Fractal and Fractional

Automatic control in biomedicine has attracted the attention of clinicians to mitigate the side effects resulting from drug overdoses administered to patients. To provide the most optimal and accurate results, the computer-controlled systems in biomedical engineering require more advanced tuning procedures that tackle patient variability and ensure the robustness of the control system. This has been enhanced over the past two decades through the replacement of standard PID controllers with fractional-order controllers. However, most of the developed fractional-order control methods address only the robustness with respect to gain variations. In this study, a novel fractional-order control algorithm that is robust to time constant variations is developed. The control algorithm is designed for second-order plus dead time systems. A graphical solution is chosen to solve the nonlinear system of equations for the proposed approach. Three biomedical applications are employed as case studies. The first one consists in the control of the bispectral index in general anesthesia, the second one refers to the blood glucose level control for diabetic patients, and finally, the third one tackles computerized control in chemotherapy. The closed-loop simulation results validate the efficiency of the tuning method according to the accepted values of the performance specifications in the scientific literature. Full article

In light of the computational efficiency bottleneck and inadequate regional feature representation in traditional global data approximation methods, this paper introduces the concept of non-uniform partition to transform global continuous approximation into multi-region piecewise approximation. Additionally, we propose an image representation algorithm based on linear canonical transformation and non-uniform partitioning, which enables the regional representation of sub-signal features while reducing computational complexity. The algorithm first demonstrates that the two-dimensional linear canonical transformation series has a least squares solution within each region. Then, it adopts the maximum likelihood estimation method and the scale transformation characteristics to achieve conversion between the nonlinear and linear expressions of the two-dimensional linear canonical transformation series. It then uses the least squares method and the recursive method to convert the image information into mathematical expressions, realize image vectorization, and solve the approximation coefficients in each region more quickly. The proposed algorithm better represents complex image texture areas while reducing image quality loss, effectively retains high-frequency details, and improves the quality of reconstructed images. Full article

Against the backdrop of global climate change, accurate prediction of carbon emissions is crucial for formulating effective emission reduction policies. Utilizing data from the China Energy Statistical Yearbook and the Shandong Statistical Yearbook between 2010 and 2022, this study estimates carbon emissions in Shandong Province from 2016 to 2022 using the carbon emission factor method and projects future trends through the fractional-order accumulated grey model FAGM(1,1). The forecast results indicate that both total carbon emissions and per capita carbon emissions in Shandong will follow a trajectory characterized by ‘slow increase-peak-steady decline’, while carbon emission intensity is expected to decrease consistently year by year. Based on these projections, this study proposes that Shandong should accelerate the optimization of its energy supply structure to establish a clean and low-carbon energy system, promote green transformation and upgrading of industries to cultivate new economic growth drivers, and enhance policy-market coordination mechanisms to strengthen institutional incentives and constraints. These findings provide a scientific basis for Shandong to achieve its carbon peak and carbon neutrality goals and also offer methodological references for other industrialized provinces facing similar challenges. Full article

This paper discusses the asymptotic controllability of fractional-order Sobolev-type perturbed stochastic control systems with Brownian motion and nonlocal fractional-order Sobolev stochastic conditions. A new set of sufficient conditions is established using the theory of semigroups together with iterative methods, with some advancements on the Brownian motion properties. Our main results are obtained assuming that the associated linear system is controllable. A most essential example is brought to illustrate the analysis obtained. Full article

The theory of integral inequalities has a wide range of applications in physics and numerical computation, and plays a fundamental role in mathematical analysis. The present study delves into the attractive domain of Hermite–Hadamard–Mercer (H–H–M)-type inequalities having a special emphasis on Wright’s general functions, referred to as Raina’s functions in the scientific literature. The main goal of our progressive study is to use Raina’s Fractional Integrals to derive two useful lemmas for second-differentiable functions. Using the derived lemmas, we proved a large number of fractional integral inequalities related to trapezoidal and midpoint-type inequalities where those that are twice differentiable in absolute values are convex. Some of these results also generalize findings from previous research. Next, we provide applications to error estimates for trapezoidal and midpoint quadrature formulas and to analytical evaluations involving modified Bessel functions of the first kind and q-digamma functions, and we show the validity of the proposed inequalities in numerical integration and analysis of special functions. Finally, the results are well-supported by numerous examples, including graphical representations and numerical tables, which collectively highlight their accuracy and computational significance. Full article

In precision agriculture, semantic segmentation enhances the crop yield by enabling precise disease monitoring, targeted herbicide application, and accurate crop–weed differentiation. This enhances yield; reduces the overuse of herbicides, water, and fertilizers; lowers labor costs; and promotes sustainable farming. Deep-learning-based methods are particularly effective for crop and weed segmentation, and achieve potential results. Typically, segmentation is performed using homogeneous data (the same dataset is used for training and testing). However, previous studies, such as crop and weed segmentation in a heterogeneous data environment, using heterogeneous data (i.e., different datasets for training and testing) remain inaccurate. The proposed framework uses patch-based augmented limited training data within a heterogeneous environment to resolve the problems of degraded accuracy and the use of extensive data for training. We propose an attention-driven and hierarchical feature fusion network (AHFF-Net) comprising a flow-constrained convolutional block, hierarchical multi-stage fusion block, and attention-driven feature enhancement block. These blocks independently extract diverse fine-grained features and enhance the learning capabilities of the network. AHFF-Net is also combined with an open-source large language model (LLM)-based pesticide recommendation system made by large language model Meta AI (LLaMA). Additionally, a fractal dimension estimation method is incorporated into the system that provides valuable insights into the spatial distribution characteristics of crops and weeds. We conducted experiments using three publicly available datasets: BoniRob, Crop/Weed Field Image Dataset (CWFID), and Sunflower. For each experiment, we trained on one dataset and tested on another by reversing the process of the second experiment. The highest mean intersection of union (mIOU) of 65.3% and F1 score of 78.7% were achieved when training on the BoniRob dataset and testing on CWFID. This demonstrated that our method outperforms other state-of-the-art approaches. Full article

This paper considers a new nonlocal fractional differential quasi-variational inequality (NFDQVI) comprising a fractional differential equation with a nonlocal condition and a time-dependent quasi-variational inequality in Hilbert spaces. Qualitative properties of the solution for the time-dependent parameterized quasi-variational inequality are investigated, which improve some known results in the literature. Moreover, the unique existence of the solution and Hyers–Ulam stability are obtained for the novel NFDQVI under mild conditions. Finally, the obtained abstract results for NFDQVI are applied to analyze the unique solvability and stability, addressing a time-dependent multi-agent optimization problem and a time-dependent price control problem. Full article

Background: Traumatic brain injury (TBI) disrupts hippocampal neurogenesis and dendritic structure. Objective: The objective was to assess whether fractal and multifractal analyses can sensitively quantify dendritic complexity changes in newly formed dentate gyrus neurons following TBI and hyperbaric oxygen therapy (HBO). Methods: Adult rats underwent sham surgery with HBO (SHBO), lesion-induced TBI (L), or lesion-induced TBI with HBO (LHBO). Dendritic morphology was evaluated using Euclidean, monofractal, and multifractal metrics. Results: Lesioned animals exhibited marked reductions in dendritic complexity across multiple metrics compared to both HBO-treated groups. HBO treatment partially restored complexity to near-sham levels, with multifractal spectra revealing subtle structural differences between SHBO and LHBO. Conclusions: Fractal and multifractal analyses provide sensitive tools for detecting TBI-induced morphological changes and therapeutic effects. Our findings support HBO as a potential neuroprotective intervention and demonstrate the utility of mathematical modeling in evaluating therapeutic efficacy in neurotrauma. Full article

Reservoirs and caprocks overlap with each other in heterogeneous carbonate rocks. The sealing capacity of caprocks and their controlling factors are not clear, which restricts the prediction, exploration, and development of carbonate hydrocarbon reservoirs. We selected core samples from the Ordovician reservoirs and caprocks in the Tarim Basin, China, for scanning electron microscopy, thin section, breakthrough pressure (BP), high-pressure mercury intrusion porosimetry (HMIP), and nitrogen adsorption method (N₂GA). The experimental results show that the reservoir and caprock can be distinguished by BP. The BP of the reservoir is less than 3.0 MPa, and the BP of the caprock is less than 3.0 Mpa. We analyzed the heterogeneity characteristics and differences in reservoirs and caprocks with different lithologies from the perspectives of monofractal and multifractal. The results indicate that the differences in pore structure of grainstone, dolomite, and micrite/argillaceous limestone result in significant heterogeneity differences between samples. The correlation analysis between the fractal parameters and BP indicates that the characteristics of reservoir microporous structures have a decisive impact on BP (correlation coefficient > 0.7). The pore structure of the carbonate reservoir–caprock system exhibits self-similarity. The heterogeneity of the caprock has no significant control effect on BP (correlation coefficient < 0.3), while the higher the heterogeneity of the reservoir, the greater the BP. The sealing capacity of the caprock depends on the heterogeneity differences in pore types and pore structures between the reservoirs and caprocks. When both the reservoir and the caprock are grainstone, the micropores in the reservoirs and caprocks are dispersed but evenly distributed, and little heterogeneous differences can achieve sealing. When the lithology of reservoirs and caprocks is different, the enhancement of heterogeneity differences in micropores will improve the sealing capacity of the caprock. In summary, fractal dimension is an effective method for studying the heterogeneous structure and sealing capacity of pore–throat in carbonate caprocks. This study proposes a new perspective that the difference between the heterogeneity of micropore structures of reservoirs and caprocks affects the sealing capacity of carbonate rocks, and provides a new explanation and model for the sealing mode of carbonate rock caprocks. Full article

Fractional calculus of variations for a broad class of fractional operators with a general analytic kernel function is considered. Using techniques from variational analysis, we derive first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the Weierstrass necessary condition, the Legendre condition, and finally the Legendre–Clebsch condition. Our results are new in the sense that the Euler–Lagrange equation is based on duality theory, and thus build up only with left fractional operators. The Weierstrass necessary condition is a variant of strong necessary optimality condition, and it is derived from maximum condition of Pontryagin for this general analytic kernels. The Legendre–Clebsch condition is obtained under normality assumptions on data because of equality constraints. Full article

The epidemic modeling of computer virus propagation is identified as an effective approach to understanding the mechanism of virus spread. Fraction-order virus spread models exhibit remarkable advantages over their integer-order counterparts. Based on a type of bursting virus, a fractional computer virus propagation model with saturation effect is suggested. The basic properties of the model are discussed. The basic reproduction number of the model is determined. The virus–endemic equilibria of the model are determined. A criterion for the global asymptotic stability of the virus-free equilibrium is derived. For a pair of potential virus–endemic equilibria, criteria for the local asymptotic stability are presented. Some interesting properties of the model, ranging from the impact of the fractional order and the saturation index on virus spread to their coupling effect, are revealed through numerical simulations. This work helps gain a deep insight into the laws governing virus propagation. Full article

Flash floods arise from the interaction of rugged topography, short-duration intense rainfall, and rapid flow concentration. Conventional risk mapping often builds empirical indices with expert-assigned weights or trains supervised models on historical event inventories—approaches that degrade in data-scarce regions. We propose a fully data-driven, unsupervised Geographic Information System (GIS) framework based on fractional order k-means, which clusters multi-dimensional geospatial features without labeled flood records. Five raster layers—elevation, slope, aspect, 24 h maximum rainfall, and distance to the nearest stream—are normalized into a feature vector for each 30 m × 30 m grid cell. In a province-scale case study of Zhejiang, China, the resulting risk map aligns strongly with the observations: 95% of 1643 documented flash flood sites over the past 60 years fall within the combined high- and medium-risk zones, and 65% lie inside the high-risk class. These outcomes indicate that the fractional order distance metric captures physically realistic hazard gradients while remaining label-free. Because the workflow uses commonly available GIS inputs and open-source tooling, it is computationally efficient, reproducible, and readily transferable to other mountainous, data-poor settings. Beyond reducing subjective weighting inherent in index methods and the data demands of supervised learning, the framework offers a pragmatic baseline for regional planning and early-stage screening. Full article

Improving the decoding accuracy of biological signals has been a research focus for decades to advance health, automation, and robotic industries. However, challenges like inter-subject variability, data scarcity, and multifunctional variability cause low decoding accuracy, thus hindering the practical deployment of biological signal paradigms. This paper proposes a multifunctional biological signals network (Multi-BioSig-Net) that addresses the aforementioned issues by devising a novel blind few-shot learning (FSL) technique to quickly adapt to multiple target domains without needing a pre-trained model. Specifically, our proposed multimodal similarity extractor (MMSE) and self-multiple domain adaptation (SMDA) modules address data scarcity and inter-subject variability issues by exploiting and enhancing the similarity between multimodal samples and quickly adapting the target domains by adaptively adjusting the parameters’ weights and position, respectively. For multifunctional learning, we proposed inter-function discriminator (IFD) that discriminates the classes by extracting inter-class common features and then subtracts them from both classes to avoid false prediction of the proposed model due to overfitting on the common features. Furthermore, we proposed a holistic-local fusion (HLF) module that exploits contextual-detailed features to adapt the scale-varying features across multiple functions. In addition, fractal dimension estimation (FDE) was employed for the classification of left-hand motor imagery (LMI) and right-hand motor imagery (RMI), confirming that proposed method can effectively extract the discriminative features for this task. The effectiveness of our proposed algorithm was assessed quantitatively and statistically against competent state-of-the-art (SOTA) algorithms utilizing three public datasets, demonstrating that our proposed algorithm outperformed SOTA algorithms. Full article

This paper investigates the complex dynamics and fractal attractors that arise in a 60-dimensional ring lattice system of electrically coupled nonchaotic Rulkov neurons. While networks of chaotic Rulkov neurons have been widely studied, systems of nonchaotic Rulkov neurons have not been extensively explored due to the piecewise complexity of the nonchaotic Rulkov map. Here, we find that rich dynamics emerge from the electrical coupling of regular-spiking Rulkov neurons, including chaotic spiking, synchronized chaotic bursting, and synchronized hyperchaos. By systematically varying the electrical coupling strength between neurons, we also uncover general trends in the maximal Lyapunov exponent across the system’s dynamical regimes. By means of the Kaplan–Yorke conjecture, we examine the fractal geometry of the ring system’s high-dimensional chaotic attractors and find that these attractors can occupy as many as 45 of the 60 dimensions of state space. We further explore how variations in chaotic behavior—quantified by the full Lyapunov spectra—correspond to changes in the attractors’ fractal dimensions. This analysis advances our understanding of how complex collective behavior can emerge from the interaction of multiple simple neuron models and highlights the deep interplay between dynamics and geometry in high-dimensional systems. Full article

This paper introduces a dynamic model that explores smoking and optimal control strategies. It shows how fractional-order (FO) analysis has uncovered hidden parts of complex systems and provides information about previously ignored elements. This paper uses the Bernoulli wavelet operational matrix method and the Adam–Bashforth–Moulton (ABM) method to analyse this model numerically. The mathematical model is segmented into five sub-classes: susceptible smokers, ingestion class, unusual smokers, regular smokers, and ex-smokers. It considers four optimal control measures: an anti-smoking education campaign, distribution of anti-smoking gum, administration of anti-nicotine drugs, and governmental restrictions on smoking in public areas. We show in this model how to control smoking in society strategically. Full article

Supercritical CO₂ (ScCO₂) injection has emerged as a promising method to enhance shale gas recovery while simultaneously achieving CO₂ sequestration. This research investigates how ScCO₂ interacts with water and shale rock, altering the pore structure characteristics of shale reservoirs. The study examines shale samples from three marine shale formations in southern China under varying thermal and pressure regimes simulating burial conditions at 1000 m (45 °C and 10 MPa) and 2000 m (80 °C and 20 MPa). The research employs multiple analytical techniques including XRD for mineral composition analysis, MICP, N₂GA, and CO₂GA for comprehensive pore characterization, FE–SEM for visual observation of mineral and pore changes, and multifractal theory to analyze pore structure heterogeneity and connectivity. Key findings indicate that ScCO₂–water–shale interactions lead to dissolution of minerals such as kaolinite, calcite, dolomite, and chlorite, and as the reaction proceeds, substantial secondary mineral precipitation occurs, with these changes being more pronounced under 2000 m simulation conditions. Mineral dissolution and precipitation cause changes in pore structure parameters of different pore sizes, with macropores showing increased PV and decreased SSA, mesopores showing decreased PV and SSA, and micropores showing insignificant changes. Moreover, mineral precipitation effects are stronger than dissolution effects. These changes in pore structure parameters lead to alterations in multifractal parameters, with mineral precipitation reducing pore connectivity and consequently enhancing pore heterogeneity. Correlation analysis further revealed that H and D₋₁₀–D₁₀ exhibit a significant negative correlation, confirming that reduced connectivity corresponds to stronger heterogeneity, while mineral composition strongly controls the multifractal responses of macropores and mesopores, with micropores mainly undergoing morphological changes. However, these changes in micropores are mainly manifested as modifications of internal space. Siliceous shale samples exhibit stronger structural stability compared to argillaceous shale, which is attributed to the mechanical strength of the quartz framework. By integrating multifractal theory with multi–scale pore characterization, this study achieves a unified quantification of shale pore heterogeneity and connectivity under ScCO₂–water interactions at reservoir–representative pressure–temperature conditions. This novelty not only advances the methodological framework but also provides critical support for understanding CO₂–enhanced shale gas recovery mechanisms and CO₂ geological sequestration in depleted shale gas reservoirs, highlighting the complex coupling between geochemical reactions and pore structure evolution. Full article

Fractional-order models provide a powerful framework for capturing memory-dependent and viscoelastic dynamics in mechanical systems, which are often inadequately represented by classical integer-order characterizations. This study addresses the identification of dynamic parameters in both single-degree-of-freedom (1-DOF) and three-degree-of-freedom (3-DOF) Duffing oscillators with fractional damping, modeled using the Grünwald–Letnikov characterization. The 1-DOF system includes a cubic nonlinear restoring force and is excited by a harmonic input to induce steady-state oscillations. For both systems, time domain simulations are conducted to capture long-term responses, followed by Fourier decomposition to extract steady-state displacement, velocity, and acceleration signals. These components are combined with a GL-based fractional derivative approximation to construct structured regressor matrices. System parameters—including mass, stiffness, damping, and fractional-order effects—are then estimated using pseudoinverse techniques. The identified models are validated through a comparison of reconstructed and original trajectories in the phase space, demonstrating high accuracy in capturing the underlying dynamics. The proposed framework provides a consistent and interpretable approach for frequency domain system identification in fractional-order nonlinear systems, with relevance to applications such as mechanical vibration analysis, structural health monitoring, and smart material modeling. Full article

This paper addresses the three-dimensional trajectory tracking control problem for the underactuated Autonomous Underwater Vehicle (AUV) operating in complex ocean environments characterized by dynamic disturbances and model uncertainties. A super-twisting extended state observer (STESO) was designed to accurately estimate and compensate for external disturbances and unmodeled dynamics in finite time. A fractional-order proportional–integral–derivative (FOPID) controller was then developed based on the disturbance estimates provided by the STESO. Leveraging the superior frequency-domain tuning flexibility of fractional calculus, the controller enhances tracking precision and robustness against dynamic disturbances. Furthermore, a strict Lyapunov-based stability analysis is presented, and the tracking error converges to zero asymptotically when disturbance estimation errors vanish. Numerical simulations validated the effectiveness and robustness of the proposed control strategy under various disturbance scenarios. Full article

The cosmic web is one of the most complex systems in nature, consisting of galaxies and clusters of galaxies joined by filaments and walls, leaving large empty regions called cosmic voids. The most common method of describing the web is a correlation function and its derivative, the fractal function. In this paper, I provide a review of the fractal properties of the cosmic web from the observational point of view within the Newtonian concordance $\mathrm{\Lambda }$CDM Universe framework. I give a brief history of fractal studies of the Universe. I then describe the derivation of the fractal function from angular and spatial distributions of galaxies and their relations. Correlation functions are not sensitive to the shape of the galaxy distribution. To improve our quantitative understanding of properties of the web, statistics must be used which are sensitive to the pattern of the web. Full article