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Fractal Fract., Volume 9, Issue 2 (February 2025) – 58 articles

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14 pages, 317 KiB  
Article
The Stability and Global Attractivity of Fractional Differential Equations with the Ψ-Hilfer Derivative in the Context of an Economic Recession
by Mdi Begum Jeelani, Farva Hafeez and Nouf Abdulrahman Alqahtani
Fractal Fract. 2025, 9(2), 113; https://doi.org/10.3390/fractalfract9020113 (registering DOI) - 13 Feb 2025
Abstract
Fractional differential equations (FDEs) are employed to describe the physical universe. This article investigates the attractivity of solutions for FDEs and Ulam–Hyers–Rassias stability, involving the Ψ-Hilfer fractional derivative. Important results are presented using Krasnoselskii’s fixed point theorem, which provides a framework for [...] Read more.
Fractional differential equations (FDEs) are employed to describe the physical universe. This article investigates the attractivity of solutions for FDEs and Ulam–Hyers–Rassias stability, involving the Ψ-Hilfer fractional derivative. Important results are presented using Krasnoselskii’s fixed point theorem, which provides a framework for analyzing the stability and attractivity of solutions. Novel results on the attractiveness of solutions to nonlinear FDEs in Banach spaces are derived, and the existence of solutions, stability properties, and behavior of system equilibria are examined. The application of Ψ-Hilfer fractional derivatives in modeling financial crises is explored, and a financial crisis model using Ψ-Hilfer fractional derivatives is proposed, providing more general and global results. Furthermore, we also perform a numerical analysis to validate our theoretical findings. Full article
(This article belongs to the Special Issue Fractional Calculus and Nonlinear Analysis: Theory and Applications)
13 pages, 1303 KiB  
Article
Scale-Free Dynamics of Resting-State fMRI Microstates
by Nurhan Erbil and Gopikrishna Deshpande
Fractal Fract. 2025, 9(2), 112; https://doi.org/10.3390/fractalfract9020112 - 12 Feb 2025
Abstract
The functional significance of RSNs is examined via simultaneous EEG-fMRI studies on the basis of the relation of RSNs with different frequency bands of EEG and EEG-based microstate analysis. In this study, we try to identify RSNs from microstates of cortical surface maps [...] Read more.
The functional significance of RSNs is examined via simultaneous EEG-fMRI studies on the basis of the relation of RSNs with different frequency bands of EEG and EEG-based microstate analysis. In this study, we try to identify RSNs from microstates of cortical surface maps of the BOLD signal. In addition, the scale-free dynamics of these map sequences were also examined. The structural and resting state functional MRI images were acquired on a 3T scanner with three different fMRI acquisition protocols from seven subjects. Microstate segmentations from EEG, fMRI, and simulated data were evaluated. Wavelet-based fractal analysis was performed on map sequence time series and the Hurst exponent (H) was calculated. By using HRF-deconvolved fMRI time series, the effect of the HRF (hemodynamic response function) on fMRI-derived microstates was tested. The fMRI map sequence has a system with a memory system smaller than 16 s. When the HRF was deconvolved, the duration of the memory of the system was reduced to 4 s. On the other hand, the results of simulation data indicated that these systems are specific to the resting state BOLD signal. Similar to EEG microstates, fMRI also has microstates and both of them have scale-free dynamics. fMRI microstate dynamics have two different components, one is related to the HRF and the other is independent of the HRF. The significance of fMRI microstates and their relation with RSNs need to be further studied. Full article
17 pages, 13169 KiB  
Article
Research on Nonlinear Dynamic Characteristics of Fractional Order Resonant DC-DC Converter Based on Sigmoid Function
by Lingling Xie and Guangwei Xu
Fractal Fract. 2025, 9(2), 111; https://doi.org/10.3390/fractalfract9020111 - 12 Feb 2025
Abstract
Resonant DC-DC converters are a class of strongly nonlinear systems with rich nonlinear phenomena. In order to describe the dynamic behavior of resonant DC-DC converters more accurately, the nonlinear dynamic behavior of fractional order (FO) resonant DC-DC converters is studied deeply, based on [...] Read more.
Resonant DC-DC converters are a class of strongly nonlinear systems with rich nonlinear phenomena. In order to describe the dynamic behavior of resonant DC-DC converters more accurately, the nonlinear dynamic behavior of fractional order (FO) resonant DC-DC converters is studied deeply, based on the fractional order nature of inductance and capacitance. Firstly, a Sigmoid function state model of the fractional order resonant converter is established and integrated with phase shift control. A discrete model of the converter is established by using an estimation correction algorithm. Secondly, the mathematical and equivalent circuit models of the fractional order converter are constructed in MATLAB. The circuit simulations and the experimental results verified the correctness of the Sigmoid function model. Thirdly, the effect of circuit parameters on the converter’s nonlinear dynamics is analyzed using bifurcation diagrams, time-domain waveforms, and phase diagrams. Finally, an experimental platform is established to validate the theoretical analysis. The results demonstrate that increasing the proportional coefficient and load resistance destabilizes the system, leading to rich nonlinear phenomena such as bifurcation and chaos. Compared to integer order converters, fractional order converters offer a broader stable operating range. Fractional order models can more accurately reflect the nonlinear dynamic characteristics of resonant DC-DC converters. Full article
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17 pages, 300 KiB  
Article
A New Result Regarding Positive Solutions for Semipositone Boundary Value Problems of Fractional Differential Equations
by Yongqing Wang
Fractal Fract. 2025, 9(2), 110; https://doi.org/10.3390/fractalfract9020110 - 12 Feb 2025
Viewed by 84
Abstract
In this paper, we discuss the positive solutions to a class of semipositone boundary value problems of fractional differential equations. The nonlinearity f(t,x) may be singular at t=0,1 and satisfies [...] Read more.
In this paper, we discuss the positive solutions to a class of semipositone boundary value problems of fractional differential equations. The nonlinearity f(t,x) may be singular at t=0,1 and satisfies f(t,x)a(t)xR(t). We derive some new properties of the Green’s function of the auxiliary problems, and discover the multiplicity and existence of the positive solutions by utilizing the fixed point index theory. Two examples are illustrated to validate the main results. Full article
22 pages, 4476 KiB  
Article
Interspecific Competition of Plant Communities Based on Fractional Order Time Delay Lotka–Volterra Model
by Jun Zhang, Yongzhi Liu, Juhong Liu, Caiqin Zhang and Jingyi Chen
Fractal Fract. 2025, 9(2), 109; https://doi.org/10.3390/fractalfract9020109 - 12 Feb 2025
Viewed by 191
Abstract
A novel time delay Lotka–Volterra (TDLV) model was developed by extending the concept of time delay from integer order to fractional order. The TDLV model was constructed to simulate the dynamics of aboveground biomass per individual of three dominant herbaceous plant species ( [...] Read more.
A novel time delay Lotka–Volterra (TDLV) model was developed by extending the concept of time delay from integer order to fractional order. The TDLV model was constructed to simulate the dynamics of aboveground biomass per individual of three dominant herbaceous plant species (Leymus chinensis, Agropyron cristatum, and Stipa grandis) in the typical grasslands of Inner Mongolia. Comparative analysis indicated that the TDLV model outperforms candidate models, such as Logistic, GM(1,1), GM(1,N), DGM(2,1), and Lotka–Volterra model, in terms of all fitting criteria. The results demonstrate that interspecies competition exhibits clear feedback and suppression effects, with Leymus chinensis playing a central role in regulating community dynamics. The system is locally stable and eventually converges to an equilibrium point, though Stipa grandis maintains relatively low biomass, requiring further monitoring. Time delays are prevalent in the system, influencing dynamic processes and causing damping oscillations as populations approach equilibrium. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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19 pages, 370 KiB  
Article
On Quantum Hermite-Hadamard-Fejer Type Integral Inequalities via Uniformly Convex Functions
by Hasan Barsam, Somayeh Mirzadeh, Yamin Sayyari and Loredana Ciurdariu
Fractal Fract. 2025, 9(2), 108; https://doi.org/10.3390/fractalfract9020108 - 12 Feb 2025
Viewed by 5
Abstract
The main goal of this study is to provide new q-Fejer and q-Hermite-Hadamard type integral inequalities for uniformly convex functions and functions whose second quantum derivatives in absolute values are uniformly convex. Two basic inequalities as power mean inequality and Holder’s [...] Read more.
The main goal of this study is to provide new q-Fejer and q-Hermite-Hadamard type integral inequalities for uniformly convex functions and functions whose second quantum derivatives in absolute values are uniformly convex. Two basic inequalities as power mean inequality and Holder’s inequality are used in demonstrations. Some particular functions are chosen to illustrate the investigated results by two examples analyzed and the result obtained have been graphically visualized. Full article
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21 pages, 1478 KiB  
Article
Exploring Fractional Damped Burgers’ Equation: A Comparative Analysis of Analytical Methods
by Azzh Saad Alshehry and Rasool Shah
Fractal Fract. 2025, 9(2), 107; https://doi.org/10.3390/fractalfract9020107 - 10 Feb 2025
Viewed by 332
Abstract
This investigation focuses on the study of the fractional damped Burgers’ equation by using the natural residual power series method coupled with the new iteration transform method in the context of the Caputo operator. The equation of Burgers under the damped context is [...] Read more.
This investigation focuses on the study of the fractional damped Burgers’ equation by using the natural residual power series method coupled with the new iteration transform method in the context of the Caputo operator. The equation of Burgers under the damped context is useful when studying one-dimensional nonlinear waves involving damping effect, and is used in fluid dynamics, among other applications. Two new mathematical methods that can be used to obtain an approximate solution to this complex non-linear problem are the natural residual power series method and the new iteration transform method. Therefore, it can be deduced that the Caputo operator aids in modeling of the fractional derivatives, as it provides a better description of the physical realities. Thus, the objective of the present work is to advance the knowledge accumulated on the behavior of solutions to the damped Burgers’ equation, as well as to check the applicability of the proposed approaches to other nonlinear fractional partial differential equations. Full article
(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
19 pages, 6533 KiB  
Article
Vibration Suppression of the Vehicle Mechatronic ISD Suspension Using the Fractional-Order Biquadratic Electrical Network
by Yujie Shen, Zhaowei Li, Xiang Tian, Kai Ji and Xiaofeng Yang
Fractal Fract. 2025, 9(2), 106; https://doi.org/10.3390/fractalfract9020106 - 10 Feb 2025
Viewed by 394
Abstract
In order to break the bottleneck of the integer-order transfer function in vehicle ISD (inerter-spring-damper) suspension design, a positive real synthesis design method of vehicle mechatronic ISD suspension based on the fractional-order biquadratic transfer function is proposed. The emergence of the fractional-order components [...] Read more.
In order to break the bottleneck of the integer-order transfer function in vehicle ISD (inerter-spring-damper) suspension design, a positive real synthesis design method of vehicle mechatronic ISD suspension based on the fractional-order biquadratic transfer function is proposed. The emergence of the fractional-order components disrupts the equivalence relationship between the passivity of components and the positive realness of integer-order transfer functions in traditional networks. In this paper, the positive real condition of the fractional-order biquadratic transfer function is given. Then, a quarter dynamic model of the vehicle mechatronic ISD suspension is established, and the parameters of the fractional-order biquadratic transfer function and vehicle suspension are obtained by an NSGA-II multi-objective genetic algorithm. Moreover, the structure of the external circuit and the parameters of the electrical components are obtained by the fractional-order passive network synthesis theory. The simulation results show that under the condition of random road input and vehicle speed of 20 m/s, the root-mean-square (RMS) value of the vehicle body acceleration and the dynamic tire load of the fractional-order ISD suspension are reduced by 7.98% and 18.75% compared with the traditional passive suspension, while under the same condition, the integer-order ISD suspension can only reduce by 5.34% and 16.07%, respectively. The results show that employing a fractional-order biquadratic electrical network in the vehicle mechatronic ISD suspension enhances vibration isolation performance compared with the suspension using an integer-order biquadratic electrical network. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)
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19 pages, 338 KiB  
Article
Exploring Impulsive and Delay Differential Systems Using Piecewise Fractional Derivatives
by Hicham Saber, Arshad Ali, Khaled Aldwoah, Tariq Alraqad, Abdelkader Moumen, Amer Alsulami and Nidal Eljaneid
Fractal Fract. 2025, 9(2), 105; https://doi.org/10.3390/fractalfract9020105 - 10 Feb 2025
Viewed by 329
Abstract
This paper investigates a general class of variable-kernel discrete delay differential equations (DDDEs) with integral boundary conditions and impulsive effects, analyzed using Caputo piecewise derivatives. We establish results for the existence and uniqueness of solutions, as well as their stability. The existence of [...] Read more.
This paper investigates a general class of variable-kernel discrete delay differential equations (DDDEs) with integral boundary conditions and impulsive effects, analyzed using Caputo piecewise derivatives. We establish results for the existence and uniqueness of solutions, as well as their stability. The existence of at least one solution is proven using Schaefer’s fixed-point theorem, while uniqueness is established via Banach’s fixed-point theorem. Stability is examined through the lens of Ulam–Hyers (U-H) stability. Finally, we illustrate the application of our theoretical findings with a numerical example. Full article
24 pages, 619 KiB  
Article
Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf Derivatives
by F. Gassem, Mohammed Almalahi, Osman Osman, Blgys Muflh, Khaled Aldwoah, Alwaleed Kamel and Nidal Eljaneid
Fractal Fract. 2025, 9(2), 104; https://doi.org/10.3390/fractalfract9020104 - 8 Feb 2025
Viewed by 295
Abstract
This paper presents a novel framework for modeling nonlinear fractional evolution control systems. This framework utilizes a power non-local fractional derivative (PFD), which is a generalized fractional derivative that unifies several well-known derivatives, including Caputo–Fabrizio, Atangana–Baleanu, and generalized Hattaf derivatives, as special cases. [...] Read more.
This paper presents a novel framework for modeling nonlinear fractional evolution control systems. This framework utilizes a power non-local fractional derivative (PFD), which is a generalized fractional derivative that unifies several well-known derivatives, including Caputo–Fabrizio, Atangana–Baleanu, and generalized Hattaf derivatives, as special cases. It uniquely features a tunable power parameter “p”, providing enhanced control over the representation of memory effects compared to traditional derivatives with fixed kernels. Utilizing the fixed-point theory, we rigorously establish the existence and uniqueness of solutions for these systems under appropriate conditions. Furthermore, we prove the Hyers–Ulam stability of the system, demonstrating its robustness against small perturbations. We complement this framework with a practical numerical scheme based on Lagrange interpolation polynomials, enabling efficient computation of solutions. Examples illustrating the model’s applicability, including symmetric cases, are supported by graphical representations to highlight the approach’s versatility. These findings address a significant gap in the literature and pave the way for further research in fractional calculus and its diverse applications. Full article
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10 pages, 286 KiB  
Article
A Short Note on Fractal Interpolation in the Space of Convex Lipschitz Functions
by Fatin Gota and Peter Massopust
Fractal Fract. 2025, 9(2), 103; https://doi.org/10.3390/fractalfract9020103 - 6 Feb 2025
Viewed by 412
Abstract
In this short note, we consider fractal interpolation in the Banach space Vθ(I) of convex Lipschitz functions defined on a compact interval IR. To this end, we define an appropriate iterated function system and exhibit the [...] Read more.
In this short note, we consider fractal interpolation in the Banach space Vθ(I) of convex Lipschitz functions defined on a compact interval IR. To this end, we define an appropriate iterated function system and exhibit the associated Read–Bajraktarević operator T. We derive conditions for which T becomes a Ratkotch contraction on a closed subspace of Vθ(I), thus establishing the existence of fractal functions of class Vθ(I). An example illustrates the theoretical findings. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractals)
35 pages, 2025 KiB  
Article
Fractional Calculus for Type 2 Interval-Valued Functions
by Mostafijur Rahaman, Dimplekumar Chalishajar, Kamal Hossain Gazi, Shariful Alam, Soheil Salahshour and Sankar Prasad Mondal
Fractal Fract. 2025, 9(2), 102; https://doi.org/10.3390/fractalfract9020102 - 5 Feb 2025
Viewed by 338
Abstract
This paper presents a contemporary introduction of fractional calculus for Type 2 interval-valued functions. Type 2 interval uncertainty involves interval uncertainty with the goal of more assembled perception with reference to impreciseness. In this paper, a Riemann–Liouville fractional-order integral is constructed in Type [...] Read more.
This paper presents a contemporary introduction of fractional calculus for Type 2 interval-valued functions. Type 2 interval uncertainty involves interval uncertainty with the goal of more assembled perception with reference to impreciseness. In this paper, a Riemann–Liouville fractional-order integral is constructed in Type 2 interval delineated vague encompassment. The exploration of fractional calculus is continued with the manifestation of Riemann–Liouville and Caputo fractional derivatives in the cited phenomenon. In addition, Type 2 interval Laplace transformation is proposed in this text. Conclusively, a mathematical model regarding economic lot maintenance is analyzed as a conceivable implementation of this theoretical advancement. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
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26 pages, 1185 KiB  
Article
Direct Synthesis of Fractional-Order Controllers Using Only Two Design Equations with Robustness to Parametric Uncertainties
by Carlos Muñiz-Montero, Jesus M. Munoz-Pacheco, Luis A. Sánchez-Gaspariano, Carlos Sánchez-López, Jesús E. Molinar-Solís and Melissa Chavez-Portillo
Fractal Fract. 2025, 9(2), 101; https://doi.org/10.3390/fractalfract9020101 - 5 Feb 2025
Viewed by 429
Abstract
This paper employs the Direct Synthesis approach to present an analytical methodology for designing fractional-order controllers, aiming to balance simplicity and robustness for practical industrial implementation. Although significant progress has been made in developing fractional-order PID controllers, the advancement of Direct Synthesis controllers [...] Read more.
This paper employs the Direct Synthesis approach to present an analytical methodology for designing fractional-order controllers, aiming to balance simplicity and robustness for practical industrial implementation. Although significant progress has been made in developing fractional-order PID controllers, the advancement of Direct Synthesis controllers has been comparatively slower. This study underscores the importance of further research on these controllers and the need for innovative approaches to enhance parameter adjustment. The proposed methodology is based on the fractional “second-order” transfer function and the solution of two equations derived from four key specifications: overshoot, settling time, and the frequency and magnitude of disturbance rejection. Additionally, the fractional order should be chosen as close as possible to 1, ensuring practical implementation and minimizing the system’s sensitivity to parameter variations. The resulting controller demonstrates strong robustness against plant parameter variations, input noise, and disturbances while achieving shorter settling times and lower overshoot. It outperforms fractional-order PID and ID controllers optimized numerically and surpasses integer-order phase lead-lag compensators designed analytically. The validation process involved Monte Carlo simulations and Kruskal–Wallis statistical analysis on a complex system characterized by closely spaced poles and significant parametric variations. Furthermore, the proposed controller effectively reduces the integral of the control signal (control effort), enhancing energy efficiency. Full article
(This article belongs to the Special Issue Design, Optimization and Applications for Fractional Chaotic System)
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24 pages, 10360 KiB  
Article
The Relationship Between the Fractal Dimension and the Evolution of Rock-Forming Minerals Crystallization on the Example of the Northwestern Part of the Lovozero Intrusion
by Miłosz Huber, Klaudia Stępniewska and Mirosław Wiktor Huber
Fractal Fract. 2025, 9(2), 100; https://doi.org/10.3390/fractalfract9020100 - 5 Feb 2025
Viewed by 391
Abstract
This article presents the results of fractal texture analyses of selected minerals (aegirine, eudialyte, orthoclase) in the northwestern part of the Lovozero intrusion. This intrusion is located in northeastern Scandinavia and is a massif made of alkaline rocks. There are rocks such as [...] Read more.
This article presents the results of fractal texture analyses of selected minerals (aegirine, eudialyte, orthoclase) in the northwestern part of the Lovozero intrusion. This intrusion is located in northeastern Scandinavia and is a massif made of alkaline rocks. There are rocks such as massive syenites and porphyrtes, as well as iiolites, urtites, and foyaites, accompanied by metasomatic rocks of the contact zone. A box-counting fractal dimension was used to numerically represent the texture of these minerals. In the further part, this coefficient was visualized in the form of maps superimposed on the study area, and some simple arithmetic calculations were performed to highlight the common features of this dimension for the selected rock-forming minerals. In conjunction with the geological interpretation of these results, rock-forming processes in this massif were depicted. This work is preliminary, showing the potential of this calculation method in petrological applications. Full article
(This article belongs to the Special Issue Fractals in Geology and Geochemistry)
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19 pages, 4734 KiB  
Article
Fractal Analysis of Volcanic Rock Image Based on Difference Box-Counting Dimension and Gray-Level Co-Occurrence Matrix: A Case Study in the Liaohe Basin, China
by Sijia Li, Zhuwen Wang and Dan Mou
Fractal Fract. 2025, 9(2), 99; https://doi.org/10.3390/fractalfract9020099 - 4 Feb 2025
Viewed by 493
Abstract
Volcanic rocks, as a widely distributed rock type on the earth, are mostly buried deep within basins, and their internal structures possess characteristics by irregularity and self-similarity. In the study of volcanic rocks, accurately identifying the lithology of volcanic rocks is significant for [...] Read more.
Volcanic rocks, as a widely distributed rock type on the earth, are mostly buried deep within basins, and their internal structures possess characteristics by irregularity and self-similarity. In the study of volcanic rocks, accurately identifying the lithology of volcanic rocks is significant for reservoir description and reservoir evaluation. The accuracy of lithology identification can improve the success rate of petroleum exploration and development as well as the safety of engineering construction. In this study, we took the electron microscope images of four types of volcanic rocks in the Liaohe Basin as the research objects and comprehensively used the differential box-counting dimension (DBC) and the gray-level co-occurrence matrix (GLCM) to identify the lithology of volcanic rocks. Obtain the images of volcanic rocks in the research area and conduct preprocessing so that the images can meet the requirements of calculations. Firstly, calculate the different box-counting dimension. Divide the grayscale image into boxes of different scales and determine the differential box-counting dimension based on the variation of grayscale values within each box. The differential box-counting dimension of basalt ranges from 1.7 to 1.75, that of trachyte ranges from 1.82 to 1.87, that of gabbro ranges from 1.76 to 1.79, and that of diabase ranges from 1.78 to 1.82. Then, the gray-level co-occurrence matrix is utilized to extract four image texture features of volcanic rock images, namely contrast, energy, entropy, and variance. The recognition of four types of volcanic rock images is achieved by combining the different box-counting dimension and the gray-level co-occurrence matrix. This method has been experimentally verified by volcanic rock image samples. It has a relatively high accuracy in identifying the lithology of volcanic rocks and can effectively distinguish four different types of volcanic rocks. Compared with single-feature recognition methods, this approach significantly improves recognition accuracy, offers reliable technical support and a data basis for volcanic rock-related geological analyses, and drives the further development of volcanic rock research. Full article
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16 pages, 5389 KiB  
Article
Control Error Convergence Using Lyapunov Direct Method Approach for Mixed Fractional Order Model Reference Adaptive Control
by Gustavo E. Ceballos Benavides, Manuel A. Duarte-Mermoud and Lisbel Bárzaga Martell
Fractal Fract. 2025, 9(2), 98; https://doi.org/10.3390/fractalfract9020098 - 4 Feb 2025
Viewed by 574
Abstract
This paper extends Lyapunov stability theory to mixed fractional order direct model reference adaptive control (FO-DMRAC), where the adaptive control parameter is of fractional order, and the control error model is of integer order. The proposed approach can also be applied to other [...] Read more.
This paper extends Lyapunov stability theory to mixed fractional order direct model reference adaptive control (FO-DMRAC), where the adaptive control parameter is of fractional order, and the control error model is of integer order. The proposed approach can also be applied to other types of model reference adaptive controllers (MRACs), provided the form of the control error dynamics and the fractional order adaptive control law are similar. This paper demonstrates that the control error will converge to zero, even if the derivative of the classical Lyapunov function V˙ is positive during a transient period, as long as V˙(e,ϕ) tends to zero as time approaches infinity. Finally, this paper provides application examples that illustrate both the convergence of the control error to zero and the behavior of V˙(e,ϕ). Full article
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14 pages, 297 KiB  
Article
Generalized Fractional Integral Inequalities Derived from Convexity Properties of Twice-Differentiable Functions
by Areej A. Almoneef, Abd-Allah Hyder, Fatih Hezenci and Hüseyin Budak
Fractal Fract. 2025, 9(2), 97; https://doi.org/10.3390/fractalfract9020097 - 4 Feb 2025
Viewed by 421
Abstract
This study presents novel formulations of fractional integral inequalities, formulated using generalized fractional integral operators and the exploration of convexity properties. A key identity is established for twice-differentiable functions with the absolute value of their second derivative being convex. Using this identity, several [...] Read more.
This study presents novel formulations of fractional integral inequalities, formulated using generalized fractional integral operators and the exploration of convexity properties. A key identity is established for twice-differentiable functions with the absolute value of their second derivative being convex. Using this identity, several generalized fractional Hermite–Hadamard-type inequalities are developed. These inequalities extend the classical midpoint and trapezoidal-type inequalities, while offering new perspectives through convexity properties. Also, some special cases align with known results, and an illustrative example, accompanied by a graphical representation, is provided to demonstrate the practical relevance of the results. Moreover, the findings may offer potential applications in numerical integration, optimization, and fractional differential equations, illustrating their relevance to various areas of mathematical analysis. Full article
(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus, 2nd Edition)
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16 pages, 5641 KiB  
Article
Multifractal Structures and the Energy-Economic Efficiency of Chinese Cities: Using a Classification-Based Multifractal Method
by Jiaxin Wang, Bin Meng and Feng Lu
Fractal Fract. 2025, 9(2), 96; https://doi.org/10.3390/fractalfract9020096 - 3 Feb 2025
Viewed by 387
Abstract
Improper urban spatial structure can lead to problems such as traffic congestion, long commuting times, and diseconomies of scale. Evaluating the efficiency of urban spatial structure is an important means to enhance the sustainable development of cities. The fractal method has been widely [...] Read more.
Improper urban spatial structure can lead to problems such as traffic congestion, long commuting times, and diseconomies of scale. Evaluating the efficiency of urban spatial structure is an important means to enhance the sustainable development of cities. The fractal method has been widely used in the identification and efficiency evaluation of urban spatial structure due to its sufficient characterization of urban complexity. However, the identification of urban fractal structures has expanded from monofractal structures to multifractal structures, while the efficiency evaluation of urban fractal structures remains limited to the single-dimensional efficiency evaluations of single fractals, seriously affecting the reliability of urban fractal structure evaluation. Therefore, this study identifies and evaluates urban spatial structure within the unified framework of multifractal analysis. Specifically, a classification-based multifractal method is introduced to identify the multifractal structure of 290 cities in China. An iterative application of the geographic detector method is used to evaluate the comprehensive energy-economic efficiency of urban multifractal structures. The results indicate that the 290 Chinese cities include 6 typical multifractal structures. The explanatory power of these six typical multifractal structures for urban energy-economic efficiency is 16.27%. The advantageous multifractal structures of cities that achieve higher energy-economic efficiency rates satisfy a cubic polynomial form. By comparing them with the advantageous multifractal structures, the main problems affecting the efficiency of urban multifractal structures in the other five types of cities are shown to include overly strong or weak concentration capacity of high-level centers, weak hierarchical structures among centers, and the spreading of low-level centers. Full article
(This article belongs to the Special Issue Fractional Processes and Systems in Computer Science and Engineering)
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25 pages, 11115 KiB  
Article
Enhancing Banking Transaction Security with Fractal-Based Image Steganography Using Fibonacci Sequences and Discrete Wavelet Transform
by Alina Iuliana Tabirca, Catalin Dumitrescu and Valentin Radu
Fractal Fract. 2025, 9(2), 95; https://doi.org/10.3390/fractalfract9020095 - 2 Feb 2025
Viewed by 592
Abstract
The growing reliance on digital banking and financial transactions has brought significant security challenges, including data breaches and unauthorized access. This paper proposes a robust method for enhancing the security of banking and financial transactions. In this context, steganography—hiding information within digital media—is [...] Read more.
The growing reliance on digital banking and financial transactions has brought significant security challenges, including data breaches and unauthorized access. This paper proposes a robust method for enhancing the security of banking and financial transactions. In this context, steganography—hiding information within digital media—is valuable for improving data protection. This approach combines biometric authentication, using face and voice recognition, with image steganography to secure communication channels. A novel application of Fibonacci sequences is introduced within a direct-sequence spread-spectrum (DSSS) system for encryption, along with a discrete wavelet transform (DWT) for embedding data. The secret message, encrypted through Fibonacci sequences, is concealed within an image and tested for effectiveness using the Mean Square Error (MSE) and Peak Signal-to-Noise Ratio (PSNR). The experimental results demonstrate that the proposed method achieves a high PSNR, particularly for grayscale images, enhancing the robustness of security measures in mobile and online banking environments. Full article
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18 pages, 424 KiB  
Article
Finite-Time Stability of Fractional-Order Switched Systems Based on Lyapunov Method
by Tian Feng, Lizhen Wang and Yangquan Chen
Fractal Fract. 2025, 9(2), 94; https://doi.org/10.3390/fractalfract9020094 - 2 Feb 2025
Viewed by 315
Abstract
This paper investigates the finite-time stability of a class of fractional-order switched systems with order 0<α<1, employing the fractional Lyapunov direct method. First, based on the Mittag-Leffler function and Gronwall inequality, two corresponding sufficient conditions are presented to [...] Read more.
This paper investigates the finite-time stability of a class of fractional-order switched systems with order 0<α<1, employing the fractional Lyapunov direct method. First, based on the Mittag-Leffler function and Gronwall inequality, two corresponding sufficient conditions are presented to ensure the finite-time stability of the considered system. Second, in consideration of the effectiveness of dwell time technique in switched systems, a sufficient condition is derived under a minimum average dwell time constraint. Finally, numerical simulations are performed to validate the effectiveness of the theoretical formulation. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Chaotic and Complex Systems)
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19 pages, 5031 KiB  
Article
Fractal Characterization and Pore Evolution in Coal Under Tri-Axial Cyclic Loading–Unloading: Insights from Low-Field NMR Imaging and Analysis
by Zelin Liu, Senlin Xie, Yajun Yin and Teng Su
Fractal Fract. 2025, 9(2), 93; https://doi.org/10.3390/fractalfract9020093 - 1 Feb 2025
Viewed by 365
Abstract
Coal resource extraction and utilization are essential for sustainable development and economic growth. This study integrates a pseudo-triaxial mechanical loading system with low-field nuclear magnetic resonance (NMR) to enable the preliminary visualization of coal’s pore-fracture structure (PFS) under mechanical stress. Pseudo-triaxial and cyclic [...] Read more.
Coal resource extraction and utilization are essential for sustainable development and economic growth. This study integrates a pseudo-triaxial mechanical loading system with low-field nuclear magnetic resonance (NMR) to enable the preliminary visualization of coal’s pore-fracture structure (PFS) under mechanical stress. Pseudo-triaxial and cyclic loading–unloading tests were combined with real-time NMR monitoring to model porosity recovery, pore size evolution, and energy dissipation, while also calculating the fractal dimensions of pores in relation to stress. The results show that during the compaction phase, primary pores are compressed with limited recovery after unloading. In the elastic phase, both adsorption and seepage pores transform significantly, with most recovering post-unloading. After yield stress, new fractures and pores form, and unloading enhances fracture connectivity. Seepage pore porosity shows a negative exponential relationship with axial strain before yielding, and a logarithmic relationship afterward. The fractal dimension of adsorption pores decreases during compaction and increases afterward, while the fractal dimension of seepage pores decreases before yielding and increases post-yielding. These findings provide new insights into the flow patterns of methane in coal seams. Full article
(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)
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31 pages, 817 KiB  
Article
Qualitative Analysis of Generalized Power Nonlocal Fractional System with p-Laplacian Operator, Including Symmetric Cases: Application to a Hepatitis B Virus Model
by Mohamed S. Algolam, Mohammed A. Almalahi, Muntasir Suhail, Blgys Muflh, Khaled Aldwoah, Mohammed Hassan and Saeed Islam
Fractal Fract. 2025, 9(2), 92; https://doi.org/10.3390/fractalfract9020092 - 1 Feb 2025
Viewed by 391
Abstract
This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD with a [...] Read more.
This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD with a tunable power parameter within a non-singular kernel, enabling a nuanced representation of memory effects not achievable with traditional fixed-kernel derivatives. This flexible framework is analyzed using fixed-point theory, rigorously establishing the existence and uniqueness of solutions for four symmetric cases under specific conditions. Furthermore, we demonstrate the Hyers–Ulam stability, confirming the robustness of these solutions against small perturbations. The versatility and generalizability of this framework is underscored by its application to an epidemiological model of transmission of Hepatitis B Virus (HBV) and numerical simulations for all four symmetric cases. This study presents findings in both theoretical and applied aspects of fractional calculus, introducing an alternative framework for modeling complex systems with memory processes, offering opportunities for more sophisticated and accurate models and new avenues for research in fractional calculus and its applications. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)
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23 pages, 13775 KiB  
Article
Physics-Informed Fractional-Order Recurrent Neural Network for Fast Battery Degradation with Vehicle Charging Snippets
by Yanan Wang, Min Wei, Feng Dai, Daijiang Zou, Chen Lu, Xuebing Han, Yangquan Chen and Changwei Ji
Fractal Fract. 2025, 9(2), 91; https://doi.org/10.3390/fractalfract9020091 - 1 Feb 2025
Viewed by 370
Abstract
To handle and manage battery degradation in electric vehicles (EVs), various capacity estimation methods have been proposed and can mainly be divided into traditional modeling methods and data-driven methods. For realistic conditions, data-driven methods take the advantage of simple application. However, state-of-the-art machine [...] Read more.
To handle and manage battery degradation in electric vehicles (EVs), various capacity estimation methods have been proposed and can mainly be divided into traditional modeling methods and data-driven methods. For realistic conditions, data-driven methods take the advantage of simple application. However, state-of-the-art machine learning (ML) algorithms are still kinds of black-box models; thus, the algorithms do not have a strong ability to describe the inner reactions or degradation information of batteries. Due to a lack of interpretability, machine learning may not learn the degradation principle correctly and may need to depend on big data quality. In this paper, we propose a physics-informed recurrent neural network (PIRNN) with a fractional-order gradient for fast battery degradation estimation in running EVs to provide a physics-informed neural network that can make algorithms learn battery degradation mechanisms. Incremental capacity analysis (ICA) was conducted to extract aging characteristics, which could be selected as the inputs of the algorithm. The fractional-order gradient descent (FOGD) method was also applied to improve the training convergence and embedding of battery information during backpropagation; then, the recurrent neural network was selected as the main body of the algorithm. A battery dataset with fast degradation from ten EVs with a total of 5697 charging snippets were constructed to validate the performance of the proposed algorithm. Experimental results show that the proposed PIRNN with ICA and the FOGD method could control the relative error within 5% for most snippets of the ten EVs. The algorithm could even achieve a stable estimation accuracy (relative error < 3%) during three-quarters of a battery’s lifetime, while for a battery with dramatic degradation, it was difficult to maintain such high accuracy during the whole battery lifetime. Full article
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30 pages, 1100 KiB  
Article
Probing Malware Propagation Model with Variable Infection Rates Under Integer, Fractional, and Fractal–Fractional Orders
by Nausheen Razi, Ambreen Bano, Umar Ishtiaq, Tayyab Kamran, Mubariz Garayev and Ioan-Lucian Popa
Fractal Fract. 2025, 9(2), 90; https://doi.org/10.3390/fractalfract9020090 - 1 Feb 2025
Viewed by 397
Abstract
Malware software has become a pervasive threat in computer and mobile technology attacks. Attackers use this software to obtain information about users of the digital world to obtain benefits by hijacking their data. Antivirus software has been developed to prevent the propagation of [...] Read more.
Malware software has become a pervasive threat in computer and mobile technology attacks. Attackers use this software to obtain information about users of the digital world to obtain benefits by hijacking their data. Antivirus software has been developed to prevent the propagation of malware, but this problem is not yet under control. To develop this software, we have to check the propagation of malware. In this paper, we explore an advanced malware propagation model with a time-delay factor and a variable infection rate. To better understand this model, we use fractal–fractional theory. We use an exponential decay kernel for this. For theoretical purposes (existence, uniqueness, and stability), we use the results from fixed-point theory, and, for numerical purposes, a Lagrange two-point interpolation polynomial is used to develop an algorithm. Matlab R2016a is used for simulation, and the physical significance is assessed. We examine the impact of different fractal and fractional orders for various parameters. Moreover, we compare four different mathematical models (classical, fractional, fractal, and fractal–fractional). Also, constant and variable fractional and fractal orders are compared using graphs. We investigate the idea that significant perturbation in infected nodes might be due to minor changes. This work may help with developing antivirus strategies in real life. Full article
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30 pages, 1986 KiB  
Article
Representation of Special Functions by Multidimensional A- and J-Fractions with Independent Variables
by Roman Dmytryshyn and Serhii Sharyn
Fractal Fract. 2025, 9(2), 89; https://doi.org/10.3390/fractalfract9020089 - 28 Jan 2025
Viewed by 528
Abstract
The paper deals with the problem of representing special functions by branched continued fractions, particularly multidimensional A- and J-fractions with independent variables, which are generalizations of associated continued fractions and Jacobi continued fractions, respectively. A generalized Gragg’s algorithm is constructed that [...] Read more.
The paper deals with the problem of representing special functions by branched continued fractions, particularly multidimensional A- and J-fractions with independent variables, which are generalizations of associated continued fractions and Jacobi continued fractions, respectively. A generalized Gragg’s algorithm is constructed that enables us to compute, by the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional A- and J-fractions with independent variables. Presented below are numerical experiments for approximating some special functions by these branched continued fractions, which are similar to fractals. Full article
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19 pages, 894 KiB  
Article
Fixed/Preassigned Time Synchronization of Impulsive Fractional-Order Reaction–Diffusion Bidirectional Associative Memory (BAM) Neural Networks
by Rouzimaimaiti Mahemuti, Abdujelil Abdurahman and Ahmadjan Muhammadhaji
Fractal Fract. 2025, 9(2), 88; https://doi.org/10.3390/fractalfract9020088 - 28 Jan 2025
Viewed by 452
Abstract
This study delves into the synchronization issues of the impulsive fractional-order, mainly the Caputo derivative of the order between 0 and 1, bidirectional associative memory (BAM) neural networks incorporating the diffusion term at a fixed time (FXT) and a predefined time (PDT). Initially, [...] Read more.
This study delves into the synchronization issues of the impulsive fractional-order, mainly the Caputo derivative of the order between 0 and 1, bidirectional associative memory (BAM) neural networks incorporating the diffusion term at a fixed time (FXT) and a predefined time (PDT). Initially, this study presents certain characteristics of fractional-order calculus and several lemmas pertaining to the stability of general impulsive nonlinear systems, specifically focusing on FXT and PDT stability. Subsequently, we utilize a novel controller and Lyapunov functions to establish new sufficient criteria for achieving FXT and PDT synchronizations. Finally, a numerical simulation is presented to ascertain the theoretical dependency. Full article
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23 pages, 1656 KiB  
Article
A Comparative Study of Fractal Models Applied to Artificial and Natural Data
by Gil Silva, Fernando Pellon de Miranda, Mateus Michelon, Ana Ovídio, Felipe Venturelli, João Parêdes, João Ferreira, Letícia Moraes, Flávio Barbosa and Alexandre Cury
Fractal Fract. 2025, 9(2), 87; https://doi.org/10.3390/fractalfract9020087 - 28 Jan 2025
Viewed by 696
Abstract
This paper presents an original and comprehensive comparative analysis of eight fractal analysis methods, including Box Counting, Compass, Detrended Fluctuation Analysis, Dynamical Fractal Approach, Hurst, Mass, Modified Mass, and Persistence. These methods are applied to artificially generated fractal data, such as Weierstrass–Mandelbrot functions [...] Read more.
This paper presents an original and comprehensive comparative analysis of eight fractal analysis methods, including Box Counting, Compass, Detrended Fluctuation Analysis, Dynamical Fractal Approach, Hurst, Mass, Modified Mass, and Persistence. These methods are applied to artificially generated fractal data, such as Weierstrass–Mandelbrot functions and fractal Brownian motion, as well as natural datasets related to environmental and geophysical domains. The objectives of this research are to evaluate the methods’ capabilities in capturing fractal properties, their computational efficiency, and their sensitivity to data fluctuations. Main findings indicate that the Dynamical Fractal Approach consistently demonstrated the highest accuracy across different datasets, particularly for artificial data. Conversely, methods like Mass and Modified Mass showed limitations in complex fractal structures. For natural datasets, including meteorological and geological data, the fractal dimensions varied significantly across methods, reflecting their differing sensitivities to structural complexities. Computational efficiency analysis revealed that methods with linear or logarithmic complexity, such as Persistence and Compass, are most suited for larger datasets, while methods like DFA and Dynamic Fractal Approaches required higher computational resources. This study provides an original comparative study for researchers to select appropriate fractal analysis techniques based on dataset characteristics and computational limitations. Full article
(This article belongs to the Section Engineering)
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16 pages, 1487 KiB  
Article
Hybrid Dynamic Event-Triggered Interval Observer Design for Nonlinear Cyber–Physical Systems with Disturbance
by Hongrun Wu, Jun Huang, Yong Qin and Yuan Sun
Fractal Fract. 2025, 9(2), 86; https://doi.org/10.3390/fractalfract9020086 - 26 Jan 2025
Viewed by 407
Abstract
This paper investigates the state estimation problem for nonlinear cyber–physical systems (CPSs). To conserve system resources, we propose a novel hybrid dynamic event-triggered mechanism (ETM) that prevents the occurrence of Zeno behavior. This work is based on designing an interval observer under the [...] Read more.
This paper investigates the state estimation problem for nonlinear cyber–physical systems (CPSs). To conserve system resources, we propose a novel hybrid dynamic event-triggered mechanism (ETM) that prevents the occurrence of Zeno behavior. This work is based on designing an interval observer under the hybrid dynamic ETM to solve the state reconstruction problem of Lipschitz nonlinear CPSs subject to disturbances. That is, the designed triggering mechanism is integrated into the design of the Interval Observer (IO), resulting in a hybrid dynamic event-triggered interval observer (HDETIO), and the system stability and robustness are proved using a Lyapunov function, demonstrating that the observer can effectively provide interval estimation for CPSs with nonlinearity and disturbances. Compared to existing work, the primary contribution of this work is its ability to pre-specify the minimum inter-event time (MIET) and apply it to interval state estimation, enhancing its practicality for real-world physical systems. Finally, the correctness and effectiveness of the designed hybrid dynamic ETM and IO framework are validated with an example. Full article
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15 pages, 303 KiB  
Article
Asymptotic Periodicity of Bounded Mild Solutions for Evolution Equations with Non-Densely Defined and Fractional Derivative
by Jiabin Zuo, Abdellah Taqbibt, Mohamed Chaib and M’hamed Elomari
Fractal Fract. 2025, 9(2), 85; https://doi.org/10.3390/fractalfract9020085 - 26 Jan 2025
Viewed by 358
Abstract
In the present article, we establish conditions for the asymptotic periodicity of bounded mild solutions in two distinct cases of evolution equations. The first class involves non-densely defined operators, while the second class incorporates densely defined operators with fractional derivatives that generate a [...] Read more.
In the present article, we establish conditions for the asymptotic periodicity of bounded mild solutions in two distinct cases of evolution equations. The first class involves non-densely defined operators, while the second class incorporates densely defined operators with fractional derivatives that generate a semigroup of contractions. Our method integrates the theory of spectral properties of uniformly bounded continuous functions defined on the positive real semi-axis. Additionally, we apply extrapolation theory to evolution equations with non-densely defined operators. To illustrate our main results, we provide a concrete example. Full article
12 pages, 3493 KiB  
Article
On a Preloaded Compliance System of Fractional Order: Numerical Integration
by Marius-F. Danca
Fractal Fract. 2025, 9(2), 84; https://doi.org/10.3390/fractalfract9020084 - 26 Jan 2025
Viewed by 389
Abstract
In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, [...] Read more.
In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, the discontinuous right-hand side into a set-valued function. Next, via Cellina’s Theorem, the obtained set-valued differential inclusion of fractional order can be restarted as a single-valued continuous differential equation of fractional order, to which the existing numerical schemes for fractional differential equations can be applied. In this way, the delicate problem of integrating discontinuous problems of fractional order, as well as integer order, is solved by transforming the discontinuous problem into a continuous one. Also, it is noted that even the numerical methods for fractional-order differential equations can be applied abruptly to the discontinuous problem, without considering the underlying discontinuity, so the results could be incorrect. The technical example of a single-degree-of-freedom preloaded compliance system of fractional order is presented. Full article
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