Fractal Fract., Volume 9, Issue 11 (November 2025) – 75 articles

This paper introduces two generalized frameworks, the extended bipolar parametric b-metricspace (EBPbMS) and the extended bipolar fuzzy b-metric space (EBFbMS), which unify and extend several existing bipolar and fuzzy metric structures. Within these settings, new fixed-point results are established for covariant and contravariant Meir–Keeler-type contractions. A fundamental correspondence between EBFbMSs and EBPbMSs is developed, providing a unified basis for analyzing convergence and stability in generalized metric environments. An illustrative example and an application to a fractional blood flow model confirm the effectiveness of the proposed approach and ensure the existence and uniqueness of the solution. These results demonstrate the capability of extended bipolar structures to model nonlinear fractional systems with memory effects. Full article

This paper develops a generalized Laplace transform theory within weighted function spaces tailored for the analysis of fractional differential equations involving the $\psi$-Hilfer derivative. We redefine the transform in a weighted setting, establish its fundamental properties—including linearity, convolution theorems, and action on ${\delta }_{\psi }$ derivatives—and derive explicit formulas for the transforms of $\psi$-Riemann–Liouville, $\psi$-Caputo, and $\psi$-Hilfer fractional operators. The results provide a rigorous analytical foundation for solving hybrid fractional Cauchy problems that combine classical and fractional derivatives. As an application, we solve a hybrid model incorporating both ${\delta }_{\psi }$ derivatives and $\psi$-Hilfer fractional derivatives, obtaining explicit solutions in terms of multivariate Mittag-Leffler functions. The effectiveness of the method is illustrated through a capacitor charging model and a hydraulic door closer system based on a mass-damper model, demonstrating how fractional-order terms capture memory effects and non-ideal behaviors not described by classical integer-order models. Full article

In this paper, we investigate Hermite–Hadamard-type inequalities for harmonically s-convex stochastic processes via Riemann–Liouville fractional integrals. We begin by introducing the notion of harmonically s-convex stochastic processes. Subsequently, we establish a variety of Riemann–Liouville fractional Hermite–Hadamard-type inequalities for harmonic s-convex stochastic. We first provide the Hermite–Hadamard inequality, then by introducing a novel identity involving mean-square stochastic Riemann–Liouville fractional integral operators, we derive several midpoint-type inequalities for harmonically s-convex stochastic processes. Illustrative example with graphical depiction and a practical application are provided. Full article

This study investigates the synchronization of coupled fractional-order Markovian reaction-diffusion neural networks (MRDNNs) with partially unknown transition rates. The novelty of this work is mainly reflected in three aspects: First, this study incorporates the Markovian switching model and reaction-diffusion term into a fractional-order system, which is a challenging and under-explored issue in existing literature, and effectively addresses the synchronization problem of fractional-order MRDNNs by introducing a continuous frequency distribution model of the fractional integrator. Second, it derives a new set of sufficient synchronization conditions with reduced conservatism; by utilizing the (extended) Wirtinger inequality and delay-partitioning techniques, abundant free parameters are introduced to significantly broaden the solution range. Third, it proposes an LMI-based optimal synchronization design by establishing an efficient offline optimization framework with semidefinite constraints, and achieves the precise solution of control gains. Finally, numerical simulations are conducted to validate the effectiveness of the proposed method. Full article

This article explores the temporal dynamics and fractal characteristics of cosmic ray (CR) intensity by conducting a comprehensive analysis of their intrinsic scaling properties. The study utilizes sophisticated methodologies, including standard Detrended Fluctuation Analysis (DFA) and the Multifractal DFA (MF-DFA) approach to robustly evaluate long-memory, self-similarity, and singularity spectra within extensive CR time series. By systematically investigating measurements from two neutron monitor stations with long data archives, the analysis demonstrates the prevalence of multifractal behavior with persistent long-range correlations. Building on the fractal regime revealed in CR time series, this work utilizes the Natural Time Analysis (NTA) tool that is based in the order of occurrence of the extreme cosmic ray events (ECREs). The operational utility of this tool is demonstrated through a case analysis of CR fluctuations during the severe geomagnetic disturbances observed from 9 to 15 May 2024, capturing early-warning signatures and complex temporal responses. Furthermore, the Modified NTA (M-NTA) is used to estimate the occurrence rate of future ECREs. Our findings contribute to a deeper understanding of the scaling laws governing CR intensity and their potential for improving the ECRE modeling, with direct implications for space weather risk mitigation and solar–terrestrial interactions. Full article

A numerical scheme utilizing shifted Bernstein polynomials is developed to address the variable fractional-order governing equation in viscoelastic fluid-conveying pipes. The pipe’s mechanical response is characterized through a variable fractional-order Kelvin–Voigt (FKV) model, which effectively captures the time-dependent and memory properties of viscoelastic materials. By coupling the FKV constitutive model with the motion equation, the governing equation for a viscoelastic pipe is obtained. The deformation field is approximated using shifted Bernstein trial functions, leading to the construction of a derivative matrix with a variable fractional order. The obtained governing relation is expressed in matrix form, and after discretization, an algebraic system is formulated that is solvable in the time domain to evaluate the pipe displacement. Moreover, a convergence investigation is carried out to examine the reliability and effectiveness of the proposed framework. Computational results demonstrate that the introduced method delivers outstanding precision and performance, while the viscoelastic pipe’s response under different scenarios—including applied loads and fluid flow rates—is comprehensively investigated. Full article

Traditional linear models struggle to capture the complex behavior of financial markets. This study revisits RMB exchange rate volatility through a nonlinear perspective based on G-expectation and multifractal theory. Using multifractal detrended fluctuation analysis (MF-DFA), we examine the scaling properties and efficiency of RMB volatility. We further apply multifractal detrended cross-correlation analysis (MF-DCCA) to explore nonlinear linkages among different RMB exchange rate volatilities. Mixing and phase randomization are employed to identify the sources of multifractality. The results reveal that adverse shocks weaken market efficiency and amplify multifractality. Significant cross-correlations are detected across RMB volatilities, with the Hurst exponent and multifractal spectrum indicating persistent long-range dependence and fat-tailed distributions. Moreover, USDCNY volatility exhibits stronger multifractality than other RMB pairs, underscoring its dominant role in volatility transmission. The time-varying Hurst exponent effectively captures nonlinear and memory effects, offering predictive value for exchange rate trends. These findings deepen our understanding of RMB exchange rate dynamics and provide implications for monetary regulation and risk management under uncertainty. Full article

This paper investigates the consensus problem in fractional-order multi-agent systems (FOMASs) under Denial of Service (DoS) attacks and actuator faults. A boundary control strategy is proposed, which reduces dependence on internal sensors and actuators by utilizing only the state information at the system boundaries, significantly lowering control costs. To address DoS attacks, a buffer mechanism is designed to store valid control signals during communication interruptions and apply them once communication is restored, thereby enhancing the system’s robustness and stability. Additionally, this study considers the impact of actuator performance fluctuations on control effectiveness and proposes corresponding adjustment strategies to ensure that the system maintains consensus and stability even in the presence of actuator failures or performance variations. Finally, the effectiveness of the proposed method is validated through numerical experiments. The results show that, even under DoS attacks and actuator faults, the system can still successfully achieve consensus and maintain good stability, demonstrating the feasibility and effectiveness of this control approach in complex environments. Full article

In dense visual simultaneous localization and mapping VSLAM (VSLAM), a fundamental challenge lies in the inability of existing loss functions to dynamically balance luminance, contrast, and structural fidelity under photometric variations, while their underlying mechanisms, particularly the conventional Gaussian kernel in SSIM, suffer from limited receptive fields due to rapid exponential decay, preventing the capture of long-range dependencies essential for global consistency. To address this, we propose a fractional Gaussian field (FGF) that synergizes Caputo derivatives with Gaussian weighting, creating a hybrid kernel that couples power-law decay for long-range memory with local smoothness. This foundational kernel serves as the core component of FGF-SSIM, a novel loss function that adaptively recalibrates luminance, contrast, and structure using fractional-order statistics. The proposed FGF-SSIM is further integrated into a complete 3D Gaussian Splatting (3DGS)-based SLAM system, named FGF-SLAM, where it is employed across both tracking and mapping modules to enhance performance. Extensive evaluations demonstrate state-of-the-art performance across multiple benchmarks. Comprehensive analysis confirms the superior long-range dependency of the fractional kernel, dedicated illumination robustness tests validate the enhanced invariance of FGF-SSIM, and quantitative results on TUM and Replica datasets show significant improvements in reconstruction quality and trajectory estimation. Ablation studies further substantiate the contribution of each proposed component. Full article

This paper analyzes the Bagley–Torvik fractional-order equation with generalized fractional Hilfer derivatives of two orders for functions in Banach spaces under conditions expressed in the language of weak topology. We develop a comprehensive theory of fractional-order differential equations of various orders. Our focus is on the equivalence results (or the lack thereof) of this new class of fractional-order Hilfer operators and on maximizing the regularity of the solution. To this end, we examine the equivalence of differential problems involving pseudo-derivatives and integral problems involving Pettis integrals. Our results are novel, even within the context of integer-order differential equations. Another objective is to incorporate fractional-order problems into the growing research field that uses weak topology and function spaces to study vector-valued functions. The auxiliary results obtained in this article are general and applicable beyond its scope. Full article

Vector-borne infectious diseases transmitted by vector organisms (e.g., mosquitoes, rodents, and ticks) are recognized as key priorities in global public health. The construction of host–vector interaction frameworks within bipartite networks enables a clearer depiction of the transmission mechanisms underlying vector-borne infectious diseases. Compared with traditional models, the effective influence of historical information on vector-borne infectious diseases is more critical. In this study, the long-term memory behavior of infected populations during the recovery phase is regarded as a power-law tail distribution, a result that is consistent with fractional calculus. Thus, a fractional-order model for vector-borne diseases on bipartite networks is established.The basic reproduction number is derived about network topology and fractional order. With stability analysis, the conditions governing the global extinction and global persistence of vector-borne infectious diseases are determined. Furthermore, the validity of the proposed model is confirmed through numerical simulation results obtained from Barabási–Albert (BA) networks and Watts–Strogatz (WS) networks. Full article

We present a theoretical examination of the fractional dynamics of information entropy within a semiconductor nanowire system influenced by Rashba spin–orbit interaction and external magnetic fields. Moreover, we determine the fractional nanowire state through the analytical solution of the fractional Schrödinger equation, considering various initial states of the nanowire system. Our research emphasizes the impact of the fractional order and the interaction parameters on the behavior of information entropy. Our findings reveal that the temporal behavior of information entropy is highly sensitive to any variations in the magnetic field length, the Rashba spin–orbit interaction, and the fractional order parameter. The results demonstrate that these parameters are pivotal in determining the coherence and correlation properties of the nanowire system. Therefore, precise control of these factors paves the way for enhancing entanglement performance and facilitating information transfer in spintronic and quantum communication applications. Full article

This paper presents a novel algorithm for the dynamic analysis of fractional-order viscoelastic spinning disks in the time domain. The novelty mainly lies in the use of the shifted Legendre polynomial algorithm for the direct time-domain numerical analysis of displacement in two directions for a three-dimensional viscoelastic rotating disk, tackling a more complex and strongly coupled problem than those addressed in previous studies. By using the fractional-order Kelvin–Voigt model to describe the viscoelastic properties of the disk, a system of governing equations with three independent variables is established. For the two ternary unknown functions in the equations, a fractional-order differential operator matrix based on Shifted Legendre polynomials is derived, transforming the original equations into two sets of algebraic equations that are easier to solve. This paper presents an in-depth analysis of the convergence of the Legendre polynomial algorithm, complemented by an investigation of its error characteristics using numerical examples, thereby verifying the method’s accuracy and feasibility. This study can be applied to the dynamic analysis of viscoelastic rotating structures under body force density. The findings provide theoretical support for the optimization and safety assessment of load-bearing rotating components in engineering. And the algorithm demonstrates high accuracy and applicability in handling fractional-order equations in science and engineering. Full article

The hydro-mechanical coupling in fractures plays a significant role in fluid transport through fracture networks. However, current studies still exhibit certain limitations in the multi-parameter characterization of fracture permeability under stress conditions. To address this, a hydro-mechanical coupling model was developed to investigate the coupled hydro-mechanical behavior of fractures under different stress states and shear displacements. The results show that fluid flow patterns within fractures exhibit notable heterogeneity and anisotropy, influenced by aperture distribution and the connectivity of preferential flow paths. High normal stress significantly reduces the mechanical aperture while enhancing its anisotropy, as the normal stress increased from 2 MPa to 8 MPa, the average mechanical aperture of the fractures decreased by 61% to 65%. With increasing shear displacement, both the mechanical aperture and its standard deviation increase, and the aperture distribution shifts from a sharply peaked pattern to a more flattened one, the maximum aperture increased by 23–38%, reflecting enhanced variability in fracture structure. Increased surface roughness amplifies the effect of shear displacement on the evolution of fracture architecture. Under low normal stress conditions, the mechanical aperture increases gradually with higher roughness, the mechanical aperture decreased more significantly in high-roughness (JRC = 17.94) fractures (28–31% greater reduction) compared to low-roughness ones (JRC = 2.01). To assess fracture permeability, a predictive model was developed and validated against further data, confirming its effectiveness in evaluating permeability. This study highlights the mechanisms by which shear displacement and normal stress influence fracture permeability. Full article

This paper investigates the chaotic and pattern dynamics of the time-fractional Ginzburg–Landau equation. First, we propose a high-precision numerical method that combines finite difference schemes with an improved Grünwald–Letnikov fractional derivative approximation. Subsequently, the effectiveness of the proposed method is validated through systematic comparisons with classical numerical approaches. Finally, numerical simulations based on this method reveal rich dynamical phenomena in the fractional Ginzburg–Landau equation: the system exhibits complex behaviors including chaotic oscillations and novel two- and three-dimensional pattern structures. This study not only advances the theoretical development of numerical solutions for fractional GLE but also provides a reliable computational tool for deeper understanding of its complex dynamical mechanisms. Full article

Motivated by the strong seasonality of East Asian meteorology and its control on pollution episodes characterized by fluctuation level, we model the season-resolved climatology of the regional PM_2.5 connectivity over the Korean Peninsula. Using daily AirKorea data for 2016–2020, we (i) remove daily climatology and the peninsula-wide background (empirical orthogonal function; EOF1) to obtain residual signals; (ii) compute the sign-preserving multifractal detrended cross-correlation coefficient MFDCCA-$\rho \left(q,s\right)$; (iii) apply iAAFT surrogate significance across scales; and (iv) construct signed, weighted networks aggregated over short (5–15 d) and mid (15–30 d) bands for DJF/MAM/JJA/SON. Our analysis targets the seasonal climatology of fluctuation-level ($q$-dependent) connectivity by pooling seasons across years; this approach increases statistical robustness at 5–30-day scales and avoids diluting season-specific organization. We find negligible connectivity for $q<0$ (small fluctuations) but dense, seasonally organized networks for $q>0$ (strongest in winter–spring and at 15–30 days). After removing the EOF1, positive subgraphs form assortative regional backbones, while negative subgraphs reveal a northwest–southeast anti-phase dipole; the connectivity around Baengnyeongdo (B) highlights a transboundary sentinel role in cool seasons. These results demonstrate that a season-resolved, fluctuation-level framework effectively isolates regional connectivity that would otherwise be masked in annual aggregates or by the peninsula-wide background. Full article

This paper deals with a new coupled system of integro-differential equations involving the generalized proportional Caputo derivatives equipped with nonlocal four-point boundary conditions. Sufficient criteria for the existence and uniqueness of solutions for the studied system are derived based on Krasnoselskii’s and Banach fixed-point theorems, respectively. Applications are constructed with three different cases to illustrate the main results. Full article

In this paper, we evaluate a general class of finite integrals involving the error function, generalized Mittag-Leffler functions, and incomplete Aleph functions. The main result provides a unified framework that extends several known formulas related to the incomplete Gamma, I-, and H-functions. Under suitable conditions, these results reduce to many classical special cases. We discuss convergence conditions that justify the validity of the obtained formulas and include explicit corollaries that highlight connections with earlier results in the literature. To illustrate applicability, we present numerical examples and graphs, demonstrating the behavior of the error function integral and Mittag-Leffler functions for specific parameter values. These integrals arise naturally in fractional calculus, probability theory, viscoelasticity, and anomalous diffusion, underscoring the importance of the present work in both mathematical analysis and applications. Full article

Gradient accumulation enables training large-scale deep learning models under GPU memory constraints by aggregating gradients across multiple microbatches before parameter updates. Standard gradient accumulation treats all microbatches uniformly through simple averaging, implicitly assuming that all stochastic gradient estimates are equally reliable. This assumption becomes problematic in non-convex optimization where gradient variance across microbatches is high, causing some gradient estimates to be noisy and less representative of the true descent direction. In this paper, FracGrad is proposed, a simple weighting scheme for gradient accumulation that biases toward recent microbatches via a power-law schedule derived from a discretized Riemann–Liouville integral. Unlike uniform summation, FracGrad reweights each microbatch gradient by $w_i\left(\alpha \right)={\displaystyle \frac{{(N-i+1)}^{\alpha }-{(N-i)}^{\alpha }}{{\sum }_{j=1}^N[{(N-j+1)}^{\alpha }-{(N-j)}^{\alpha }]}}$, controlled by $\alpha \in (0,1]$. When $\alpha =1$, standard accumulation is recovered. In experiments on mini-ImageNet with ResNet-18 using up to $N=32$ accumulation steps, the best FracGrad variant with $\alpha =0.1$ improves test accuracy from 16.99% to 31.35% at $N=16$. Paired t-tests yield $p\approx 2\times {10}^{-6}$. Full article

This study models customer patterns using fractal analysis and examines how touchpoints and customer journey mapping shape value to enhance sustainable organizational and customer performance. The analysis is made at cell-level using mathematical modelling and tactical indicators within a top-three retail organization with national presence. This study reveals that only half of the customers engage with multiple touchpoints, with specific departments experiencing low interaction levels. The company’s customer concentration is well-balanced, reducing dependency on a single client, and mutual dependency suggests robust customer–company sustainable relationships. Results underline the significance of the proposed tools in evaluating the value of the customers for the company. Practical implications include guiding managers to use modelling in strengthening customer touchpoints to enhance qualitative relationships and profitability. This research contributes to customer relationship management and fractal analysis literature, demonstrating their role in maximizing bi-directional value and performance for success. Full article

Organic pores serve as crucial storage spaces for shale gas, whose morphology and structure vary significantly among different types of organic matter, directly influencing the storage and seepage capacity of shale gas. The Upper Permian shale in the Western Hubei Trough formed in diverse sedimentary facies and has undergone multiple geological activities, resulting in strong heterogeneity of organic pores across different strata and regions. To figure out the heterogeneous characteristics of organic pores and the forming reason, the occurrence state of organic matter, pore morphology, and structural parameters (pore size, specific surface area, pore volume, and fractal dimension) of the Upper Permian shale in Western Hubei, have been discussed in detail, based on the data of field emission scanning electron microscopy and low-temperature nitrogen adsorption experiments conducted on the extracted organic matter. On this basis, fractal dimension theory was applied to discuss the heterogeneity of organic pores in different layers, and the reason for heterogeneity has been analyzed in detail. The results indicate that the occurrence mode of organic matter in different layers presents various characteristics: in the Gufeng Formation, the organic matters distribute primarily dispersed in flocculent state; at the bottom of Wujiaping Formation, they occur as isolated individuals, while the organic matters turn into discontinuous laminated distribution in the middle and upper Wujiaping Formation; in the Dalong Formation, the organic matters show continuous parallel banded distribution. Moreover, the morphology and structural parameters of organic pores exhibit obvious changes from the Gufeng Formation to the Dalong Formation: (a) the pore morphology shows the changed trend as extremely complex-simple-complex; (b) the specific surface area and pore volume follow the trend as large-small-large; (c) the pore size distribution displays in the pattern of bimodal-unimodal-bimodal; (d) the data of fractal dimension show the variation of high–low–high. Overall, the various sedimentary environments during the Upper Permian shale depositional period determined the differences in organic sources, which dominated the heterogeneity of organic pores in shale. These data clarify the development and variation characteristics of organic matter pores under different depositional environments, providing a theoretical basis for shale gas exploration and development during the transition from marine to marine–continental facies. Full article

In this paper, we introduce the notion of fuzzy exponential contractions within the framework of fuzzy metric spaces. These mappings, which involve point-dependent exponential terms, are studied under the assumptions of either fuzzy continuity or the weaker condition of fuzzy Picard continuity. We establish corresponding existence and uniqueness theorems, and we further demonstrate the scope of the theory through illustrative examples and by applying it to prove an existence and uniqueness result for a class of nonlinear fractional differential equations. Full article

In the observation of tight sandstone cores, the variations in the hydrocarbon charging usually can be observed in the same geological age reservoirs, which manifest as differential oil staining on the core surface. In order to clarify the micro total pore–throat size distribution (TPSD) characteristics and oil content differences of different oil-bearing tight reservoirs, we drilled two types of oil-bearing cores in the Chang6 formation of Xiasiwan Oilfield, conducted casting thin section (CTS), scanning electron microscopy (SEM), and X-ray diffraction (XRD) to qualitatively and quantitatively analyze petrological and pore–throat characteristics. The TPSD of different oil-bearing cores were quantitatively characterized and compared by combining high-pressure mercury injection (HPMI) and constant rate mercury injection (CRMI). Meanwhile, we quantitatively evaluated the complexity of the pore–throat structure based on fractal theory. Our results reveal significant difference in the clay mineral contents between the two types of cores, despite both being classified as arkose. Due to higher contents of illite, calcite, and chlorite, the pores of oil-smelling sandstone are obviously affected by cementation. The result of TPSD characteristics shows that the oil-appearing sandstone samples exhibit well-developed big pores and throats, displaying bimodal distribution, and three-stage fractal characteristics in the TPSD curves. Conversely, oil-smelling sandstone samples manifesting a left-skewed bimodal, pore space contribution of the samples is more dependent on pores and throats smaller than 0.12 μm. The TPSD curves exhibit three-stage and four-stage fractal characteristics. Therefore, the differences in oil-bearing properties between the two types of cores are attributed to variations in mineral composition, diagenesis, clay mineral content, pore types, pore–throat size distribution (PSD), and pore–throat complexity. Our results provide crucial guidance for subsequent reservoir quality assessment in this study area and the development of tight sandstone reservoirs with similar geological characteristics. Full article

We study stochastic variants of the Kairat-II and Kairat-X equations in (3 + 1) dimensions, two canonical models in soliton theory. Random fluctuations are incorporated through a Wiener process, yielding a multiplicative stochastic embedding of the wave fields. By combining the enhanced direct algebraic technique with the new projective Riccati equation approach, we obtain closed-form stochastic soliton solutions and analyze how noise modulates their amplitude and localization. The solutions are illustrated with consistent 3D surface plots (mean field vs. sample paths) and 2D time traces to highlight wave geometry and variability. In addition, we employ the energy balance approach to separate kinetic and potential contributions and to verify an energy balance relation for the derived solutions, thereby clarifying their physical plausibility and stability under noise. The results provide exact, easily verifiable benchmarks for stochastic nonlinear wave models and a practical template for incorporating randomness into nonlinear dispersive systems. Full article

In this paper, we present a collocation algorithm for numerically treating the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Certain orthogonal polynomials, which are expressed as combinations of Chebyshev polynomials, and their shifted polynomials are introduced. Some new theoretical formulas regarding these polynomials have been developed, including their operational matrices of both integer and fractional derivatives. The derived formulas will be the foundation for designing the proposed numerical algorithm, which relies on converting the governing problem with its underlying conditions into a nonlinear algebraic system, which can be solved using Newton’s iteration technique. A rigorous error analysis for the proposed combined Chebyshev expansion is presented. Some numerical examples are given to ensure the applicability and efficiency of the presented algorithm. These results demonstrate that the proposed algorithm attains superior accuracy with fewer expansion terms. Full article

The nonlocality (capable of associating target dynamics across multiple time moments) and memory properties (able to retain historical trajectories) of fractional calculus serve as the core theoretical approach to resolving the “dynamic information association deficiency” in UAV mobile target search. This paper proposes the Fractional-order Black-winged Kite Algorithm (FOBKA), which transforms the search problem into an adaptability function optimization model aimed at “maximizing target capture probability” based on Bayesian theory. Addressing the limitations of the standard Black-winged Kite Algorithm (BKA), the study incorporates fractional calculus theory for enhancement: A fractional-order operator is embedded in the migration behavior phase, leveraging the memory advantage of fractional-orders to precisely capture the temporal span, spatial position, and velocity evolution of targets, thereby enhancing global detection capability and convergence accuracy. Simultaneously, population individuals are initialized using motion-encoding, and the attack behavior phase combines alternating updates with a Lévy flight mechanism to balance local exploration and global search performance. To validate FOBKA’s superiority, comparative experiments were conducted against eight newly proposed meta-heuristic algorithms across six distinct test scenarios. Experimental data demonstrate that FOBKA significantly outperforms the comparison algorithms in convergence accuracy, operational robustness, and target capture probability. Full article

We first investigate the dynamical behavior of an active Brownian particle influenced by a viscoelastic memory effect characterized by a power-law kernel, under the effects of thermal and active noises. We then analyze the dynamics of an active Brownian particle confined in a harmonic trap in the presence of the same noise sources. To derive the Fokker–Planck equation for the joint probability density of the active particle, we obtain analytical solutions for the joint probability density and its moments using double Fourier transforms in the limits $t\ll \tau$, $t\gg \tau$, and $\tau =0$. As a result, the mean squared displacement of an active Brownian particle driven by thermal noise exhibits a super-diffusive scaling of $t^{2h+1}$ in the short-time regime ($t\ll \tau$). In contrast, for a particle in a harmonic trap driven by active noise, the mean squared velocity scales linearly with $t$ when $\tau =0$. Moreover, the higher-order moments of an active Brownian particle in a harmonic trap with thermal noise scale with $t^{4h+2}$ in the long-time limit ($t\gg \tau$) and for $\tau =0$, consistent with our analytical results. Full article

Fractional calculus in discrete-time is a recent field that has drawn much interest for dealing with multidisciplinary systems. A result of this tremendous potential, researchers have been using constant and variable-order fractional discrete calculus in the modelling of financial and economic systems. This paper explores the emergence of chaotic and regular patterns of the fractional four-dimensional (4D) discrete economic system with constant and variable orders. The primary aim is to compare and investigate the impact of two types of fractional order through numerical solutions and simulation, demonstrating how modifications to the order affect the behavior of a system. Phase space orbits, the 0-1 test, time series, bifurcation charts, and Lyapunov exponent analysis for different orders all illustrate the constant and variable-order systems’ behavior. Moreover, the spectral entropy (SE) and $C_0$ complexity exhibit fractional-order effects with variations in the degree of complexity. The results provide new insights into the influence of fractional-order types on dynamical systems and highlight their role in promoting chaotic behavior. Additionally, two types of control strategies are devised to guide the states of a 4D fractional discrete economic system with constant and variable orders to the origin within a specified amount of time. MATLAB simulations are presented to demonstrate the efficacy of the findings. Full article

Macroeconomic mathematical models are practical instruments structured to carry out theoretical analyses of macroeconomic developments. In this manuscript, the Caputo-like fractional operator of variable order is used to introduce and investigate the mechanism of the discrete macroeconomic model. The nature of the dynamics was established, and the emergence of chaos using a distinct variable fractional order, especially the stability of the trivial solution, is examined. The findings reveal that the variable-order discrete macroeconomic model displays chaotic dynamics employing bifurcation, the Largest Lyapunov exponent ($L{E}_{max}$), the phase portraits, and the 0–1 test. Furthermore, a complexity analysis is performed to demonstrate the existence of chaos and quantify its complexity using $C_0$ complexity and spectral entropy ($SE$). These studies show that the suggested variable-order fractional discrete macroeconomic model has more complex features than integer and constant fractional orders. Finally, MATLAB R2024b simulations are run to exemplify the outcomes of this study. Full article