Fractal Fract., Volume 9, Issue 12 (December 2025) – 76 articles

Semi-supervised learning (SSL) is critical for medical image segmentation but often struggles with network dependency and pseudo-label error accumulation. To address these issues, we propose a fractal-dimension-guided hierarchical contrastive learning dual-student framework(FD-HCL). We extend the Mean Teacher architecture with a dual-student design and introduce an independence-aware exponential moving average (I-EMA) update mechanism to mitigate model coupling. For enhanced feature learning, we devise a hierarchical contrastive learning (HCL) mechanism guided by voxel uncertainty, spanning global, high-confidence, and low-confidence regions. We further improve structural integrity by incorporating a fractal-dimension (FD)-weighted consistency loss and integrating a novel uncertainty-aware bidirectional copy–paste (UB-CP) augmentation. Extensive experiments on the LA and BraTS 2019 datasets demonstrate the state-of-the-art performance of our framework across 10% and 20% labeled data settings. On the LA dataset with 10% labeled data, our method achieved a Dice score that outperformed the best existing approach by 0.68%. Similarly, under the 10% labeling setting on the BraTS 2019 dataset, we surpassed the state-of-the-art Dice score by 0.55%. Full article

Short-duration extreme rainfall is a major trigger of flash floods and urban inundation, yet its quantification remains a profound challenge due to the scarcity of high-resolution observations. This review synthesizes how three central paradigms of nonlinear science, multifractal cascade theory, self-organized criticality (SOC) and chaos theory, provide critical insights and practical methodologies for bridging this observational gap. We examine how multifractal temporal downscaling leverages scale-invariance to derive sub-hourly rainfall statistics from coarser data. The SOC paradigm is discussed for its ability to explain the power-law statistics of rainfall extremes and cluster properties, offering a physical basis for estimating rare events. The role of chaos theory and its modern evolution into complex network analysis is explored for diagnosing predictability and spatiotemporal organization. By comparing and integrating these perspectives plus recent developments in stochastic hydrology, this review highlights their collective potential to advance the estimation, understanding, and prediction of short-duration extreme rainfall, ultimately informing improved risk assessment and climate resilience strategies. Full article

Theaim of this research is to establish existence and uniqueness results for the Riemann–Liouville fractional integral equation of order $\alpha$$\varkappa \left(t\right)=f\left(t\right)+{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g\left(s,\varkappa \left(s\right)\right)ds,t\in [0,1],$ by developing common fixed point theorems for generalized contractions involving control functions of two variables in the framework of complex valued suprametric spaces. The proposed results extend and generalize several existing findings in the literature, and some illustrative examples are provided to demonstrate the novelty and applicability of the main theorem. Full article

This paper proposes a novel fractional-order African vulture optimization algorithm (FO-AVOA) for solving the optimal reactive power dispatch (ORPD) problem. By integrating fractional calculus into the conventional AVOA framework, the proposed method enhances the exploration–exploitation balance, accelerates convergence, and improves solution robustness. The ORPD problem is formulated as a constrained optimization task with the objective of minimizing real power losses while satisfying generator voltage limits, transformer tap ratios, and reactive power compensator constraints. The general optimization capability of the FO-AVOA is verified using the CEC 2017, 2020, and 2022 benchmark functions. In addition, the method is applied to the IEEE 30-bus and IEEE 57-bus test systems. The results demonstrate significant power loss reductions of up to 15.888% and 24.39% for the IEEE 30-bus and IEEE 57-bus systems, respectively, compared with the conventional AVOA and other state-of-the-art optimization algorithms, along with strong robustness and stability across independent runs. These findings confirm the effectiveness of the FO-AVOA as a reliable optimization tool for modern power system applications. Full article

This paper investigates the synchronization problem of fractional-order coupled neural networks (FOCNNs) featuring higher-order interactions and mixed delays under an event-dependent intermittent control framework. The proposed model incorporates higher-order interactions to more accurately capture the complex cooperative behaviors observed in real neural systems. To facilitate analysis, several novel theoretical tools are developed, which extend the existing framework of fractional-order control and provide upper bounds for the solutions of delayed fractional-order systems. Furthermore, a new event-dependent intermittent controller is designed, and sufficient synchronization criteria are rigorously derived. Finally, numerical simulations are presented to verify the effectiveness and robustness of the proposed control strategy. Full article

This paper develops a unified synchronization framework for octonion-valued fractional-order neural networks (FOOVNNs) subject to mixed delays, Lévy disturbances, and topology switching. A fractional sliding surface is constructed by combining $I^{1-\mu }{e}_g$ with integral terms in powers of $|e_g|$. The controller includes a nonsingular term $-{\rho }_{2g}{s}_g^{c_2}\mathrm{sign}\left(s_g\right)$, a disturbance-compensation term $-{\widehat{\theta }}_g\mathrm{sign}\left(s_g\right)$, and a delay-feedback term $-{\lambda }_g{e}_g(t-\tau )$, while dimension-aware adaptive laws ${\phantom{,}}^C{D}_t^{\mu }{\rho }_g=k_{1g}N{\parallel s_g\parallel }^{c_2}$ and ${\phantom{,}}^C{D}_t^{\mu }{\widehat{\theta }}_g=k_{2g}N\parallel s_g\parallel$ ensure scalability with network size. Fixed-time convergence is established via a fractional stochastic Lyapunov method, and predefined-time convergence follows by a time-scaling of the control channel. Markovian switching is treated through a mode-dependent Lyapunov construction and linear matrix inequality (LMI) conditions; non-Gaussian perturbations are handled using fractional Itô tools. The architecture admits observer-based variants and is implementation-friendly. Numerical results corroborate the theory: (i) Two-Node Baseline: The fixed-time design drives ${\parallel e\left(t\right)\parallel }_1$ to $\mathcal{O}\left({10}^{-4}\right)$ by $t\approx 0.94\mathrm{s}$, while the predefined-time variant meets a user-set $T_p=0.5\mathrm{s}$ with convergence at $t\approx 0.42\mathrm{s}$. (ii) Eight-Node Scalability: Sliding surfaces settle in an $\mathcal{O}\left(1\right)$ band, and adaptive parameter means saturate well below their ceilings. (iii) Hyperspectral (Synthetic): Reconstruction under Lévy contamination achieves a competitive PSNR consistent with hypercomplex modeling and fractional learning. (iv) Switching Robustness: under four modes and twelve random switches, the error satisfies ${max}_t{\parallel e\left(t\right)\parallel }_1\le 0.15$. The results support octonion-valued, fractionally damped controllers as practical, scalable mechanisms for robust synchronization under non-Gaussian noise, delays, and time-varying topologies. Full article

Fractional-order chaotic systems describe complex dynamic processes with memory effects and long-range correlations, while integer-order chaotic systems are generally viewed as a special case of fractional-order counterparts. This close relationship often renders the two difficult to distinguish in practice. However, existing studies mostly design analytical methods for integer-order or fractional-order chaotic systems separately, lacking a unified classification framework that does not rely on prior assumptions about the system order. In this paper, we propose a multi-branch deep learning model integrating a multi-scale convolutional neural network, time–frequency analysis, Transformer blocks, and dynamic memory network to classify integer-order chaos, fractional-order chaos, and steady states. Experiments are conducted on time series from canonical chaotic systems—including the Lorenz, Rössler, Lü, and Chen systems—in both integer- and fractional-order formulations, under two data generation protocols: varying initial conditions and varying system parameters. We evaluate the model under two scenarios: (1) assessing baseline classification performance on noise-free data and (2) testing robustness against increasing levels of Gaussian, salt-and-pepper and Rayleigh noise. The model achieves classification accuracy above 99% on noise-free data across all tested systems. Under noise interference, it demonstrates strong robustness: accuracy remains above 89.7% under high-intensity Gaussian noise. Moreover, we demonstrate that the model trained with fixed system parameters but varying initial conditions generalizes poorly to unseen parameter settings, whereas training with diverse system parameters—while fixing initial conditions—markedly improves generalization. This work offers a data-driven framework for distinguishing integer- and fractional-order chaos and highlights the critical role of training data diversity in building generalizable classifiers for dynamical systems. Full article

Understanding the temporal dynamics of urban air pollution is essential for effective environmental management, yet the nonlinear and scale-dependent behavior of key pollutants remains insufficiently explored. This study examines the multifractal properties of fine particulate matter ($P{M}_{2.5}$), ozone ($O_3$), and the Air Quality Index (AQI) across four major urban locations in California—Los Angeles, Orange, Riverside–Rubidoux, and Riverside–Mira Loma—regions characterized by persistent air-quality challenges and high population exposure. Using Multifractal Detrended Fluctuation Analysis (MF-DFA), we assess long-range dependence, heterogeneity, and cross-pollutant interactions to address the central question of whether these pollutants exhibit genuine multifractal behavior and how it varies across locations. The results reveal strong multifractality in all series, with spectrum widths ranging from 0.42 to 0.71 for $P{M}_{2.5}$-AQI and from 0.28 to 0.46 for $O_3$-AQI, indicating pronounced scale-dependent variability. Los Angeles consistently exhibited the widest spectra, reflecting greater temporal complexity. The generalized Hurst exponent at $q=2$ remained between 0.52 and 0.58 across all pollutant pairs, indicating persistent dynamics. Surrogate-data testing further confirmed that 60–75% of the observed multifractality arises from intrinsic long-range correlations rather than distributional effects. Overall, this study demonstrates that urban air pollutants in California display rich multifractal structures that differ systematically across regions, reflecting local emission patterns and atmospheric processes. These findings highlight the relevance of multifractal analysis as a powerful tool for improving air-quality modeling, forecasting, and policy design in densely populated environments. Full article

This study presents a fractional-order dynamical model for diabetes progression, formulated by extending an existing obesity model using the Atangana–Baleanu fractional derivative, termed the Atangana–Baleanu Fractional Diabetes Model (ABFDM). We rigorously establish the existence, uniqueness, positivity, and boundedness of solutions, ensuring the model’s epidemiological and biological validity. The Ulam–Hyers $\left(\mathbb{UH}\right)$ stability of the ABFDM is also demonstrated, confirming the system’s robustness against perturbations in initial conditions and parameter uncertainties. Numerical simulations, informed by population data from Saudi Arabia, indicate that increasing treatment coverage fourfold reduces uncontrolled diabetes ($D_U$) by approximately 73% and diabetes with complications ($D_W$) by about 68%. The greatest improvements occur when treatment is increased tenfold, further lowering prediabetes ($D_P$) by approximately 89% and diabetic complications ($D_W$) by about 73%. These results highlight that optimized, targeted interventions effectively control diabetes progression and mitigate the burden of related complications. These findings demonstrate that targeted treatment strategies can effectively mitigate diabetes progression within the fractional-order modeling framework. Full article

This study proposes an automatic denatured recognition method of biological tissue during high-intensity focused ultrasound (HIFU) therapy. The technique integrates ultrasonic phase space reconstruction (PSR) with a convolutional block attention mechanism-enhanced EfficientNet-B0 model (CBAM-EfficientNet-B0). Ultrasonic echo signals are first transformed into high-dimensional phase space reconstruction trajectory diagrams using PSR, which reveal distinct fractal and chaotic characteristics to analyze tissue complexity. The CBAM module is incorporated into EfficientNet-B0 to enhance feature extraction from these nonlinear dynamic representations by focusing on critical channels and spatial regions. The network is further optimized with Dropout and Scaled Exponential Linear Units (SeLUs) to prevent overfitting, alongside a cosine annealing learning rate scheduler. Experimental results demonstrate the superior performance of the proposed CBAM-EfficientNet-B0 model, achieving a high recognition accuracy of 99.57% and outperforming five benchmark CNN models (EfficientNet-B0, ResNet101, DenseNet201, ResNet18, and VGG16). The method avoids the subjectivity and uncertainty inherent in traditional manual feature extraction, enabling effective identification of HIFU-induced tissue denaturation. This work confirms the significant potential of combining nonlinear dynamics, fractal analysis, and deep learning for accurate, real-time monitoring in HIFU therapy. Full article

The present study explores bone healing patterns induced by orthodontic (OE) and surgical extrusion (SE) of structurally compromised teeth, where extrusion techniques are commonly used in rehabilitation. Changes in the trabecular bone were assessed by means of fractal analysis (FA) of consecutive periapical radiographs. (2) The present study is a retrospective case–control study. Pre- and post-treatment periapical radiographs from 44 adults undergoing orthodontic (OE) or surgical extrusion (SE) were retrieved. The radiographs were taken at T0 (pre-treatment), T1 (post-treatment), T2 (3-month follow-up), and T3 (6-month follow-up). Bone density (fractal dimension, FD) was analyzed in the apical and proximal bone regions (ROIs) of the extruded teeth, and both intra-group and inter-group differences were examined. (3) In all the regions of interest (ROIs), statistically significant intra-group differences in terms of bone density (FD) for both groups were found. In the OE group, the FD value increased, respectively, at T1, T2, and T3 in the apical ROI, compared to T0. For the proximal ROI, nearly the same trend was observed, respectively, at T1, T2, and T3 versus T0. As for the SE group, a statistically significant increase in the apical ROI was noted at T1, T2, and T3 when compared to T0. The same trend was registered in the proximal ROI compared to T0. However, no statistically significant inter-group differences in FD were detected between the two groups. (4) Orthodontic extrusion and surgical extrusion both resulted in an increased bone density (FD) despite the different healing patterns. Further prospective studies with a longer follow-up in this field are required. Full article

The pore complexity and heterogeneity in porous media display obvious fractal characteristics, which can be characterized by the fractal dimension for the pore tortuosity (D_T) and the fractal dimension for the pore size (D_f). Correspondingly, a three-dimensional (3D) fractal permeability model for porous media is proposed based on the D_T and D_f. The accuracy of the proposed model is verified by the classical theoretical relation of the permeability versus porosity, the measured permeability, and the previous study. The sensitivity analysis of model parameters (D_f, D_T, λ_min and λ_max) based on elasticity coefficient indicates that the proposed model is much more sensitive to D_f and D_T than λ_min and λ_max, and more sensitive to D_f than D_T. The proposed model is much more sensitive to λ_min than λ_max. Furthermore, the proposed model is compared with the modified Kozeny–Carman equation. The root mean square error (RMSE) analysis shows that the RMSE of the proposed model and the modified Kozeny–Carman equation in predicting permeability are 8.9857 × 10⁻⁴ and 0.5082, exhibiting high prediction accuracy of the proposed model. The proposed fractal permeability model achieves a more accurate characterization of the fluid transport by more comprehensively describing the complexity and tortuosity of pore structure, which can also provide the prospective theoretical significance and method reference for predicting the permeability of 3D porous media. Full article

Difference schemes for the numerical solution of fractional differential equations rely on discretizations of the fractional derivative. In this paper, we obtain the second-order expansion formula for the L1 approximation of the Caputo fractional derivative. Second-order approximations of the fractional derivative are constructed based on the expansion formula and parameter-dependent discretizations of the second derivative. Examples illustrating the application of these approximations to the numerical solution of ordinary and partial fractional differential equations are presented, and the convergence and order of the difference schemes are proved. Numerical experiments are also provided, confirming the theoretical predictions for the accuracy of the numerical methods. Full article

This study presents an efficient numerical scheme for solving the generalized nonlinear time-fractional Klein–Gordon equation. The Caputo time-fractional derivative is discretized using a conventional finite-difference approach, while the spatial domain is approximated with uniform hyperbolic polynomial B-splines. These discretizations are coupled through the $\theta$-weighted scheme. The uniform hyperbolic polynomial B-spline framework extends classical spline theory by incorporating hyperbolic functions, thereby enhancing flexibility and smoothness in curve and surface representations—features particularly useful for problems exhibiting hyperbolic characteristics. A rigorous stability and convergence analysis of the proposed method is provided. The effectiveness of the scheme is further validated through numerical experiments on benchmark problems. The results demonstrate up to two orders of magnitude improvement in $L_{\infty }$ error norms compared to prior spline methods. This substantial accuracy enhancement highlights the robustness and efficiency of the proposed approach for fractional partial differential equations. Full article

The outbreaks of large-scale epidemics, such as COVID-19 in 2019–2022, challenge modelers. Beside the effect of the incubation period of the virus, the delay property of detection should be also stressed. This kind of memory effect affects the entire change rate, which cannot be reflected by the conventional instantaneous derivative. The fractional derivative (FD) meets this request to some extent. Yet the shortcoming of it limits its usage. Through a strict modeling approach, a new susceptible–infective–removed (SIR) model with the memory-dependent derivative (MDD) has been constructed. The numerical simulations indicate that (1) the neglecting of the incubation period may underestimate the number of susceptible individuals and overestimate the infected ones; (2) the neglecting of the treatment period may badly overestimate the removed individuals; (3) the consequence of tardy detection intervention may be very serious, and the infectious rate may increase rapidly with a postponed peak time; and (4) the SIR model with the FD yields bad estimations, not only in the primary stage but also in the subsequent evolution. Due to the reasonability of the new SIR model with the MDD, it is suggested to epidemic researchers. Full article

In this manuscript, we investigate a fractional-order conformable three-dimensional chaotic financial model with interest rate, investment demand, and price index compartments. On the application of fixed-point theorems and nonlinear analysis, we establish theoretical results regarding the existence and uniqueness of a solution and also study Ulam–Hyers criteria for the stability of the solution of the considered system. Further, we use the fractional-order Runge–Kutta (RK-4) method to approximate the solution of our problem. Also, deep neural network (DNN) techniques are applied to investigate the model from artificial intelligence (AI) perspectives. Numerical simulation shows that it reproduces accurately the qualitative dynamics and confirms the theoretical stability results of the mentioned system. Subsequently, for the DNN analysis, we follow the Levenberg–Marquardt algorithm using Matlab 2023. Different quantities like the root-mean-square error (RMSE), mean squared error (MSE), and regression coefficient and a comparison with numerical data are presented graphically. Also, absolute errors between numerical values and those predicted by DNNs corresponding to different fractional orders are presented. Full article

In this work, we propose a simple analytical/semi-analytical approach to solve various φ,ψ-fractional partial differential equations (φ,ψ-FPDEs) using initial and boundary conditions (ICs, BCs) depending on the φ,ψ-Double Elzaki transform (φ,ψ-DET) method. The suggested approach takes advantage of a DPET modification that works well with φ,ψ-fractional operators. The proposed method not only solves the φ,ψ-FPDEs but also reduces them to a more straightforward algebraic recurrence issue. This simple yet powerful idea can be used to solve φ,ψ-FPDEs in science and engineering. We contrast the outcomes of the stated computational examples with exact solutions in order to verify the exactness and efficacy of this methodology. Full article

A modified Enzyme Action Optimizer (mEAO) is proposed to tune a Fractional-Order Proportional–Integral–Derivative (FOPID) controller for precise temperature regulation of a nonlinear continuous stirred tank reactor (CSTR). The nonlinear reactor model, adopted from a standard benchmark formulation widely used in CSTR control studies, is employed as the simulation reference. The tuning framework operates in a simulation-based manner, as the optimizer relies solely on the time-domain responses to evaluate a composite cost function combining overshoot, settling time, rise time, and steady-state error. Comparative simulations involving EAO, Starfish Optimization Algorithm (SFOA), Success History-based Adaptive Differential Evolution with Linear population size reduction (L-SHADE), and Particle Swarm Optimization (PSO) demonstrate that the proposed mEAO achieves the lowest cost value, the fastest convergence, and superior transient performance. Further comparisons with classical tuning methods, Rovira 2DOF-PID, Ziegler–Nichols PID, and Cohen–Coon PI, confirm improved tracking accuracy and smoother actuator behavior. Robustness analyses under varying set-points, feed-temperature disturbances, and measurement noise confirm stable temperature regulation without retuning. These findings demonstrate that the mEAO-based FOPID controller provides an efficient and reliable optimization framework for a nonlinear thermal-process control, with strong potential for future real-time and multi-reactor applications. Full article

The fractal extension of the time-dependent Ginzburg–Landau (TDGL) equation, formulated within the framework of Scale Relativity, generalizes superconducting dynamics to non-differentiable space–time. Although analytically well established, its numerical solution remains difficult because of the strong coupling between amplitude and phase curvature. Here we develop two complementary deep learning solvers for the fractal TDGL (FTDGL) system. The Fractal Physics-Informed Neural Network (F-PINN) embeds the Scale-Relativity covariant derivative through automatic differentiation on continuous fields, whereas the Fractal Graph Neural Network (F-GNN) represents the same dynamics on a sparse spatial graph and learns local gauge-covariant interactions via message passing. Both models are trained against finite-difference reference data, and a parametric study over the dimensionless fractality parameter $\mathcal{D}$ quantifies its influence on the coherence length, penetration depth, and peak magnetic field. Across multivortex benchmarks, the F-GNN reduces the relative L₂ error on ${\left|\psi \right|}^2$ from 0.190 to 0.046 and on B_z from approximately 0.62 to 0.36 (averaged over three seeds). This ≈4× improvement in condensate-density accuracy corresponds to a substantial enhancement in vortex-core localization—from tens of pixels of uncertainty to sub-pixel precision—and yields a cleaner reconstruction of the 2π phase winding around each vortex, improving the extraction of experimentally relevant observables such as ξ_eff, λ_eff, and local B_z peaks. The model also preserves flux quantization and remains robust under 2–5% Gaussian noise, demonstrating stable learning under experimentally realistic perturbations. The $\mathcal{D}$—scan reveals broader vortex cores, a non-monotonic variation in the penetration depth, and moderate modulation of the peak magnetic field, while preserving topological structure. These results show that graph-based learning provides a superior inductive bias for modeling non-differentiable, gauge-coupled systems. The proposed F-PINN and F-GNN architectures therefore offer accurate, data-efficient solvers for fractal superconductivity and open pathways toward data-driven inference of fractal parameters from magneto-optical or Hall-probe imaging experiments. Full article

This research employs a Caputo fractional-derivative model to investigate the effects of magnetic field inclination and thermal radiation on the unsteady flow of a Casson fluid over an inclined plate in a porous medium. The model incorporates memory effects to generalize the classical formulation, while also accounting for internal heat generation and a chemical reaction. The governing equations are solved analytically using the Laplace transform, yielding power-series solutions in the time domain. Convergence analysis and benchmarking confirm the reliability and accuracy of the derived solutions. Key physical parameters are analyzed, and their impacts on the system are presented both graphically and in tabular form. The results indicate that increasing the inclination of the plate and magnetic field significantly suppresses the velocity distribution and reduces the associated boundary-layer thickness. Conversely, a higher fractional-order parameter enhances the velocity, temperature, and species concentration profiles by reducing memory effects. This study makes a significant contribution to the fractional modeling of unsteady heat and mass transfer in complex non-Newtonian fluids and provides valuable insights for the precise control of transport processes in industrial, chemical, and biomedical applications. Full article

We extend the classical fractal uncertainty principle (FUP) framework in time-frequency analysis by exploring several novel directions. First, we generalize the FUP beyond the classical Gaussian window by investigating non-Gaussian windows and the corresponding generalized Fock space techniques. Second, we develop uncertainty estimates in alternative joint representations, including the continuous wavelet transform and directional representations such as shearlets. Third, we study fractal uncertainty on random and anisotropic fractal sets, providing probabilistic and geometric refinements of the FUP. Fourth, we connect these results with semiclassical and microlocal analysis, thereby elucidating the role of fractal geometry in resonance theory and pseudodifferential operators. Finally, we extend the analysis beyond Gaussian Gabor multipliers by considering non-Gaussian generating functions and irregular lattice samplings. Our results yield new operator norm estimates and spectral properties, with potential applications in signal processing, quantum mechanics, and numerical analysis. Full article

Researchers have devised numerous methods to model intricate behaviors in phenomena that unfold in multiple stages. This work focuses on a specific category of piecewise hybrid terminal systems characterized by delay. To account for hereditary memory effects, which are absent in standard integer-order systems, our framework partitions the time interval into two distinct phases. The initial phase employs the classical derivative, while the subsequent phase utilizes the Atangana–Baleanu–Caputo (ABC) fractional derivative. We establish conditions that guarantee both the existence and uniqueness of solutions through the application of suitable fixed-point arguments. Furthermore, Hyers–Ulam (H-U) stability is investigated to ascertain the robustness and reliability of the derived solutions. To exemplify these theoretical findings, we present a fractional-order tuberculosis treatment model that incorporates the development of drug resistance, alongside a general numerical example. Numerical simulations reveal that changes in the fractional order influence the dynamic behavior of the disease. Full article

This study employs the modified extended direct algebraic method (MEDAM) to investigate the generalized nonlinear fractional $(3+1)$-dimensional wave equation with gas bubbles. This advanced analytical framework is used to construct a comprehensive class of exact wave solutions and explore the associated dynamical characteristics of diverse wave structures. The analysis yields several categories of soliton solutions, including rational, hyperbolic (sech, tanh), and trigonometric (sec, tan) function forms. To the best of our knowledge, these soliton solutions have not been previously documented in the existing literature. By selecting appropriate standards for the permitted constraints, the qualitative behaviors of the derived solutions are illustrated using polar, contour, and two- and three-dimensional surface graphs. Furthermore, a stability analysis is performed on the obtained soliton solutions to ascertain their robustness and dynamical stability. The suggested analytical approach not only deepens the theoretical understanding of nonlinear wave phenomena but also demonstrates substantial applicability in various fields of applied sciences, particularly in engineering systems, mathematical physics, and fluid mechanics, including complex gas–liquid interactions. Full article

This work focuses on the finite-time synchronization control (FTSC) for fractional-order chaotic systems (FOCSs) subject to uncertain dynamics, unknown parameters and input nonlinearities. In the control law design, the uncertain dynamics of the FOCSs are addressed by using fuzzy logic systems (FLSs), while the unknown control direction caused by unknown input nonlinearity is handled through applying the Nussbaum gain function (NGF) method. Parameter adaptive laws are derived to estimate the unknown parameters of the given FOCSs, the parameter vectors of the FLSs, and unknown bounded constants, respectively. By integrating these parameter-adaptive laws with the FT backstepping control framework and FO Lyapunov direct method, an adaptive fuzzy FTSC strategy is developed. This strategy ensures that the synchronization error (SE) can converge to a small neighborhood of zero (SNoZ) within a FT and all signals of the closed-loop system (CLS) remain ultimately bounded. In the end, three simulation cases are utilized to demonstrate the efficiency of the proposed control method. Full article

In this paper, we consider the family of parameterized Mandelbrot-like sets generated as any point $c\in \mathbb{C}\setminus \left\{0\right\}$ of the complex plane belongs to any member of this family for a real parameter $t\ge 1$, provided that its corresponding orbit of 0 does not escape to infinity under iteration $f_c^n\left(0\right)={\left[f_c^{n-1}\left(0\right)\right]}^2+log{c}^t$; otherwise, it is not a member of this set. This classically means there is only a binary membership possibility for all points. Here, we call this type of fractal set a Mandelblog set, and then we introduce a membership function that assigns a degree to each c to be an element of a fuzzy Mandelblog set under the iterations, even if the orbits of the points are not limited. Moreover, we provide numerical examples and gray-scale graphics that illustrate the membership degrees of the points of the fuzzy Mandelblog sets under the effects of iteration parameters. This approach enables the formation of graphs for these fuzzy fractal sets by representing points that belong to the set as white pixels, points that do not belong as black pixels, and other points, based on their membership degrees, as gray-toned pixels. Furthermore, the membership function facilitates the direct proofs of the symmetry criteria for these fractal sets. Full article

In this paper, we study the local stability of an incommensurate fractional reaction–diffusion glycolysis model. The glycolysis process, fundamental to cellular metabolism, exhibits complex dynamical behaviors when formulated as a nonlinear reaction–diffusion system. To capture the heterogeneous memory effects often present in biochemical and chemical processes, we extend the classical model by introducing incommensurate fractional derivatives, where each species evolves with a distinct fractional order. We linearize the system around the positive steady state and derive sufficient conditions for local asymptotic stability by analyzing the eigenvalues of the associated Jacobian matrix under fractional-order dynamics. The results demonstrate how diffusion and non-uniform fractional orders jointly shape the stability domain of the system, highlighting scenarios where diffusion destabilizes homogeneous equilibria and others where incommensurate memory effects enhance stability. Numerical simulations are presented to illustrate and validate the theoretical findings. Full article

Extracting high-quality surface wave dispersion curves from crosscorrelation functions (CCFs) of ambient noise data is critical for seismic velocity inversion and subsurface structure interpretation. However, the non-uniform spatial distribution of noise sources may introduce spurious noise into CCFs, significantly reducing the signal-to-noise ratio (SNR) of empirical Green’s functions (EGFs) and degrading the accuracy of dispersion measurement and velocity inversion. To mitigate this issue, this study aims to develop an effective denoising approach that enhances the quality of CCFs and facilitates more reliable surface wave extraction. We propose a fractional Laplacian-based adaptive progressive denoising (FLAPD) method that leverages local gradient information and a fractional Laplacian mask to estimate noise variance and construct a fractional bilateral kernel for iterative noise removal. We applied the proposed method to the CCFs from 79 long-period seismic stations in Shandong, China. The results demonstrate that the denoising method enhanced the data quality substantially, increasing the number of reliable dispersion curves from 1094 to 2196, and allowing an increased number of temporal sampling points to be retrieved from previously low-SNR curves. This helps to expand the spatial coverage and results in a more accurate velocity inversion result than that without denoising. This study provides a robust denoising solution for ambient noise tomography in regions with complex noise source distributions. Full article

This study presents a novel softsign-function-based fractional-order proportional–integral–derivative (softsign-FOPID) controller optimized using the fungal growth optimizer (FGO) for the stabilization and precise position control of an unstable magnetic ball suspension system. The proposed controller introduces a smooth nonlinear softsign function into the conventional FOPID structure to limit abrupt control actions and improve transient smoothness while preserving the flexibility of fractional dynamics. The FGO, a recently developed bio-inspired metaheuristic, is employed to tune the seven controller parameters by minimizing a composite objective function that simultaneously penalizes overshoot and tracking error. This optimization ensures balanced transient and steady-state performance with enhanced convergence reliability. The performance of the proposed approach was extensively benchmarked against four modern metaheuristic algorithms (greater cane rat algorithm, catch fish optimization algorithm, RIME algorithm and artificial hummingbird algorithm) under identical conditions. Statistical analyses, including boxplot comparisons and the nonparametric Wilcoxon rank-sum test, demonstrated that the FGO consistently achieved the lowest objective function value with superior convergence stability and significantly better (p < 0.05) performance across multiple independent runs. In time-domain evaluations, the FGO-tuned softsign-FOPID exhibited the fastest rise time (0.0089 s), shortest settling time (0.0163 s), lowest overshoot (4.13%), and negligible steady-state error (0.0015%), surpassing the best-reported controllers in the literature, including the sine cosine algorithm-tuned PID, logarithmic spiral opposition-based learning augmented hunger games search algorithm-tuned FOPID, and manta ray foraging optimization-tuned real PIDD². Robustness assessments under fluctuating reference trajectories, actuator saturation, sensor noise, external disturbances, and parametric uncertainties (±10% variation in resistance and inductance) further confirmed the controller’s adaptability and stability under practical non-idealities. The smooth nonlinearity of the softsign function effectively prevented control signal saturation, while the fractional-order dynamics enhanced disturbance rejection and memory-based adaptability. Overall, the proposed FGO-optimized softsign-FOPID controller establishes a new benchmark in nonlinear magnetic levitation control by integrating smooth nonlinear mapping, fractional calculus, and adaptive metaheuristic optimization. Full article

Pore–throat structure and gas distribution are critical factors in evaluating the quality of tight sandstone reservoirs and hydrocarbon resource potential. Twelve tight sandstone samples from the Lower Permian Shihezi Formation in Hangjin Banner, Ordos Basin, were selected for CTS, X-ray diffraction, HPMI, and gas displacement NMR analyses. By converting the T₂ spectra into pore–throat distributions and applying fractal methods, we quantitatively analyzed the influences of multiple factors on gas distribution characteristics across different pore–throat sizes. The main results are as follows: All samples exhibit a three-stage pore–throat distribution, defining mesopores, micropores, and nanopores; quartz content mainly influences the fractal dimension of mesopores by enhancing structural stability and gas storage capacity, whereas clay minerals control the fractal characteristics of nanopores by increasing pore–throat complexity. An increase in clay mineral content increases the fractal dimension, indicating stronger reservoir heterogeneity and consequently poorer gas-bearing capacity. Larger pore–throat parameters (R_m, S_k, and S_max) correspond to lower fractal dimensions, indicating better connectivity and greater gas storage capacity. Among these factors, pore–throat parameters exert the most significant influence on the fractal dimensions of mesopores and micropores, jointly determining the overall connectivity and the upper limit of the reservoir’s gas-bearing capacity. The results demonstrate that larger pore–throat parameters and higher quartz content help reduce the fractal dimension and enhance the gas-bearing capacity of tight reservoirs. This research enhances understanding of pore–throat structures and gas-bearing capacity in low-permeability reservoirs and provides a theoretical basis for exploration, development, and enhanced recovery in the study area. Full article