Fractal and Fractional

Fractal dimension, as a common nonlinear dynamics metric, is extensively applied in biomedicine, fault diagnosis, underwater acoustics, etc. However, traditional fractal dimension can only analyze the complexity of the time series given a single channel at a particular scale. To characterize the complexity of multichannel time series, multichannel information processing was introduced, and multivariate Higuchi fractal dimension (MvHFD) was proposed. To further analyze the complexity at multiple scales, multivariate multiscale Higuchi fractal dimension (MvmHFD) was proposed by introducing multiscale processing algorithms as a technology that not only improved the use of fractal dimension in the analysis of multichannel information, but also characterized the complexity of the time series at multiple scales in the studied time series data. The effectiveness and feasibility of MvHFD and MvmHFD were verified by simulated signal experiments and real signal experiments, in which the simulation experiments tested the stability, computational efficiency, and signal separation performance of MvHFD and MvmHFD, and the real signal experiments tested the effect of MvmHFD on the recognition of multi-channel mechanical signals. The experimental results show that compared to other indicators, A achieves a recognition rate of 100% for signals in three features, which is at least 17.2% higher than for other metrics. Full article

Remaining useful life prediction guarantees a reliable and safe operation of turbofan engines. Long-range dependence (LRD) and heavy-tailed characteristics of degradation modeling make this method advantageous for the prediction of RUL. In this study, we propose fractional Lévy stable motion for degradation modeling. First, we define fractional Lévy stable motion simulation algorithms. Then, we demonstrate the LRD and heavy-tailed property of fLsm to provide support for the model. The proposed method is validated with the C-MAPSS dataset obtained from the turbofan engine. Principle components analysis (PCA) is conducted to extract sources of variance. Experimental data show that the predictive model based on fLsm with exponential drift exhibits superior accuracy relative to the existing methods. Full article

This paper proposes an active disturbance rejection control (ADRC) architecture for a permanent magnet synchronous motor (PMSM) position servo system. The presented method achieved enhanced tracking and disturbance rejection performance with a limited observer bandwidth. The model-aided extended state observer (MESO)-based ADRC was designed for the current, speed, and position loops of the PMSM position servo system. By integrating known plant information, the MESO improved disturbance estimation with a limited observer bandwidth without amplifying the noise. Additionally, a fractional-order proportional-derivative (FOPD) controller was designed as the feedback controller for the speed loop to further enhance the disturbance rejection. A simulation and experimental tests were conducted on a PMSM servo platform. The results demonstrate not only that the proposed method achieved superior tracking performance but also that the position error of the proposed strategy decreases to 2.25% when the constant disturbance was input, significantly improving the disturbance rejection performance. Full article

In this paper, we use the finite volume element method (FVEM) to approximate a one-dimensional, time fractional generalized Burgers’ equation. We construct the fully discrete finite volume element scheme for this equation by approximating the time fractional derivative term by the $L1$ formula and approximating the spatial terms using FVEM. The convergence of the scheme is proven. Finally, numerical examples are provided to confirm the scheme’s validity. Full article

In this paper, we investigate the controllability conditions of linear control systems involving distinct local fractional derivatives. Sufficient conditions for controllability using Kalman rank conditions and the Gramian matrix are presented. We show that the controllability of the local fractional system can be determined by the invertibility of the Gramian matrix and the full rank of the Kalman matrix. We also show that the local fractional system involving distinct orders is controllable if and only if the Gramian matrix is invertible. Illustrative examples and an application are provided to demonstrate the validity of the theoretical findings. Full article

In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known $L1$ formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution is proved by using the matrix theory, and the unconditional stability is also discussed in detail. By introducing the mixed elliptic projection, the error estimates for the unknown variable u in the discrete $L^{\infty }\left(L^2(\Omega )\right)$ norm and for the auxiliary variable $\mathit{\lambda }$ in the discrete $L^{\infty }\left({\left(L^2(\Omega )\right)}^2\right)$ and $L^{\infty }\left(\mathit{H}(\mathrm{div},\Omega )\right)$ norms are obtained. Finally, three numerical examples are given to demonstrate the theoretical results. Full article

The fractal dimension (D) is a very useful indicator for recognizing images. The fractal dimension increases as the pattern of an image becomes rougher. Therefore, images are frequently described as certain models of fractal geometry. Among the models, two-dimensional fractional Brownian motion (2D FBM) is commonly used because it has specific physical meaning and only contains the finite-valued parameter (a real value from 0 to 1) of the Hurst exponent (H). More usefully, H and D possess the relation of D = 3 − H. The accuracy of the maximum likelihood estimator (MLE) is the best among estimators, but its efficiency is appreciably low. Lately, an efficient MLE for the Hurst exponent was produced to greatly improve its efficiency, but it still incurs much higher computational costs. Therefore, in the paper, we put forward a deep-learning estimator through classification models. The trained deep-learning models for images of 2D FBM not only incur smaller computational costs but also provide smaller mean-squared errors than the efficient MLE, except for size 32 × 32 × 1. In particular, the computational times of the efficient MLE are up to 129, 3090, and 156248 times those of our proposed simple model for sizes 32 × 32 × 1, 64 × 64 × 1, and 128 × 128 × 1. Full article

The quantification of the irregular morphology and distribution pattern of mineral grains is an essential but challenging task in ore-related mineralogical research, allowing for tracing the footprints of pattern-forming geological processes that are crucial to understanding mineralization and/or diagenetic systems. In this study, a large model, namely, the Segmenting Anything Model (SAM), was employed to automatically segment and annotate quartz, lepidolite and albite grains derived from Yichun rare-metal granite (YCRMG), based on which a series of fractal and multifractal methods, including box-counting calculation, perimeter–area analysis and multifractal spectra, were implemented. The results indicate that the mineral grains from YCRMG show great scaling invariance within the range of 1.04~52,300 μm. The automatic annotation of mineral grains from photomicrographs yields accurate fractal dimensions with an error of only 0.6% and thus can be utilized for efficient fractal-based grain quantification. The resultant fractal dimensions display a distinct distribution pattern in the diagram of box-counting fractal dimension (D_b) versus perimeter–area fractal dimension (D_PA), in which lepidolites are sandwiched between greater-valued quartz and lower-valued albites. Snowball-textured albites, i.e., concentrically arranged albite laths in quartz and K-feldspar, exhibit characteristic D_b values ranging from 1.6 to 1.7, which coincide with the fractal indices derived from the fractal growth model. The zonal albites exhibit a strictly increasing trend regarding the values of fractal and multifractal exponents from core to rim, forming a featured “fractal-index banding” in the radar diagram. This pattern suggests that the snowball texture gradually evolved from rim to core, thus leading to greater fractal indices of outer zones, which represent higher complexity and maturity of the evolving system, which supports a metasomatic origin of the snowball texture. Our study demonstrates that fractal analyses with the aid of a large model are effective and efficient in characterizing and understanding complex patterns of mineral grains. Full article

Deep low-rank coalbed methane (CBM) resources are numerous and widely distributed in China, although their exploration remains in its infancy. In this work, gas adsorption (N₂/CO₂), mercury intrusion porosimetry, and 3D CT reconstruction were performed on five coal samples of deep and shallow low-rank coal from northeast China to analyze their pore structure. The impact of the features in the pore structure at full scale on the capacity for methane adsorption and seepage is discussed. The results indicate that there are significant differences between deep low-rank coal and shallow low-rank coal in terms of porosity, permeability, composition, and adsorption capacity. The full-scale pore distribution was dispersed over a broad range and exhibited a multi-peak distribution, with the majority of the peak concentrations occurring between 0.45–0.7 nm and 3–4 nm. Mesopores are prevalent in shallow coal rock, whereas micropores are the most numerous in deep coal rock. The primary contributors to the specific surface area of both deep and superficial coal rock are micropores. Three-dimensional CT reconstruction can characterize pores with pore size greater than 1 μm, while the dominating equivalent pore diameters (D_eq) range from 1 to 10 μm. More mini-scale pores and fissures are observed in deep coal rock, while shallow coal rock processes greater total and connection porosity. Multifractal features are prevalent in the fractal qualities of all the numbered samples. An enhancement in pore structure heterogeneity occurs with increasing pore size. The pore structure of deep coal rock is more heterogeneous. Furthermore, methane adsorption capacity is favorably connected with D₁ (0.4 nm < pore diameter ≤ 2 nm), D₂ (2 nm < pore diameter ≤ 5 nm), micropore volume, and specific surface area and negatively correlated with D₃ (5 nm < pore diameter ≤ 50 nm), showing that methane adsorption capability is primarily controlled by micropores and mesopores. Methane seepage capacity is favorably connected with the pore volume and connected porosity of macropores and negatively correlated with D₄ (pore diameter > 50 nm), indicating that the macropores are the primary factor influencing methane seepage capacity. Full article

This article aims to use various fixed-point techniques to study the stability issue of the impulsive Volterra integral equation in the sense of Ulam–Hyers (sometimes known as Hyers–Ulam) and Hyers–Ulam–Rassias. By eliminating key assumptions, we are able to expand upon and enhance some recent findings. Full article

This article investigates the differential privacy of the initial state for nabla discrete fractional-order dynamic systems. A novel differentially private Gaussian mechanism is developed which enhances the system’s security by injecting random noise into the output state. Since the existence of random noise gives rise to the difficulty of analyzing the nabla discrete fractional-order systems, to cope with this challenge, the observability of nabla discrete fractional-order systems is introduced, establishing a connection between observability and differential privacy of initial values. Based on it, the noise magnitude required for ensuring differential privacy is determined by utilizing the observability Gramian matrix of systems. Furthermore, an optimal Gaussian noise distribution that maximizes algorithmic performance while simultaneously ensuring differential privacy is formulated. Finally, a numerical simulation is provided to validate the effectiveness of the theoretical analysis. Full article

Exploration of mineral resources in the deep sea has become an international trend. However, deep-sea mineral exploration faces challenges such as complex offshore drilling and the weak and mixed signals of ore deposits. Therefore, studying methods for identifying weak and mixed anomalies and extracting composite information in the deep sea is crucial for innovative prediction and evaluation of deep-sea mineral resources. In this study, the Central Pacific Ocean, Northwestern Pacific Ocean, and Eastern Pacific Ocean were selected as research areas. Drawing upon the fractal self-similarity exhibited by rare earth minerals in the deep-sea sediments within the Pacific Ocean, we conducted an analysis and comparison of the fractal geochemical characteristics in various regions of the Pacific Ocean’s deep-sea sediments. Thereafter, we studied the spatial distribution of rare earth elements (REEs) in deep-sea sediments in these regions to explore the mechanisms responsible for rare earth enrichment in the Pacific Ocean. The results revealed that the geochemical fractal characteristics of deep-sea sediments in the Northwestern Pacific Ocean Basin and the Central Pacific Ocean Basin were similar, whereas there were slight differences in the fractal characteristics observed in the Eastern Pacific Ocean Basin. By calculating the singularity index of CaO/P₂O₅, it was found that the singularity index in the Central and Northwestern Pacific Ocean basins was lower than that in the Eastern Pacific Ocean Basin, suggesting that the phosphorus content in the Eastern Pacific Ocean Basin was lower than that in the Central and Northwestern Pacific Ocean basins. In the Eastern Pacific Ocean, we found that phosphorus content in deep-sea sediments was the primary controlling factor for REE enrichment. Conversely, in the Central and Northwestern Pacific Ocean, both the phosphorus and calcium content in deep-sea sediments played significant roles in REE enrichment. Full article

In this paper, the Caputo-based fractional derivative optimal control model is looked at to learn more about how the human respiratory syncytial virus (RSV) spreads. Model solution properties such as boundedness and non-negativity are checked and found to be true. The fundamental reproduction number is calculated by using the next-generation matrix’s spectral radius. The fractional optimal control model includes the control functions of vaccination and treatment to illustrate the impact of these interventions on the dynamics of virus transmission. In addition, the order of the derivative in the fractional optimal control problem indicates that encouraging vaccination and treatment early on can slow the spread of RSV. The overall analysis and the simulated behavior of the fractional optimum control model are in good agreement, and this is due in large part to the use of the MATLAB platform. Full article

Controllability is a fundamental issue in the field of fractional complex network control, yet it has not received adequate attention in the past. This paper is dedicated to exploring the controllability of complex networks involving the Caputo fractional derivative. By utilizing the Cayley–Hamilton theorem and Laplace transformation, a concise proof is given to determine the controllability of linear fractional complex networks. Subsequently, leveraging the Schauder Fixed-Point theorem, controllability Gramian matrix, and fractional calculus theory, we derive controllability conditions for nonlinear fractional complex networks with a weighted adjacency matrix and Laplacian matrix, respectively. Finally, a numerical method for the controllability of fractional complex networks is obtained using Matlab (2021a)/Simulink (2021a). Three examples are provided to illustrate the theoretical results. Full article

With the gradual depletion of natural sand due to over-exploitation, alternative building materials, such as manufactured sand aggregate (MSA), have attracted much attention. In order to interpret the evolution of pore structure and fractal characteristics in MSA mortar over long-term water saturation, the 1H low-field nuclear magnetic resonance (LF-NMR) relaxation method was used to investigate the temporal evolution of the pore structure in five single-graded MSA mortars and synthetic-graded mortars with small amplitudes in particle size. MSA presents a fresh rock interface characterized by a scarcity of pores, which significantly reduces the porosity of the mortar. The surface-to-volume ratio (SVR) is employed for characterizing the MSA gradation. Through an analysis of parameters, such as total porosity, pore gradation, pore connectivity, and pore fractal dimension of mortar, a correlation model between pore structure parameters and aggregate SVR is constructed. The fractal characteristics of pores and their variations are discussed under three kinds of pore gradations, and the correlation model between fractal dimension and porosity is established. These results demonstrate the high impermeability and outstanding corrosion resistance of synthetic-graded mortar. The fractal model of the pore structure evolution of MSA mortar has a guiding effect on the pore distribution evolution and engineering permeability evaluation of MSA mortar in engineering. Full article

This study presents multistability mechanisms that can enhance the energy harvesting performance of a piezoelectric energy harvester (PEH) with geometrical nonlinearities. To configure triple potential wells, static bifurcation diagrams in the structural parameter plane are depicted. On this basis, the key structural parameter is considered, of which three reasonable values are then chosen for comparing and evaluating the performances of the triple-well PEH under them. Then, intra-well responses and the corresponding voltages of the system are investigated qualitatively. A preliminary analysis of the suitable energy-harvesting conditions is carried out, which is then validated by numerical simulations of the evolution of coexisting attractors and their basins of attraction with variations in the excitation level and frequency. It follows that, under a low-level ambient excitation, the intra-well responses around the trivial equilibrium dominate the energy-harvesting performance. When the level of the environmental excitation is very low, which one of the three values of the key structural parameter is the best for improving the performance of the PEH system depends on the range of the excitation frequency; when the excitation level increases sufficiently to induce inter-well responses, the maximum one is the best for improving the performance of the PEH. The findings provide valuable insights for researchers working in the structure optimization and practical applications of geometrically nonlinear PEHs. Full article

This study introduces two innovative methods, the new transform iteration method and the residual power series transform method, to solve fractional nonlinear system Korteweg–de Vries (KdV) equations. These equations, fundamental in describing nonlinear wave phenomena, present complexities due to the involvement of fractional derivatives. In demonstrating the application of the new transform iteration method and the residual power series transform method, computational analyses showcase their efficiency and accuracy in computing solutions for fractional nonlinear system KdV equations. Tables and figures accompanying this research present the obtained solutions, highlighting the superior performance of the new transform iteration method and the residual power series transform method compared to existing methods. The results underscore the efficacy of these novel methods in handling complex nonlinear equations involving fractional derivatives, suggesting their potential for broader applicability in similar mathematical problems. Full article

The temporal evolution of the global mean sea level (GMSL) is investigated in the present analysis using the monthly mean values obtained from two sources: a reconstructed dataset and a satellite altimeter dataset. To this end, we use two well-known techniques, detrended fluctuation analysis (DFA) and multifractal DFA (MF-DFA), to study the scaling properties of the time series considered. The main result is that power-law long-range correlations and multifractality apply to both data sets of the global mean sea level. In addition, the analysis revealed nearly identical scaling features for both the 134-year and the last 28-year GMSL-time series, possibly suggesting that the long-range correlations stem more from natural causes. This demonstrates that the relationship between climate change and sea-level anomalies needs more extensive research in the future due to the importance of their indirect processes for ecology and conservation. Full article

The fractional-order LCL filter, composed of two fractional-order inductors and one fractional-order capacitor, is a novel fractional-order π-type circuit introduced in recent years. Based on mathematical modeling, this article comprehensively studies the principles and frequency characteristics of fractional-order LCL filters. Five critical properties are derived and rigorously demonstrated. One of the most significant findings is that we identify the necessary and sufficient condition for resonance in fractional-order LCL filters when the sum of the orders of the fractional-order inductors and the fractional-order capacitor is equal to 2, which provides a theoretical foundation for effectively avoiding resonance in fractional-order LCL filters. The correctness of our theoretical derivation and analysis was confirmed through digital simulations. This study reveals that fractional-order LCL filters exhibit more versatile operational characteristics than traditional integer-order LCL filters, paving the way for broader application prospects. Full article