Fractal Fract., Volume 9, Issue 10 (October 2025) – 59 articles

Using a Donsker-type construction, we prove the existence of a new class of processes, which we call the self-stabilizing processes. These processes have a particular property: the “local intensities of jumps” vary with the values. Moreover, we also show that the self-stabilizing processes have many other good properties, such as stochastic Hölder continuity and strong localizability. Such a self-stabilizing process is simultaneously a Markov process, a martingale (when the local index of stability is greater than 1), a self-scaling process and a self-regulating process. Full article

In response to the growing complexity of global exploration targets, traditional Euclidean geometric and linear statistical methods reveal inherent theoretical limitations in characterizing hydrocarbon reservoirs as complex geological bodies that exhibit simultaneous local disorder and global order. Fractal theory, with its core parameter systems such as fractal dimension and scaling exponents, provides an innovative mathematical–physics toolkit for quantifying spatial heterogeneity and resolving the multi-scale characteristics of reservoirs. This review systematically consolidates recent advancements in the application of fractal theory to oil and gas resource assessment, with the aim of elucidating its transition from a theoretical concept to a practical tool. We conclusively demonstrate that fractal theory has driven fundamental methodological progress across four critical dimensions: (1) In reservoir classification and evaluation, fractal dimension has emerged as a robust quantitative metric for heterogeneity and facies discrimination. (2) In pore structure characterization, the theory has successfully uncovered structural self-similarity across scales, from nanopores to macroscopic vugs, enabling precise modeling of complex pore networks. (3) In seepage behavior analysis, fractal-based models have significantly enhanced the predictive capacity for non-Darcy flow and preferential migration pathways. (4) In fracture network modeling, fractal geometry is proven pivotal for accurately characterizing the spatial distribution and connectivity of natural fractures. Despite significant progress, current research faces challenges, including insufficient correlation with dynamic geological processes and a scarcity of data for model validation. Future research should focus on the following directions: developing fractal parameter inversion methods integrated with artificial intelligence, constructing dynamic fractal–seepage coupling models based on digital twins, establishing a unified fractal theoretical framework from pore to basin scale, and expanding its application in low-carbon energy fields such as carbon dioxide sequestration and natural gas hydrate development. Through interdisciplinary integration and methodological innovation, fractal theory is expected to advance hydrocarbon resource assessment toward intelligent, precise, and systematic development, providing scientific support for the efficient exploitation of complex reservoirs and the transition to green, low-carbon energy. Full article

Uncertainty is the biggest issue when modeling real-world multi-level fractional optimization problems. In this paper, a fully intuitionistic fuzzy multi-level multi-objective fractional programming problem (FIF-MLMOFPP) is tackled via two different approaches. Because of the ambiguity introduced in the model, all the parameters and decision variables in each objective function and feasible domain are intuitionistic fuzzy numbers (IFNs). Firstly, FIF-MLMOFPP is converted into a non-fractional fully intuitionistic fuzzy multi-level multi-objective programming problem (FIF-MLMOPP) utilizing a series of transformations. The accuracy functions and ordering relations of IFNs are employed to transform the non-fractional FIF-MLMOPP into a deterministic variant. An interactive approach is first applied to solve the problem by transforming it into discrete multi-objective optimization problems (MOOPs). Each separate MOOP addresses the ϵ-constraint methodology and the goal of satisfactoriness. Neutrosophic fuzzy goal programming (NFGP) is the second approach applied to solve the FIF-MLMOFPP, as the marginal evaluations of predetermined neutrosophic fuzzy objectives for all functions at each level are attained through various membership functions, including degrees of truth, indeterminacy, and falsehood, within neutrosophic uncertainty. The NFGP algorithm is presented to achieve optimal levels for each marginal evaluation objective by minimizing their deviation variables, thus yielding a suitable solution. To confirm and approve the two suggested approaches, a numerical example and a comparison between them are presented. Finally, recommendations for additional research are given. Full article

This work investigates how fractionality affects the dynamical behavior of dust-acoustic shock waves that arise and propagate in a depleted-electron complex plasma. This model consists of inertial negatively charged dust grains and inertialess nonextensive distributed ions. Initially, the fluid model equations that govern the propagation of nonlinear dust-acoustic shock waves are reduced to the integer Burgers-type equations using the reductive perturbation method. Thereafter, the integer Burgers-type equations are converted to the fractional cases using a suitable transformation. For analyzing this fractional family, both the Tantawy technique and the new iterative method are implemented within the Caputo sense framework. These methods can produce highly accurate analytical approximations without necessitating stringent assumptions or intricate computational processes, in contrast to other similar methods. Numerical examples and the calculation of the absolute error demonstrate the efficacy of the suggested methodologies, emphasizing their superior precision and swift convergence. Full article

With the rise of embodied agents and indoor service robots, object detection has become a critical component supporting semantic mapping, path planning, and human–robot interaction. However, indoor scenes often face challenges such as severe occlusion, large-scale variations, small and densely packed objects, and complex textures, making existing methods struggle in terms of both robustness and accuracy. This paper proposes MDF-YOLO, a multi-domain fusion detection framework based on Hölder regularity guidance. In the backbone, neck, and feature recovery stages, the framework introduces the CrossGrid Memory Block, Hölder-Based Regularity Guidance–Hierarchical Context Aggregation module, and Frequency-Guided Residual Block, achieving complementary feature modeling across the state space, spatial domain, and frequency domain. In particular, the HG-HCA module uses the Hölder regularity map as a guiding signal to balance the dynamic equilibrium between the macro and micro paths, thus achieving adaptive coordination between global consistency and local discriminability. Experimental results show that MDF-YOLO significantly outperforms mainstream detectors in metrics such as mAP@0.5, mAP@0.75, and mAP@0.5:0.95, achieving values of 0.7158, 0.6117, and 0.5814, respectively, while maintaining near real-time inference efficiency in terms of FPS and latency. Ablation studies further validate the independent and synergistic contributions of CGMB, HG-HCA, and FGRB in improving small-object detection, occlusion handling, and cross-scale robustness. This study demonstrates the potential of Hölder regularity and multi-domain fusion modeling in object detection, offering new insights for efficient visual modeling in complex indoor environments. Full article

The aim of this paper is to establish the existence and uniqueness of positive solutions to the non-local Brézis–Oswald-type fractional problems that involve fractional $(r,q)$-Laplace operators and Hardy potentials. In particular, we observe an eigenvalue problem associated with the fractional $(r,q)$-Laplacian to determine the existence of at least one positive weak solution for our problem. The main features of this paper are the lack of the semicontinuity property of an energy functional related to our problem and the presence of a singular coefficient. The decisive tool for overcoming this technical difficulty is the concentration–compactness principle in fractional, critical and Hardy terms. Moreover, we establish the uniqueness results of Brézis–Oswald–type by exploiting a generalization of the discrete Picone inequality. Full article

In this work, the advection-dispersion model (ADM) is time-fractionalized by the exploitation of Atangana-Baleanu (AB) differential operator to describe contaminant transport in a geological environment. Dispersion, adsorption, and decay, which are known as the foremost transport mechanisms, are considered. The exact solutions of the suggested Atangana-Baleanu advection-dispersion models (AB-ADMs) are acquired using Fourier sine transform and Laplace transform. The classical ADMs are demonstrated to be the special limiting cases of the suggested models. The high consistency among the suggested models and experimental data denotes that the AB-ADMs characterize contaminant transport more effectively. Additionally, the corresponding numerical and graphical results are explored to demonstrate the necessity, effectiveness, and suitability of the suggested models. Full article

In DC microgrids, the combination of pulsed loads and renewable energy sources significantly impairs system stability, especially in highly dynamic operating environments. The resilience and reaction time of conventional proportional–integral (PI) controllers are often inadequate when managing the nonlinear dynamics of hybrid energy storage systems. This research suggests a frequency-decomposed fractional-order control strategy for stabilizing DC microgrids with solar, batteries, and supercapacitors. The control architecture divides system disturbances into low- and high-frequency components, assigning high-frequency compensation to the ultracapacitor (UC) and low-frequency regulation to the battery, while a fractional-order controller (FOC) enhances dynamic responsiveness and stability margins. The proposed approach is implemented and assessed in MATLAB/Simulink (version R2023a) using comparison simulations against a conventional PI-based control scheme under scenarios like pulsed load disturbances and fluctuations in renewable generation. Grey Wolf Optimizer (GWO), a metaheuristic optimization procedure, has been used to tune the parameters of the FOPI controller. The obtained results using the same conditions were compared using an optimal fractional-order PI controller (FOPI) and a conventional PI controller. The microgrid with the best FOPI controller was found to perform better than the one with the PI controller. Consequently, the objective function is reduced by 80% with the proposed optimal FOPI controller. The findings demonstrate that the proposed method significantly enhances DC bus voltage management, reduces overshoot and settling time, and lessens battery stress by effectively coordinating power sharing with the supercapacitor. Also, the robustness of the proposed controller against parameters variations has been proven. Full article

The complexity of geological structures significantly impacts both mining production efficiency and operational safety, making its quantitative assessment a core issue in ensuring coal’s safe production and coalbed methane development. Focusing on the Wugou Coal Mine in Anhui Province, which exhibits multi-phase tectonic superposition, modification, and relatively complex structural characteristics, this study integrates stereographic projection analysis, fractal theory, and multiple structural curvature methods to quantitatively characterize structural types and evaluate complexity. The results show that the Wugou Coal Mine has undergone four main stages of tectonic deformation since the formation of the coal seam. The superposition and modification of tectonic events of different periods and properties have led to a complex structural pattern. The fractal dimension effectively characterizes the development degree and distribution density of faults. Structural curvature not only intuitively reflects the deformation extent of fold bending and fault separation, but also provides valuable insights into the structural types, structural positions, and the characteristics of superimposed folds. By combining the strengths of fractal analysis and curvature characterization, a fractal-curvature integrated evaluation model was developed to assess structural complexity. This model facilitates a high-resolution quantitative evaluation, delineating the geological structures of the Wugou Coal Mine into zones of extremely complex, complex, moderately complex, and simple structures. The findings not only provide accurate geological guidance for mine design and hazard prevention but also offer a quantitative evaluation methodology for the optimal selection of favorable areas for coalbed methane development. Full article

The aim of this article is to introduce the distributed-order hyperchaotic detuned (DOHD) laser model. Its dissipative dynamics, invariance, and fixed points (FPs) and their stability are investigated. Numerical solutions of the DOHD laser model are computed using the modified Predictor–Corrector approach. Its viscoelasticity is described by the so-called DO derivative, allowing for the study of different technical systems and materials, and the model is found to have a whole circle of FPs as a hyperchaotic attractor. We discuss the coexistence of more attractors under various initial conditions and the same sets of parameters for our model (multistability). We also introduce the notion of dual combination synchronization (DCS), using four integer-order drive models and two DO response models. A theorem is stated and proved to obtain an analytical control function that ensures DCS for our models. Numerical simulations are presented to support these analytical results. Regarding the use of the well–known Caputo derivative, the results are very similar to those of DO, except when the Caputo order, $0<\sigma \le 1$, is very close to 1, where the dynamics shows a “spiralling behavior” towards a fixed point. In all other cases, both Caputo and DO exhibit a very similar behavior. Full article

We develop a convergence framework for Grünwald–Letnikov (GL) fractional and classical integer difference operators acting on sequences in fuzzy-paranormed (fp) spaces, motivated by data that are imprecise and contain sporadic outliers. Fuzzy paranorms provide a resolution-dependent notion of proximity, while statistical and lacunary statistical convergence downweight sparse deviations by natural density; together, they yield robust criteria for difference-filtered signals. Within this setting, we establish uniqueness of fp–${\mathsf{\Delta }}^m$ statistical limits; an equivalence between fp-statistical convergence of ${\mathsf{\Delta }}^m$ (and its GL extension ${\mathsf{\Delta }}^{\alpha }$) and fp-strong p-Cesàro summability; an equivalence between lacunary fp-${\mathsf{\Delta }}^m$ statistical convergence and blockwise strong p-Cesàro summability; and a density-based decomposition into a classically convergent part plus an fp-null remainder. We also show that GL binomial weights act as an ${\ell }^1$ convolution, ensuring continuity of ${\mathsf{\Delta }}^{\alpha }$ in the fp topology, and that nabla/delta forms are transferred by the discrete Q–operator. The usefulness of the criteria is illustrated on simple engineering-style examples (e.g., relaxation with memory, damped oscillations with bursts), where the fp-Cesàro decay of difference residuals serves as a practical diagnostic for Cesàro compliance. Beyond illustrative mathematics, we report engineering-style diagnostics where the fuzzy Cesàro residual index correlates with measurable quantities (e.g., vibration amplitude and energy surrogates) under impulsive disturbances and missing data. We also calibrate a global decision threshold ${\tau }_{\mathrm{glob}}$ via sensitivity analysis across $(\alpha ,p,m)$, where $m\in \mathbb{N}$ is the integer difference order, $\alpha >0$ is the fractional order, and $p\ge 1$ is the Cesàro exponent, and provide quantitative baselines (median/M-estimators, ${\ell }_1$ trend filtering, Gaussian Kalman filtering, and an $\alpha$-stable filtering structure) to show complementary gains under bursty regimes. The results are stated for integer m and lifted to fractional orders $\alpha >0$ through the same binomial structure and duality. Full article

The primary objective of this study is to provide a more precise and beneficial mathematical model for assessing corruption dynamics by utilizing non-local derivatives. This research aims to provide solutions that accurately capture the complexities and practical behaviors of corruption. To illustrate how corruption levels within a community change over time, a non-linear deterministic mathematical model has been developed. The authors present a non-integer order model that divides the population into five subgroups: susceptible, exposed, corrupted, recovered, and honest individuals. To study these corruption dynamics, we employ a new method for solving a time-fractional corruption model, which we term the q-homotopy analysis transform approach. This approach produces an effective approximation solution for the investigated equations, and data is shown as 3D plots and graphs, which give a clear physical representation. The stability and existence of the equilibrium points in the considered model are mathematically proven, and we examine the stability of the model and the equilibrium points, clarifying the conditions required for a stable solution. The resulting solutions, given in series form, show rapid convergence and accurately describe the model’s behaviour with minimal error. Furthermore, the solution’s uniqueness and convergence have been demonstrated using fixed-point theory. The proposed technique is better than a numerical approach, as it does not require much computational work, with minimal time consumed, and it removes the requirement for linearization, perturbations, and discretization. In comparison to previous approaches, the proposed technique is a competent tool for examining an analytical outcomes from the projected model, and the methodology used herein for the considered model is proved to be both efficient and reliable, indicating substantial progress in the field. Full article

The goal of this paper is to give an error analysis of a finite difference method for a time-fractional Black–Scholes equation with weakly singular solutions. The time Gerasimov-Caputo derivative is discretized by the L1 scheme on a graded mesh designed to compensate for the initial singularities, and a standard finite difference method is used for spatial discretization on a uniform mesh. A discrete comparison principle is presented for the fully discrete scheme, and stability and convergence of the scheme in maximum norm are established by constructing some appropriate barrier functions. Furthermore, an $\alpha$-robust pointwise error estimate of the fully discrete scheme on a uniform mesh is given. Finally, some numerical results are presented to show the sharpness of the error estimate. Full article

Fractional differential equations are used to model complex systems where present dynamics depend on past states. In this work, we study a linear fractional model with two Caputo orders that captures long-term memory together with short-term adaptation. The existence and uniqueness of solutions are established using Banach and Krasnoselskii’s fixed-point theorems. A parameter study isolates the roles of the fractional orders and coefficients, yielding an explicit stability region in the $(\alpha ,\beta )$–plane via computable contraction bounds. For computation, we implement the Adams–Bashforth–Moulton (ABM) and fractional linear multistep (FLM) methods, comparing accuracy and convergence. As an application, we model animal learning in which proficiency evolves under memory effects and pulsed stimuli. The results quantify the impact of feedback timing on trajectories within the admissible region, thereby illustrating the suitability of dual-order fractional models for memory-driven behavior. Full article

The Yang local fractional setting provides the generalized framework to explore the non-differentiable mappings considering the local properties. Due to the dominance of these concepts, mathematicians have investigated multiple problems, including mathematical modelling, optimization, and inequalities. Incorporating these useful concepts, this study aims to derive Weddle-type integral inequalities within the context of fractal space. To achieve the intended results, we establish a new local fractional identity. By using this identity, the convexity property, the bounded property of mappings, the L-Lipschitzian property of mappings, and other famous inequalities, we develop numerous upper bounds. Additionally, we provide 2D and 3D graphical representations and numerous applications, which show the significance of our main findings. To the best of our knowledge, this is the first study concerning error inequalities of Weddle’s quadrature formulation within the fractal space. Full article

This paper presents a novel fractional-order field-oriented control (FO-FOC) strategy for induction motor drives in electric vehicles (EVs) powered by an autonomous photovoltaic (PV) battery energy system. The proposed control approach integrates a fractional-order sliding mode controller (FO-SMC) into the conventional FOC framework to enhance dynamic performance, improve robustness, and reduce sensitivity to parameter variations. The originality of this work lies in the combined use of fractional-order control and real-time adaptive parameter updating, applied within a PV battery-powered EV platform. This dual-layer control structure allows the system to effectively reject disturbances, maintain torque and flux tracking, and mitigate the effects of component aging or thermal drift. Furthermore, to address the chattering phenomenon typically associated with sliding mode control, a continuous saturation function was employed, resulting in smoother voltage and current responses more suitable for real-time implementation. Extensive simulation studies were conducted under ideal conditions, with parameter mismatch, and with the proposed adaptive update laws. Results confirmed the superiority of the FO-based approach over classical integer-order designs in terms of speed tracking, flux regulation, torque ripple reduction, and system robustness. The proposed methodology offers a promising solution for next-generation sustainable mobility systems requiring high-performance, energy-efficient, and fault-tolerant electric drives. Full article

Rod-pumping systems represent complex nonlinear systems. Traditional soft-sensing methods used for efficiency prediction in such systems typically rely on complicated fractional-order partial differential equations, severely limiting the real-time capability of efficiency estimation. To address this limitation, we propose an approximate efficiency prediction model for nonlinear fractional-order differential systems based on selective heterogeneous ensemble learning. This method integrates electrical power time-series data with fundamental operational parameters to enhance real-time predictive capability. Initially, we extract critical parameters influencing system efficiency using statistical principles. These primary influencing factors are identified through Pearson correlation coefficients and validated using p-value significance analysis. Subsequently, we introduce three foundational approximate system efficiency models: Convolutional Neural Network-Echo State Network-Bidirectional Long Short-Term Memory (CNN-ESN-BiLSTM), Bidirectional Long Short-Term Memory-Bidirectional Gated Recurrent Unit-Transformer (BiLSTM-BiGRU-Transformer), and Convolutional Neural Network-Echo State Network-Bidirectional Gated Recurrent Unit (CNN-ESN-BiGRU). Finally, to balance diversity among basic approximation models and predictive accuracy, we develop a selective heterogeneous ensemble-based approximate efficiency model for nonlinear fractional-order differential systems. Experimental validation utilizing actual oil-well parameters demonstrates that the proposed approach effectively and accurately predicts the efficiency of rod-pumping systems. Full article

This paper innovatively achieves finite-time modified function projection synchronization (MFPS) for different fractional-order chaotic systems. By leveraging the advantages of radial basis function (RBF) neural networks in nonlinear approximation, this paper proposes a novel fractional-order sliding-mode controller. It is designed to address the issues of system model uncertainty and external disturbances. Based on Lyapunov stability theory, it has been demonstrated that the error trajectory can converge to the equilibrium point along the sliding surface within a finite time. Subsequently, the finite-time MFPS of the fractional-order hyperchaotic Chen system and fractional-order chaotic entanglement system are realized under conditions of periodic and noise disturbances, respectively. The effects of the neural network parameters on the performance of the MFPS are then analyzed in depth. Finally, a color image encryption scheme is presented integrating the above MFPS method and exclusive-or operation, and its effectiveness and security are illustrated through numerical simulation and statistical analysis. In the future, we will further explore the application of fractional-order chaotic system MFPS in other fields, providing new theoretical support for interdisciplinary research. Full article

The current study explores the fully distributed containment control problem of fractional-order time-delay multi-agent systems by introducing a novel pull-based dynamic event-triggered approach. Firstly, to reduce communication overhead and mitigate time delays in controller updates, a pull-based dynamic event-triggered strategy is proposed. Secondly, in virtue of a Lyapunov candidate function, the proposed pull-based dynamic event-triggered control protocol exhibits inherent distributed properties enabling agents to operate independently and cooperatively without global information. Thirdly, we design adaptive parameters to ensure containment control convergence and provide a rigorous proof to preclude Zeno behavior. Eventually, numerical simulations are performed to verify the validity of the theoretical analysis. Full article

Hemodynamic management is extremely important in cardiac patients undergoing surgery. Traditionally, the approach towards hemodynamic stabilization included the control of both mean arterial pressure (MAP) and cardiac output (CO) using Sodium Nitroprusside and Dopamine. More efficient and safer drugs have been introduced, such as Norepinephrine. The focus of this manuscript is to provide some preliminary results regarding the closed loop control of MAP using Norepinephrine. However, to design a dedicated control system, a mathematical model describing the effect of Norepinephrine on mean arterial pressure is required. Only a handful of papers describe a pharmacokinetic–pharmacodynamic (PK-PD) model. In this paper, a simplified model suitable for designing a controller is determined based on PK-PD insights and existing clinical data. Existing closed loop controllers are based on the simple proportional integral derivative (PID) controller, with limited robustness to patient variability. In this paper, two advanced control strategies are proposed to replace PID. The closed loop simulation results include reference tracking and disturbance rejection and show the efficiency and robustness of the proposed control algorithms. The preliminary results set the background for further research in this area. Full article

In this work, we numerically investigate the fractional clannish random walker’s parabolic equations (FCRWPEs) and the nonlinear fractional Cahn–Allen (NFCA) equation using the Hybrid Decomposition Method (HDM). The analysis uses the Atangana–Baleanu fractional derivative (ABFD) in the Caputo sense, which has a nonsingular and nonlocal Mittag–Leffler kernel (MLk) and provides a more accurate depiction of memory and heredity effects, to examine the dynamic behavior of the models. Using nonlinear analysis, the uniqueness of the suggested models is investigated, and distinct wave profiles are created for various fractional orders. The accuracy and effectiveness of the suggested approach are validated by a number of example cases, which also support the approximate solutions of the nonlinear FCRWPEs. This work provides significant insights into the modeling of anomalous diffusion and complex dynamic processes in fields such as phase transitions, biological transport, and population dynamics. The inclusion of the ABFD enhances the model’s ability to capture nonlocal effects and long-range temporal correlations, making it a powerful tool for simulating real-world systems where classical derivatives may be inadequate. Full article

Exoskeleton-assisted bodyweight support training (BWST) has demonstrated enhanced neurorehabilitation outcomes in which joint motion prediction serves as the critical foundation for adaptive human–machine interactive control. However, joint angle prediction under dynamic unloading conditions remains unexplored. This study introduces an adaptive wavelet-denoising fusion (AWDF) model to predict lower-limb joint angles during BWST. Utilizing a custom human-tracking bodyweight support system, time series data of surface electromyography (sEMG), and inertial measurement unit (IMU) from ten adults were collected across graded bodyweight support levels (BWSLs) ranging from 0% to 40%. Systematic comparative experiments evaluated joint angle prediction performance among five models: the sEMG-based model, kinematic fusion model, wavelet-enhanced fusion model, late fusion model, and the proposed AWDF model, tested across prediction time horizons of 30–150 ms and BWSL gradients. Experimental results demonstrate that increasing BWSLs prolonged gait cycle duration and modified muscle activation patterns, with a concomitant decrease in the fractal dimension of sEMG signals. Extended prediction time degraded joint angle estimation accuracy, with 90 ms identified as the optimal tradeoff between system latency and prediction advancement. Crucially, this study reveals an enhancement in prediction performance with increased BWSLs. The proposed AWDF model demonstrated robust cross-condition adaptability for hip and knee angle prediction, achieving average root mean square errors (RMSE) of 1.468° and 2.626°, Pearson correlation coefficients (CC) of 0.983 and 0.973, and adjusted R² values of 0.992 and 0.986, respectively. This work establishes the first computational framework for BWSL-adaptive joint prediction, advancing human–machine interaction in exoskeleton-assisted neurorehabilitation. Full article

The study of fractals has a long history in mathematics and signal analysis, providing formal tools to describe self-similar structures and scale-invariant phenomena. In recent years, cognitive science has developed a set of powerful theoretical and experimental tools capable of probing the representations that enable humans to extend hierarchical structures beyond given input and to generate fractal-like patterns across multiple domains, including language, music, vision, and action. These paradigms target recursive hierarchical embedding (RHE), a generative capacity that supports the production and recognition of self-similar structures at multiple scales. This article reviews the theoretical framework of RHE, surveys empirical methods for measuring it across behavioral and neural domains, and highlights their potential for cross-domain comparisons and developmental research. It also examines applications in linguistic, musical, visual, and motor domains, summarizing key findings and their theoretical implications. Despite these advances, the computational and biological mechanisms underlying RHE remain poorly understood. Addressing this gap will require linking cognitive models with algorithmic architectures and leveraging the large-scale behavioral and neuroimaging datasets generated by these paradigms for fractal analyses. Integrating theory, empirical tools, and computational modelling offers a roadmap for uncovering the mechanisms that give rise to recursive generativity in the human mind. Full article

Fractional heat conduction models extend classical formulations by incorporating fractional differential operators that capture multiscale relaxation effects. In this work, we introduce an electrical analogy that represents the action of these operators via generalized longitudinal impedance and admittance elements, thereby clarifying their physical role in energy transfer: fractional derivatives account for the redistribution of heat accumulation and dissipation within micro-scale heterogeneous structures. This analogy unifies different classes of fractional models—diffusive, wave-like, and mixed—as well as distinct fractional operator types, including the Caputo and Atangana–Baleanu forms. It also provides a general computational methodology for solving heat conduction problems through the concept of thermal impedance, defined as the ratio of surface temperature variations (relative to ambient equilibrium) to the applied heat flux. The approach is illustrated for a semi-infinite sample, where different models and operators are shown to generate characteristic spectral patterns in thermal impedance. By linking these spectral signatures of microstructural relaxation to experimentally measurable quantities, the framework not only establishes a unified theoretical foundation but also offers a practical computational tool for identifying relaxation mechanisms through impedance analysis in microscale thermal transport. Full article

This study introduces a new class of coupled differential systems described by fractal–fractional Caputo derivatives with both constant and state-dependent delays. In contrast to traditional delay differential equations, the proposed framework integrates memory effects and geometric complexity while capturing adaptive feedback delays that vary with the system’s state. Such a formulation provides a closer representation of biological and physical processes in which delays are not fixed but evolve dynamically. Sufficient conditions for the existence and uniqueness of solutions are established using fixed-point theory, while the stability of the solution is investigated via the Hyers–Ulam (HU) stability approach. To demonstrate applicability, the approach is applied to two illustrative examples, including a predator–prey interaction model. The findings advance the theory of fractional-order systems with mixed delays and offer a rigorous foundation for developing realistic, application-driven dynamical models. Full article

We introduce a class of triply coupled systems of differential equations with fractal–fractional Caputo derivatives and time-dependent delays. This framework captures long-memory effects and complex structural patterns while allowing delays to evolve over time, offering greater realism than constant-delay models. The existence and uniqueness of solutions are established using fixed point theory, and Hyers–Ulam stability is analyzed. A numerical scheme based on the Adams–Bashforth method is implemented to approximate solutions. The approach is illustrated through a numerical example and applied to a three-species food-chain model, comparing scenarios with and without time-dependent delays to demonstrate their impact on system dynamics. Full article

The predator–prey model is a fundamental mathematical tool in ecology used to understand the dynamic relationship between predator and prey populations. This study develops a fractional-order delayed dynamical model for a four-species food web, which includes an intermediate predator feeding on two prey species and a top predator preying on all three species. The boundedness of the system’s solutions is first rigorously established using the Laplace transform method. Next, a nonlinear dynamical analysis is performed to determine the existence conditions and local stability of both the trivial and positive equilibrium points. In particular, by treating the time delay as a bifurcation control parameter, explicit criteria for the onset of Hopf bifurcation are derived. Theoretically, when the delay magnitude exceeds a critical threshold, the system loses stability and exhibits sustained oscillatory behavior. Finally, systematic numerical simulations are performed under specific parameter settings. The effects of varying fractional orders and delay magnitudes on the system’s dynamics are quantitatively explored, and the results show strong agreement with the theoretical predictions. Full article

Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern the asymptotics of the heat content on self-similar fractals. We consider the Dirichlet problem for the heat equation on bounded open regions whose boundaries are self-similar fractals. The class of self-similar domains we consider allows for non-disjoint overlap of the self-similar copies, provided some control over the separation. The possible complex dimensions, determined strictly by the similitudes that define the self-similar domain, control the scaling exponents of the asymptotic expansion for the heat content. We illustrate our method in the case of generalized von Koch snowflakes and, in particular, extend known results for these fractals with arithmetic scaling ratios to the generic (in the topological sense), non-arithmetic setting. Full article

This paper is concerned with the global funnel control (FC) issue of the nonlinear systems with unknown dynamics and time-varying fractional powers. An FC strategy is proposed in this paper, not only the barrier functions but also the tracking and intermediate errors are introduced to our control law in a proportional feedback way, which not only guarantees uniform performance insurance under any initial condition of the control system but also leads to about 50% reduction in control amplitude with respect to the existing solutions. Moreover, it exhibits notable simplicity, with no need for parametric details of time-varying fractional powers, adding a power integrator technique, parameter identification, function approximation or derivative calculation. A comparative simulation demonstrates the effectiveness and superiority of the developed method. Full article