Fractal Fract., Volume 10, Issue 7 (July 2026) – 79 articles

Constitutive models for clays have historically treated yield geometry, plastic-flow direction, and compression as separate problems, with little regard for their interaction. This paper presents a controlled numerical assessment of how three components—Chatwong et al.’s verified teardrop yield surface, a stress-fractional flow rule, and an AJOP-derived hardening modulus— interact when coupled in a 2 × 2 × 2 factorial design. The components are integrated incrementally along one idealized shear-strain-controlled constant-p′ path with an approximate undrained variant for two independently calibrated clays (Boston Blue Clay and London Clay) under a specified state-dependent fractional order. Within this scope, the main flow effect is consistently the largest single quantity for both soils, and the flow × compression interaction is comparably large wherever defined. Compression’s role grows substantially with the overconsolidation ratio, and the main geometry effect is markedly soil-dependent, scaling with the surface-shape parameter. Two structural singularities are identified: a phase-transformation point in the teardrop surface’s non-associated flow rule, absent from the fractional rule, and a hardening singularity in the AJOP-based modulus, whose tangent falls to the swelling index at a finite, soil-dependent preconsolidation stress, bounding the evaluable overconsolidation range of the compression-related interactions; a proportional-κ variant removes this singularity by construction while preserving the factorial ranking, identifying it as a property of the constant-κ embedding, not of AJOP itself. Under an approximate undrained path, the geometry × flow interaction carries over unchanged, while compression’s role is suppressed several-fold. The borrowed yield surface and flow rule are validated independently against 379 points from real undrained triaxial tests across four calibrated soils using this paper’s own re-calibrated predictions; the fractional–AJOP framework itself is assessed for internal consistency only, and its laboratory validation, together with K₀, cyclic and multi-axial paths, remains for future work. Full article

A fractional-order equivalent circuit model (FOECM) provides a compact and physically interpretable representation of the memory-dependent polarization behavior of lithium-ion batteries. Leveraging this property, a fractional order model-guided residual-correction framework is proposed for state-of-charge (SOC) estimation, in which the FOECM, unscented Kalman filter (UKF), and long short-term memory (LSTM) residual learner are integrated into a unified estimation chain rather than treated as separate modules. In this framework, the FOECM is parameterized using Dynamic Stress Test (DST) data and incorporated into the UKF to construct the FOECM + UKF estimator. The LSTM learns the history-dependent SOC residual from sequences of measured operating signals and FOECM + UKF SOC estimates, and its output is added to the UKF estimate without replacing the fractional-order physical model. The proposed hybrid estimator is trained and configured using the available DST, Supplemental Federal Test Procedure (US06), and Federal Urban Driving Schedule (FUDS) data, and is independently evaluated on the US06 and FUDS profiles of Cell 008. Compared with the FOECM + UKF estimator, the proposed hybrid estimator reduces the SOC root mean square error (RMSE) from 2.70% to 0.90% on US06 and from 2.93% to 1.20% on FUDS, with mean absolute error (MAE) values of 0.66% and 1.03%, respectively. These results demonstrate the effectiveness of coupling fractional-order memory modeling with sequence-based residual correction under the tested dynamic operating profiles. Full article

Aiming at the limitation that the traditional integer-order dynamic model of a liquid crystal optical phased array (LCOPA) cannot simultaneously characterize the liquid crystal deformation memory effect, cross-coupling of two-dimensional deflection channels and time-delay characteristics, this paper proposes a two-dimensional coupled fractional-order dynamic modeling and parameter identification method based on fractional calculus. First, the two-dimensional beam deflection mechanism of LCOPA is analyzed to establish a static beam propagation model. Then, a fractional-order generalized Kelvin constitutive equation is adopted to construct a coupled dynamic model integrating multi-element coupling, channel cross-coupling and time-delay characteristics. A Legendre wavelet integral operational matrix is constructed to simplify fractional operations, combined with the least squares algorithm to achieve synchronous high-precision identification of multiple parameters. Finally, an experimental platform is built for verification. The results show that the proposed model achieves a fitting degree of 98% for beam deflection dynamics, with steady-state prediction error less than 0.15% and dynamic RMSE no more than 0.00020 rad, significantly outperforming integer-order models. This research provides theoretical support and engineering reference for modeling and controller design of high-precision non-mechanical beam steering systems. Full article

The main aim of this work is to introduce the notion of a tvs-valued cone suprametric space, which unifies and generalizes cone suprametric spaces, tvs-valued cone metric spaces, and cone metric spaces, and to establish new fixed point theorems for single-valued and multivalued generalized contractive mappings within this framework. The presented results extend and unify some well-known outcomes available in the existing literature. To demonstrate the applicability and significance of the proposed theory, a number of illustrative examples are provided. In addition, the obtained fixed point results are employed to investigate the existence of solutions for a Caputo fractional differential equation as well as a Volterra–Fredholm integral equation. Full article

This paper is a first step toward providing answers to the question of whether fractal pattern formation gives rise to power-law (fractional) kinetics and how such kinetics relate to geometric properties such as fractal dimension. The study focuses on Lichtenberg figures produced by high-voltage discharges on wood, a heterogeneous dielectric medium with anisotropic conductivity and variable moisture content. During breakdown, the discharge propagates through branching streamers and carbonization fronts, exhibiting scale-free growth, long-tailed waiting times, and memory effects. The associated current signals are analyzed using the theory of fractal elements developed by Nigmatullin and Chen. This framework allows complex self-similar waveforms to be decomposed into elementary fractal modes characterized by power-law exponents and amplitudes. The results show that the electrical response is governed by fractional dynamics encoded in these modes. However, no direct one-to-one relationship is found between the fractal dimension of the discharge patterns and the kinetic power-law exponents. This decoupling is attributed to the influence of the heterogeneous medium and the percolation pathways through which the discharge propagates. Full article

Kadomtsev–Petviashvili (KP)-type nonlinear dispersive wave equations play a fundamental role in the description of weakly nonlinear waves in plasmas, fluids, and nonlinear optical systems. In strongly turbulent or heterogeneous media, however, transport processes often become nonlocal and exhibit anomalous scaling behavior. Such phenomena are naturally described using fractional differential operators. Recent work has significantly advanced the mathematical understanding of fractional Kadomtsev–Petviashvili models by proving the existence of periodically modulated solitary waves and lump solutions, and by analyzing instability and other significant properties. The present paper complements this line of research by combining a plasma-oriented modeling motivation with a variational existence framework, structural properties of traveling profiles, and a Fourier spectral computational pipeline for profile construction and dynamical validation. The mathematical properties of the resulting equation are investigated. In particular, we prove the existence of traveling-wave solutions using a variational formulation and concentration-compactness arguments. To compute these coherent structures numerically, we develop a Fourier pseudospectral Petviashvili iteration scheme adapted to the fractional KP operator. The computed profiles are validated through direct time integration of the governing equation using an exponential time-differencing spectral method. The results demonstrate that fractional dispersive effects (e.g., the dependence on $\alpha$) significantly modify the structure of nonlinear plasma waves and provide a natural framework for describing wave dynamics in turbulent plasma environments. The numerical results include a detailed verification of the predicted algebraic decay law, a convergence study of the Petviashvili iteration, validation against the exact KP soliton, and dynamical stability tests over long time intervals. A quantitative comparison with existing results in the literature is also provided. Full article

This paper investigates the existence of pseudosolutions for a class of fractional retarded dynamic equations on time scales in Banach spaces endowed with the weak topology. The proposed model combines fractional dynamics, explicit delay effects, and hybrid continuous–discrete temporal structures within a unified analytical framework. The analysis is performed by means of the $\Delta$-Henstock–Kurzweil–Pettis integral, allowing significantly weaker regularity assumptions than those required by classical integration theories. The existence result is established using the De Blasi measure of weak noncompactness together with Kubiaczyk’s fixed point theorem for weakly sequentially continuous operators. The obtained theorem extends several existing results on fractional differential equations and dynamic equations on time scales by incorporating explicit delays and generalized integration into a common framework. The obtained results provide a unified analytical framework for studying hereditary systems evolving on hybrid time domains. Full article

The interleaved parallel Buck–Boost converter can reduce output voltage ripple and has been widely used in engineering practice. The application of fractional-order theory has a significant influence on model accuracy and power converter performance. Based on fractional calculus theory and the state space averaging method, this paper establishes a fractional-order mathematical model of the CCM interleaved parallel Buck–Boost converter. The steady-state operating point and ripple characteristics of the converter under the Caputo fractional-order definition are analyzed and compared with those under other fractional-order definitions. Fractional-order energy storage elements are constructed, and a fractional-order circuit simulation model of the converter is established for comparative simulation analysis. Finally, experiments are carried out to verify the effectiveness of the theoretical analysis. Full article

Thermal therapies require precise prediction of the temperature and stress distributions in skin tissue to ensure efficacy while minimizing tissue damage. Classical bioheat models rely on oversimplified boundary assumptions, such as thermally insulated surfaces and mechanically free membranes, that fail to capture the physiological environment. This study develops a fractional-order dual-phase-lag bioheat model that incorporates clinically realistic conditions, namely simultaneous convective, radiative, and evaporative heat losses, active epidermal cooling, and subcutaneous mechanical restraint modeled through a Winkler elastic foundation. Both the Caputo and the Atangana–Baleanu (ABC) fractional derivatives are employed to represent memory effects in biological tissues. Analytical solutions in the Laplace–Fourier domain are obtained using displacement potential functions, with numerical inversion carried out via the Stehfest algorithm and Gaussian quadrature. The results show that realistic boundary conditions substantially alter the thermomechanical response: convective and evaporative cooling reduce surface temperatures and penetration depths, whereas active cooling permits deeper heating without epidermal damage. The Winkler foundation yields higher compressive stresses than traction-free models, and the ABC operator produces smoother responses than the Caputo operator. Overall, the model reveals the trade-offs between thermal efficacy and mechanical safety, thereby bridging bioheat modeling and clinical practice. Full article

Regression methods are widely used to investigate macroeconomic relationships; however, they are generally estimated without first examining whether the underlying variables exhibit fractal structures, persistence, and chaotic dynamics. Although nonlinear regression models relax the assumption of linearity, they rarely account for the complex geometric, long-memory, and dynamical properties that characterize macroeconomic time series. Motivated by this limitation, this study proposes a fractal-oriented Lie regression framework that integrates fractional persistence and Lie algebra to model nonlinear macroeconomic interactions within a unified analytical structure. For Türkiye, the empirical analysis employs monthly data on inflation, interest rates, exchange rates and oil prices covering the period 2000M1–2026M1, encompassing major economic crises and structural breaks. Prior to model estimation, the dynamical characteristics of the variables are examined using entropy measures, long-range dependency analysis, Lyapunov exponents and attractors. The results reveal persistent fractal structures, significant fractional dependence and chaotic behavior, indicating that macroeconomic variables evolve within a complex nonlinear dynamical system rather than around a conventional equilibrium. Based on these results, the variables are represented within a Lie algebra framework in which nonlinear transformation matrices preserve the underlying geometric structure while simultaneously capturing both self-dynamics and cross-variable interactions. The proposed Lie regression model demonstrates substantial improvements over standard regression methods in both model adequacy and forecasting performance. Oil prices emerge as the dominant transmitter of shocks by generating pronounced asymmetric effects on inflation, exchange rates and overall macroeconomic stability. The model achieves remarkable forecasting accuracy by reducing RMSE, MAE, and MAPE from 18.58, 13.61, and 69.92 under a standard regression model to 0.27, 0.22 and 16.4, respectively. Finally, the estimated Lie transformation matrix is employed as a policy-simulation mechanism to evaluate the transmission of alternative oil-price shocks. Scenarios based on 5%, 10%, and 20% increases in oil prices quantify the resulting adjustments in inflation, interest rates, and exchange rates by providing forward-looking assessments of macroeconomic vulnerability. The proposed framework extends standard regression analysis by explicitly incorporating fractional persistence and chaotic dynamics into a Lie algebra representation, thereby offering a more accurate and theoretically consistent approach for modeling complex macroeconomic systems. Full article

Flux linkage observers have been widely employed in permanent magnet synchronous motor (PMSM) sensorless drives, and the back electromotive force (BEMF) estimated by an extended state observer (ESO) can be used to compensate for the position estimation error of the flux linkage observer. However, inverter dead time introduces periodic disturbances into the estimated synchronous reference frame, thereby contaminating the BEMF estimation and degrading the accuracy of position compensation. To this end, a position estimation error compensation strategy based on a fractional-order quasi-resonant extended state observer (FOQR-ESO) is proposed. First, a PMSM voltage model incorporating the inverter dead-time effect is established, and the resulting sixth-order harmonic component in the estimated synchronous reference frame is analyzed. Then, a fractional-order quasi-resonant element is embedded into the ESO to enhance its capability to estimate harmonic components. The proposed FOQR-ESO separates the BEMF-related component from the sixth-order harmonic disturbance, while the fractional-order parameter provides an additional degree of freedom for shaping the observer frequency response. Simulation and experimental results demonstrate that the proposed FOQR-ESO effectively suppresses the sixth-order harmonic disturbance and achieves higher rotor position estimation accuracy than the conventional ESO and the integer-order quasi-resonant extended state observer (IOQR-ESO). Full article

This paper studies mild solutions and Bolza optimal control for Riemann–Liouville fractional stochastic integro-differential systems incorporating fourth-order diffusion, time-varying control, non-instantaneous impulses, and infinite delay. Based on our self-developed $(\mu ,\nu ,\xi ,e,k)$-resolvent family, we derive the mild solution formulation and prove its existence via the Krasnoselskii–Schaefer fixed-point theorem. Using the Arzel$\acute{a}$-Ascoli theorem, Mazur’s lemma, and Balder’s lower semicontinuity principle, we further establish the existence of optimal control pairs. A numerical example from Euler–Bernoulli beam dynamics illustrates the theoretical results. Full article

This paper investigates the task-space position tracking problem of a redundant manipulator under multiple disturbances. Different from conventional joint-space tracking schemes, the considered task only constrains the end-effector position, leaving redundant degrees of freedom to be exploited for secondary optimization. However, conventional computed torque control is sensitive to dynamic-model mismatch, while integer-order equivalent input disturbance compensation has limited flexibility in balancing disturbance tracking and noise attenuation. To address these limitations, a composite control framework is proposed by integrating task-space position error regulation, null-space redundancy optimization, and a fractional-order equivalent input disturbance compensation (FEIDC) strategy. The task-space controller generates the desired acceleration, which is mapped to the joint acceleration command through a damped pseudoinverse Jacobian, and a null-space term is incorporated to optimize secondary criteria. For the feedback-linearized joint dynamics, the proposed FEIDC introduces a fractional-order filter into the equivalent input disturbance estimation channel, providing an additional order parameter for shaping disturbance attenuation and noise sensitivity. Simulation validation on a UR5e manipulator compares the effectiveness of the proposed method with sliding mode control (SMC), active disturbance rejection control (ADRC) and integer-order equivalent input disturbance compensation strategy (IEIDC). In comparisons with SMC, ADRC, and IEIDC, the proposed FEIDC achieves the lowest joint and Cartesian RMSEs, namely, $7.2863\times {10}^{-4}$ rad and $3.8035\times {10}^{-4}$ m, respectively. Full article

Urban systems exhibit multifractal scaling behaviors arising from their hierarchical and heterogeneous spatial organization. Yet the Green View Index (GVI), a widely used indicator of street-level green visibility, is conventionally measured as a static value, overlooking how green exposure is reorganized across scales, which may bias greening evaluations and planning decisions. This study proposes an “X-minute Greenery” framework to examine GVI as a scale-dependent spatial process. Street view imagery and walking isochrone data from Hong Kong and Shenzhen, China, are integrated to measure GVI across 5, 10, 15, and 20 min walking ranges, while a multifractal-inspired approach is developed to characterize its scale-dependent variation structure. Empirical findings show that (1) GVI variation is not random but associated with urban governance and spatial morphology; (2) a staged pattern of variation is identified, with a GVI level around 0.2 marking an approximate shift in configurations of greening dynamics; (3) even under similar initial GVI conditions, areas can still follow divergent variation patterns across walking scales, exposing latent spatial inequalities aligned with institutional and morphological factors. By framing urban green visibility as a scale-aware complex-systems phenomenon, this study adapts multifractal reasoning to street-level greenery, challenges the “equal green, equal policies” assumption, and supports context-sensitive planning interventions. Full article

In the present research, necessary and sufficient conditions for the univalence of the fractional integral of the Gaussian hypergeometric function are obtained by employing specific approaches from differential subordination theory. The first theorem provides the best dominant for a particular differential subordination, which is proved to be a Carathéodory function. These results facilitate the establishment of several corollaries that give the necessary and sufficient conditions for this fractional operator to be a function with a positive real part of the first derivative. The univalence properties obtained through this research enhance the potential for new applications in geometric function theory and related research domains. Full article

The Yushupo Coal Mine in the Ningwu Coalfield is characterized by well-developed faults and pronounced folds, making it a representative site for testing a comprehensive structural evaluation model. Based on a systematic analysis of the structural deformation, an integrated fractal and curvature analysis is used to evaluate faults and analyze structural curvature and to establish a comprehensive quantitative evaluation model combining both methods. The results show that fault structures and their associated and derived folds together determine the tectonic deformation characteristics and structural complexity of the study area. Using refined evaluation unit grids, a high-precision fractal quantitative characterization of fault distribution density is achieved. High-value zones are concentrated in banded and beaded patterns in areas of dense fault development and fault intersections, but this method cannot identify fault throw or the degree of fold deformation. Structural curvature effectively reflects fold curvature and fault throw, with its absolute value increasing with both. The NE-trending high-value or low-value curvature belts can precisely indicate fold cores and areas near fault planes where structural deformation is strong. By utilizing the complementary characteristics of fractal dimension and mean curvature, a joint quantitative characterization of fault scale and fold deformation is achieved, effectively overcoming the limitation that a single fractal parameter cannot reflect fault throw and fold deformation and is easily disturbed by local small faults. Based on this comprehensive evaluation model and combined with structural genetic analysis, three zones are recognized in the study area: the northern NE-trending high-steep monocline zone, the central box-shaped hinge NE-trending fault zone, and the southern EW-trending steeply dipping fault block zone. The evaluation results facilitate risk assessment of gas outbursts and water inrushes and provide a geological basis for targeted mitigation measures across different structural zones. Full article

By redesigning the definition of energy in the discrete sense, we construct structure-preserving alternating direction implicit (ADI) difference schemes to solve the space fractional nonlinear damped wave equation, where the temporal and spatial discretization is accomplished via two types of approximation formulas under the shifted convolution quadrature (SCQ) framework. The theoretical analysis includes error estimation, stability analysis, dissipation law, and approximate conservation properties of energy, which is the key focus of this study. We provide several numerical examples to verify the feasibility and computational efficiency of these schemes, and illustrate the dynamic diffusion behaviors of the circular ring solitons, thereby elucidating the close relationship between the space fractional parameters and the diffusion phenomenon. Full article

This study proposes a hybrid multi-domain color image encryption algorithm that integrates a 2D Fractional-Order Chaotic Map (2D-FOCM), a 5D Gauss-Logistic Hyperchaotic System (5D-GLHS), two-level Discrete Wavelet Transform (DWT), and iris-biometric key distribution within a unified framework. The proposed architecture addresses three fundamental challenges simultaneously: the key distribution vulnerability of symmetric encryption, the limited dynamic complexity of integer-order chaotic systems, and the inadequacy of single-domain encryption approaches. The key distribution problem inherent in symmetric encryption is resolved through biometric uniqueness: rather than transmitting the encryption key directly, only an iris image is exchanged over a secure channel, and each party independently derives the chaotic control parameters from the iris ring region. Statistical features extracted from the iris ring region determine the control parameters of both the 2D-FOCM and the 5D-GLHS, establishing a user-specific, biometrically grounded dynamic key structure with an effective key space of 2¹⁴⁹. The 2D-FOCM, constructed via the piecewise constant arguments method with a Caputo fractional-order derivative, exhibits a broader chaotic parameter range, higher Lyapunov exponents (LE₁ ≈ 16.90, LE₂ ≈ 16.96), and higher approximate entropy than its integer-order counterparts, thereby expanding the key space and suppressing periodic tendencies. The encryption pipeline combines two-level DWT-based frequency-domain subband permutation using 2D-FOCM sequences with five sequential spatial-domain operations directed by the 5D-GLHS: chaotic sequence sorting-based permutation, forward chaining diffusion, inter-block scrambling, intra-block permutation, and XOR diffusion. Comprehensive security evaluations demonstrate that the proposed method achieves the highest average information entropy (7.9975) among 14 compared methods, near-ideal differential attack resistance (NPCR: 99.6109–99.6292%; UACI: 33.3291–33.4308%), and near-zero pixel correlation coefficients across all channels and spatial directions. Chi-square test results confirm superior histogram uniformity in the G and B channels over all 12 compared methods. These results collectively validate the proposed algorithm as a robust and competitive solution for color image security. Full article

We study a fractional Schrödinger–Poisson system involving the spectral fractional Laplacian on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$$(N>2s)$ subject to homogeneous Dirichlet boundary conditions. The model consists of a fractional Schrödinger equation coupled with a fractional Poisson equation through a self-consistent potential. Using the spectral Green operator associated with ${(-\Delta )}^t$, the coupled system is reduced to a single nonlocal integro-differential equation. The associated Green kernel admits a spectral representation in terms of the Dirichlet eigenpairs of the Laplacian. Under suitable assumptions on the Green kernel and Lipschitz conditions on the nonlinearities, we establish the existence of weak solutions together with local uniqueness within the contraction framework via the Banach fixed point theorem for sufficiently small coupling parameters. We further investigate the regularity of the self-consistent potential, continuous dependence on the model parameters, and a conditional convergence to the classical Schrödinger–Poisson system as $s,t\to 1^{-}$. A numerical illustration based on truncated spectral expansions is presented to demonstrate the practical implementation of the proposed framework. Unlike the predominantly variational methods available in the literature, the proposed framework combines a spectral Green-operator reduction with an operator-theoretic fixed-point analysis, providing a constructive formulation that is directly amenable to numerical implementation. Full article

Fractal geometry has recently been advanced through fixed-point iteration schemes, enabling new analytical approaches. In this paper, we apply the Picard–Abbas iteration process to study Mandelbrot sets, Julia sets, and biomorphs for polynomials of the form $z^{k+1}+c$, where $z,c\in \mathbb{C}$ and $k\ge 1$. We develop computational algorithms to generate visual representations of these fractals and examine their dynamic behavior, geometric patterns, and color distributions. We also study standard numerical measures for Mandelbrot and Julia sets (the average escape time and the non-escaping area index). Additionally, we introduce a new numerical measure, relative area, to quantify the impact of iteration parameters on biomorph size. The proposed measure can be used to analyze biomorphs generated using other iteration schemes. Full article

Continental shale oil reservoirs contain multiscale pore–fracture systems with strong heterogeneity and fractal characteristics, which complicate oil mobilization during post-fracturing shut-in imbibition. In this study, shale cores from the LGS Formation (a lacustrine continental shale oil formation in China) were used to investigate capillary-force-regulated pressurized shut-in imbibition by integrating interfacial tension measurements, apparent contact angle tests, capillary pressure calculation, time-lapse nuclear magnetic resonance (NMR), NMR-based fractal characterization, visual observations, and pre-/post-imbibition permeability measurements. Two surfactant-based imbibition agents with different capillary-force regulation mechanisms were compared to represent different capillary-force regulation pathways. Agent 1 mainly modified apparent wettability, increasing the contact angle from 51.0° to 66.1°, whereas Agent 2 reduced the oil–water interfacial tension from 31.85 to 22.12 mN/m while maintaining a favorable apparent contact angle of 49.3°. Time-lapse NMR results showed that oil recovery increased with shut-in time and reached approximately 12–30% after 144 h. Agent 2 generally produced higher recovery than Agent 1, with the optimum response at 0.15 wt%. NMR-derived fractal dimensions ranged mainly from 2.32 to 2.61, confirming the multiscale heterogeneity of the LGS shale pore system. Pore-size-resolved recovery further showed that oil mobilization was dominated by pores larger than 20 nm and microfracture-related spaces, whereas pores smaller than 20 nm contributed only limited bulk recovery. This indicates an apparent nanopore mobilization threshold near 20 nm, controlled by fractal pore complexity, pore-throat connectivity, oil adsorption, capillary pressure, and molecular accessibility of imbibition agents. Visual and permeability evidence further showed that pressurized imbibition can selectively activate connected pore–fracture pathways. Post-imbibition dry-core permeability increased in all tested samples, although the enhancement was highly heterogeneous. These results demonstrate that shut-in imbibition in LGS shale is governed by coupled interfacial regulation, fractal pore heterogeneity, pore-size-dependent oil accessibility, and selective pore–fracture structural modification. Full article

In order to improve the performance of three-phase NPC inverters under unbalanced working conditions, a quasi-proportional complex integral controller (QPCI) three-phase inverter control method based on an adaptive fractional-order algorithm is proposed. Firstly, the reasons for the poor performance of conventional control methods under unbalanced load conditions are analyzed. Subsequently, a fractional-order quasi-proportional complex integral (FO-QPCI) controller is proposed, and the effects of its control parameters, including proportional gain (KP), integral gain (KI), resonant frequency (ω_c), and fractional order (μ), are systematically investigated. Furthermore, the optimal control parameter dataset is utilized to train a Generalized Regression Neural Network (GRNN), through which an adaptive parameter-tuning model is established. As a result, the proposed FO-QPCI controller can dynamically adjust its control parameters according to different voltage references and load unbalanced levels. Finally, to verify the effectiveness of the proposed control strategy, both simulation models and an experimental platform based on a three-level inverter are developed. The results show that the proposed control method has good control ability for the output voltage and harmonics under unbalanced working conditions. Full article

Fractional-order chaotic systems have gained immense popularity in image encryption due to their complex nonlinear dynamics and infinite memory effects. However, their excessive computational complexity remains a persistent bottleneck, severely hindering real-time engineering applications. To address this issue, a lightweight Two-Dimensional Fractional-Order Discrete Cross-Coupled Map (2D-FODCCM) is constructed based on the Short Memory Principle (SMP). This simplified map exhibits excellent dynamical behaviors and cryptographic properties, while its physical realizability and computational efficiency are successfully verified on an STM32 microcontroller. Furthermore, guided by modern cryptanalysis, an efficient and robust image cryptosystem is developed. It integrates a plaintext-associated mechanism to thwart chosen-plaintext attacks, a dynamic fractal Dragon Curve for global spatial permutation, and an innovative pixel fusion strategy to significantly boost encryption throughput. Comprehensive experimental validations demonstrate that the proposed scheme achieves a massive key space of $2^{511}$, exceptional execution efficiency of 123.61 Mbit/s for standard $512\times 512\times 3$ color images, and outstanding robustness against severe noise and data occlusion, offering a highly practical solution for secure real-time multimedia communication. Full article

Renewable energy producers in coupled electricity and carbon markets face multi-dimensional uncertainties with cross-domain correlations. This paper proposes T-CMRS, an integrated forecasting optimization system for wind-solar-storage portfolios enhanced by fractional-order memory-aware uncertainty characterization. T-CMRS comprises (1) a transformer-based probabilistic forecasting engine capturing long-range temporal dependencies; (2) a fractional-order uncertainty modeling module employing Caputo derivatives to capture memory effects in price dynamics; and (3) a CVaR-based risk-averse optimization module reformulated as a linear program. Case studies on modified IEEE systems demonstrate that T-CMRS achieved 8.7% higher average daily profit than deterministic methods with 33.3% lower volatility, 15% VaR improvement over stochastic programming, and 12.3% scenario generation error reduction via fractional-order modeling. Scalability analysis confirms tractable computation for 500-bus systems, verifying applicability to large-scale deployments. Full article

This article investigates a coupled system of wave equations in ${\mathbb{R}}^n$ featuring a delay term within internal fractional feedback. Using semigroup theory, we establish the existence and uniqueness of solutions under a suitable condition relating the weight of the delay term in the fractional feedback to the weight of the term without delay. Second, we demonstrate that the system is strongly stable. Finally, we establish a polynomial decay rate using multiplier techniques combined with the frequency domain method. Full article

The aim of this paper is to introduce an inertial Picard–CR iterative method for approximating fixed points of a family of nonexpansive mappings in real Hilbert spaces. The proposed method is designed to improve the convergence performance of existing schemes while maintaining simplicity in implementation. Under suitable conditions, we establish strong convergence results for the proposed iterative process. We apply the algorithm to solve split feasibility problems, signal processing problems, polynomiography problems, fractional differential, and integral equations. Several numerical experiments demonstrating the superiority of the proposed method over existing methods are presented. The obtained results extend and improve several known results in the literature. Full article

This paper aims to extend the averaging principle for first order fuzzy stochastic differential equations to Riemann–Liouville fuzzy fractional stochastic differential equations, addressing the research gap in this field. This extension enables the averaging principle for fuzzy stochastic systems to cover fractional scenarios with memory, providing a simplified analytical tool for practical systems involving both “randomness” and “fuzziness” uncertainties. Full article

We present newly derived approximate analytical solutions (AAS) of the motion, free and forced modes of dynamics of two viscoelastic rheological Maxwell–Faraday discrete dynamic systems (RMFDDSFTPEP), creeper type, and fractional type, with piezoelectric polarization property of the Faraday piezoelectric element. They always occur in paired rheological discrete dynamic systems depending on the order of sparse coupling of rheological basic light elements in standard light-binding structures and their connections with the rigid body and the fixed point. The research results presented in the paper AAS for the creep-flow EFM of creep-flow dynamics of a RMFDDSFTPEP, which characterizes the SLFTMF coupling set of models, which includes piezoelectric effects. To avoid the inversion of complex, fractional-order expressions of the Laplace transforms (LT), the terms of the complex expressions are expanded into power orders, based on the assumption that certain ratios are small relative to unity. The inverse LT is performed term by term to obtain AAS in the time domain. Full article

Visibility graphs map time series to networks whose combinatorial structure encodes fractality, recovering the Hurst exponent of self-affine processes. We ask what the visibility construction carries beyond this fractal content. We lift the visibility graph to a 1-varifold, a measure on position and direction space from geometric measure theory, and equip it with a multiscale positive definite kernel. The lift embeds visibility graphs of unequal size in a common Hilbert space and yields a channel-resolved measure of cross-series geometric alignment. On a 25.8-year daily panel of thirteen commodity and currency layers, we define a relative alignment contrast that compares commodity currencies and safe haven currencies in their geometric alignment with the commodity complex. During global risk-off episodes the contrast is large and positive: commodity currencies import commodity shock geometry far beyond a time-shift independence benchmark, while the Japanese yen remains near geometric independence and the franc is confounded by a managed regime. The contrast is significant under three stress definitions with autocorrelation robust inference, holds as a continuous dose response, survives the removal of any single crisis, withstands moment, fractal, and topological controls, is direction-consistent across sixteen specifications, and collapses under a time-shift placebo. Detrended fluctuation analysis explains only two percent of it, so the reconfiguration is geometric information beyond fractality at this horizon, and a scaling exponent of the kernel mass separates a fractal-free component from a fractal-driven one. For investors, financial institutions, and policymakers, the contrast is a real-time structural diagnostic of flight to safety: it marks when commodity currencies stop diversifying the commodity complex while genuine safe havens still do, signaling through a channel that second-moment risk measures are built to miss. Full article

Currently, there are numerous studies on the gas–water relative permeability model. However two key mechanisms are often overlooked: including water film thickness as a function of driving pressures, and the gas slippage effect, which can be neglected under high-temperature and high-pressure conditions. To address this gap, a mathematical model for relative permeability based on fractal theory is proposed, explicitly accounting for the effect of water film thickness as a function of driving pressure. The validity of the proposed model is verified through comparison with laboratory experimental results, followed by a sensitivity analysis. The findings indicate that under elevated pressure and temperature conditions, gas viscosity increases, which reduces the flow capacity of the gas phase. Consequently, gas’ relative permeability decreases, while water’s relative permeability increases correspondingly. The pore fractal dimension exhibits a negligible impact on relative permeability. As the displacement pressure gradient decreases, water’s relative permeability declines, and gas’ relative permeability rises correspondingly. This work has significant implications for the development of gas reservoirs under high-temperature and high-pressure conditions. Full article