b>) +350m middle and its microseismic events.; (<b>c</b>) +300m middle and its microseismic events.</p>
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<p>A flow diagram of the research method and fractal dimension analysis.</p>
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<p>The frequency distribution sand stage divisions of daily microseismic activities: (<b>a</b>) overall mine events; and (<b>b</b>) the distribution of microseismic events near mining-induced faults.</p>
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<p>The distribution and location relationships of microseismic events in the four different stages. (<b>a</b>) overall situation; (<b>b</b>) Localized enlargement.</p>
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<p>A 3D spatial fractal dimension representation frame model, based on the locations of the mine’s earthquake sources. (<b>a</b>) Time scale division; (<b>b</b>) The nesting process of events and spatial units; (<b>c</b>) Nesting results of events and spatial units.</p>
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<p>The distributions of the microseismic events around mining-induced faults in the four time stages.</p>
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<p>A comparison of the fitting coefficients of the maximum events and fractal dimension values in microcells using different grids.</p>
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<p>The fractal dimension distributions.</p>
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<p>A perspective cloud chart of the fractal dimension interpolation under different perspectives in each time stage.</p>
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<p>A schematic diagram of the fractal dimension value distributions in overall space in the different time stages.</p>
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<p>A contour map of the middle-level fractal dimension interpolation in the different time stages.</p>
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<p>A distribution diagram of the fractal dimension values in the middle levels of the different time stages.</p>
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<p>A comparison chart of the fractal dimension values and the number of events in the middle level of each time stage. (<b>a</b>) Changes in fractal dimension at different stages; (<b>b</b>) Changes in the number of events at different stages.</p>
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<p>A comparison diagram of the changes in the main parameter of the microseismic events. (<b>a</b>) radiant energy; (<b>b</b>) richter magnitude; (<b>c</b>) seismic moment; (<b>d</b>) source radius.</p>
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18 pages, 4434 KiB  
Article
Fractal Characteristics of Pore Structure of Longmaxi Shales with Different Burial Depths in Southern Sichuan and Its Geological Significance
by Manping Yang, Yanyan Pan, Hongye Feng, Qiang Yan, Yanjun Lu, Wanxin Wang, Yu Qi and Hongjian Zhu
Fractal Fract. 2025, 9(1), 2; https://doi.org/10.3390/fractalfract9010002 - 24 Dec 2024
Viewed by 388
Abstract
Burial depth can significantly impact the pore structure characteristics of shale. The Lower Silurian Longmaxi Shale in the Weiyuan block of the Sichuan Basin is a marine formation that we studied for deep shale gas exploration. We used two groups of Longmaxi samples, [...] Read more.
Burial depth can significantly impact the pore structure characteristics of shale. The Lower Silurian Longmaxi Shale in the Weiyuan block of the Sichuan Basin is a marine formation that we studied for deep shale gas exploration. We used two groups of Longmaxi samples, outcrop shale and middle-deep shale, to investigate the pore structure fractal features at varying burial depths using a combination of mineralogy, organic geochemistry, scanning electron microscopy (SEM), and low-temperature gas (CO2, N2) adsorption. The V-S fractal model was used to determine the fractal dimension (Dc) of micropores, and the FHH fractal model was used to determine the fractal dimension (DN) of mesopores. The findings indicate that the pore morphology of organic matter becomes irregular and more broken as the burial depth increases, as does the content and maturity of organic matter. The pore size of organic matter gradually decreases, the SSA (BET, DR) and PV (BJH, DA) of shale pores increase, the pore structure becomes more complex, and the average shale pore size decreases. According to this study, the organic matter content and its maturity show an increasing trend as burial depth increases. Meanwhile, the organic matter’s pore morphology tends to be irregular, and fracture rates rise, which causes the organic matter’s pore size to gradually decrease. In addition, the SSA (comprising the values assessed by BET and DR techniques) and PV (evaluated by BJH and DA methods) of shale pores grew, suggesting that the pore structure became more complex. Correspondingly, the average pore size of the shale decreased. The fractal dimensions of the micropores (DC), mesoporous surface (DN1), and mesoporous structure (DN2) of outcrop shale are 2.6728~2.7245, 2.5612~2.5838, and 2.7911~2.8042, respectively. The mean values are 2.6987, 2.5725, and 2.7977, respectively. The DC, DN1, and DN2 of middle-deep shale are 2.6221~2.7510, 2.6085~2.6390, and 2.8140~2.8357, respectively, and the mean values are 2.7050, 2.6243, and 2.8277, respectively. As the fractal dimension grows, the shale’s pore structure becomes more intricate, and the heterogeneity increases as the buried depth increases. The fractal dimension has a positive association with the pore structure parameters (SSA, PV), TOC, and Ro and a negative association with the mineral component (quartz, feldspar, clay mineral) contents. Minerals like quartz, feldspar, and clay will slow down the expansion of pores, but when SSA and PV increase, the pore heterogeneity will be greater and the pore structure more complex. Full article
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<p>Mineral composition (<b>A</b>) and the relative contents of clay mineral (<b>B</b>). Data from Hao et al. [<a href="#B22-fractalfract-09-00002" class="html-bibr">22</a>].</p>
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<p>SEM images of Longmaxi shales at different burial depths. (<b>A</b>): Sample D10, organic pores, clay mineral interlayer pores, microcracks; (<b>B</b>): Sample M1, organic pores; (<b>C</b>): Sample O1, intergranular pores; (<b>D</b>): Sample D5, intergranular pores; (<b>E</b>): Sample D7, intergranular pores; (<b>F</b>): Sample M1, clay mineral interlaminar pores, calcite intergranular dissolution pores; (<b>G</b>): Sample O1, clay mineral interlaminar pores; (<b>H</b>): Sample O1, microcracks; (<b>I</b>): Sample M2, dolomite intergranular dissolution pores, microcracks.</p>
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<p>CO<sub>2</sub> adsorption curves (<b>A</b>) and PSD (<b>B</b>) of samples based on CO<sub>2</sub> adsorption experiments.</p>
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<p>N<sub>2</sub> adsorption curves of shale with different burial depths. Note: The blue dotted line represents a relative pressure of 0.45.</p>
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<p>Pore size distribution curve of shale based on N<sub>2</sub> adsorption experiments.</p>
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<p>Shale pore fractal fitting curves based on CO<sub>2</sub> adsorption experiments.</p>
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<p>Shale pore fractal fitting curves based on N<sub>2</sub> adsorption experiments. Note: Green represents the fractal dimension D<sub>N1</sub> of relative pressure &lt; 0.45; red represents the fractal dimension D<sub>N2</sub> with relative pressure &gt; 0.45; the blue dotted line represents relative pressure = 0.45.</p>
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<p>Correlation between fractal dimension and SSA (<b>A</b>) and PV (<b>B</b>) of shale samples.</p>
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<p>Correlation between fractal dimension and TOC content (<b>A</b>) and thermal maturity of organic matter (<b>B</b>) in Longmaxi shales.</p>
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<p>Correlation between fractal dimension and mineral composition of Longmaxi shales.</p>
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<p>Relationship between clay mineral composition and fractal dimension of Longmaxi shales.</p>
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14 pages, 364 KiB  
Article
A Bicomplex Proportional Fractional (ϑ,φ)-Weighted Cauchy–Riemann Operator Using Riemann–Liouville Derivatives with Respect to an Hyperbolic-Valued Function
by José Oscar González-Cervantes, Juan Bory-Reyes and Juan Adrián Ramírez-Belman
Fractal Fract. 2025, 9(1), 1; https://doi.org/10.3390/fractalfract9010001 - 24 Dec 2024
Viewed by 367
Abstract
Based on the Riemann–Liouville derivatives with respect to functions taking values in the set of hyperbolic numbers, we consider a new bicomplex proportional fractional (ϑ,φ)-weighted Cauchy–Riemann operator, involving orthogonal bicomplex functions as weights, and its associated fractional Borel–Pompeiu [...] Read more.
Based on the Riemann–Liouville derivatives with respect to functions taking values in the set of hyperbolic numbers, we consider a new bicomplex proportional fractional (ϑ,φ)-weighted Cauchy–Riemann operator, involving orthogonal bicomplex functions as weights, and its associated fractional Borel–Pompeiu formula is proved as the main result. Full article
(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)
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