Search Results (647)

Fractal geometry offers a mathematical framework to quantify the complexity of brain structure and function. The fractal dimension (FD) captures self-similarity and irregularity across spatial and temporal scales, surpassing the limits of traditional Euclidean metrics. In neuroscience, FD serves as a key descriptor of the brain’s hierarchical organization—from dendritic arborization and cortical folding to neural dynamics measured by diverse neuroimaging techniques. This review summarizes theoretical foundations and methodological advances in FD estimation, including the box-counting approach for imaging, and Higuchi’s and Katz’s algorithms for electrophysiological data, addressing reliability and reproducibility issues. In addition, we illustrate how fractal analysis characterizes brain complexity in health and disease. Clinical applications include detecting white matter alterations in multiple sclerosis, atypical maturation in intrauterine growth restriction, reduced cortical complexity in Alzheimer’s disease, and altered neuroimaging patterns in schizophrenia. Emerging evidence highlights FD’s potential for distinguishing consciousness states and quantifying neural integration and differentiation. Bridging mathematics, physics, and neuroscience, fractal analysis provides a quantitative lens on the brain’s multiscale organization and pathological deviations. FD thus stands as both a theoretical descriptor and a translational biomarker whose standardization could advance precision diagnostics and understanding of neural dynamics. Full article

Interfaces of rather different natures—as, e.g., bacterial colony or forest fire boundaries, or semiconductor layers grown by different methods (MBE, sputtering, etc.)—are self-affine fractals, and feature scaling with universal exponents (depending on the substrate’s dimensionality d and global topology, as well as on the driving randomness’ spatial and temporal correlations but not on the underlying mechanisms). Adding lateral growth as an essential (non-equilibrium) ingredient to the known equilibrium ones (randomness and interface relaxation), the Kardar–Parisi–Zhang (KPZ) equation succeeded in finding (via the dynamic renormalization group) the correct exponents for flat $d=1$ substrates and (spatially and temporally) uncorrelated randomness. It is this interplay which gives rise to the unique, non-Gaussian scaling properties characteristic of the specific, universal type of non-equilibrium roughening. Later on, the asymptotic statistics of process $h\left(x\right)$ fluctuations in the scaling regime was also analytically found for $d=1$ substrates. For $d>1$ substrates, however, one has to rely on numerical simulations. Here we review a variational approach that allows for analytical progress regardless of substrate dimensionality. After reviewing our previous numerical results in $d=1$, 2, and 3 on the time evolution of one of the functionals—which we call the non-equilibrium potential (NEP)—as well as its scaling behavior with the nonlinearity parameter $\lambda$, we discuss the stochastic thermodynamics of the roughening process and the memory of process $h\left(x\right)$ in KPZ and in the related Golubović–Bruinsma (GB) model, providing numerical evidence for the significant dependence on initial conditions of the NEP’s asymptotic behavior in both models. Finally, we highlight some open questions. Full article

The reliability of mechanical systems hinges on analyzing the actual surface-to-surface contact area, which critically influences dynamic behavior, friction, material performance, and thermal dissipation. Uneven surfaces lead to incomplete contact, where only a fraction of asperities touch, creating a nominal contact area. This study proposes a novel fractal contact model for various mechanical behaviors between mechanical contact surfaces, integrating the Weierstrass–Mandelbrot fractal function and nonuniform rational B-spline interpolation (NURBS) to model material-dependent actual contact conditions. Furthermore, this research delved into the changes in thermal conductivity across the surfaces of metal materials within a simulated setting. It maintained a contact ratio ranging from 0.038% to 15.2%, a factor that remained unaffected by contact pressure. Both experimental and simulated findings unveiled an actual contact rate spanning from 0.44% to 1.06%, thereby underscoring the distinctive interface behaviors specific to different materials. The proposed approach provides fresh perspectives for investigating material–contact interactions and tackling associated engineering hurdles. Full article

The development of shale gas reservoirs relies on complex fracture networks created via multistage hydraulic fracturing, yet most existing models still use oversimplified fracture geometries and therefore cannot fully capture the coupled effects of multiscale fracture topology on flow and production. To address this gap, in this study, we combine fractal geometry with the Embedded Discrete Fracture Model (EDFM) to analyze the production dynamics of hydraulically fractured horizontal wells in shale gas reservoirs. A tree-like fractal fracture network is first generated using a stochastic fractal growth algorithm, where the iteration number, branching number, scale factor, and deviation angle control the self-similar hierarchical structure and spatial distribution of fractures. The resulting fracture network is then embedded into an EDFM-based, fully implicit finite-volume simulator with Non-Neighboring Connections (NNCs) to represent multiscale fracture–matrix flow. A synthetic shale gas reservoir model, constructed using representative geological and engineering parameters and calibrated against field production data, is used for all numerical experiments. The results show that increasing the initial water saturation from 0.20 to 0.35 leads to a 26.4% reduction in cumulative gas production due to enhanced water trapping. Optimizing hydraulic fracture spacing to 200 m increases cumulative production by 3.71% compared with a 100 m spacing, while longer fracture half-lengths significantly improve both early-time and stabilized gas rates. Increasing the fractal iteration number from 1 to 3 yields a 36.4% increase in cumulative production and markedly enlarges the pressure disturbance region. The proposed fractal–EDFM framework provides a synthetic yet field-calibrated tool for quantifying the impact of fracture complexity and design parameters on shale gas well productivity and for guiding fracture network optimization. Full article

Deep learning for semantic segmentation has made significant advances in recent years, achieving state-of-the-art performance. Medical image segmentation, as a key component of healthcare systems, plays a vital role in the diagnosis and treatment planning of diseases. Due to the fractal and scale-invariant nature of biological structures, effective medical image segmentation requires models capable of capturing hierarchical and self-similar representations across multiple spatial scales. In this paper, a Recurrent Neural Network (RNN) is explored within the Convolutional Neural Network (CNN) and Vision Transformer (ViT)-based hybrid U-shape network, named RCV-UNet. First, the ViT-based layer was developed in the bottleneck to effectively capture the global context of an image and establish long-range dependencies through the self-attention mechanism. Second, recurrent residual convolutional blocks (RRCBs) were introduced in both the encoder and decoder to enhance the ability to capture local features and preserve fine details. Third, by integrating the global feature extraction capability of ViT with the local feature enhancement strength of RRCBs, RCV-UNet achieved promising global consistency and boundary refinement, addressing key challenges in medical image segmentation. From a fractal–fractional perspective, the multi-scale encoder–decoder hierarchy and attention-driven aggregation in RCV-UNet naturally accommodate fractal-like, scale-invariant regularity, while the recurrent and residual connections approximate fractional-order dynamics in feature propagation, enabling continuous and memory-aware representation learning. The proposed RCV-UNet was evaluated on four different modalities of images, including CT, MRI, Dermoscopy, and ultrasound, using the Synapse, ACDC, ISIC 2018, and BUSI datasets. Experimental results demonstrate that RCV-UNet outperforms other popular baseline methods, achieving strong performance across different segmentation tasks. The code of the proposed method will be made publicly available. Full article

This paper uses the theory of self-affine fractal functions to model the dynamic flight graphs of starling flocks, integrating the fractional calculus of self-affine fractal functions to quantitatively characterize the intrinsic nonlinear dynamics and memory effects within the system, employing statistical inference methods to find the fractal fit for the images. The changes in box dimensions over time could characterize the phase transition process of the starling flight flocks. By analyzing the rate of change of fractal dimensions, we identify critical points corresponding to phase transitions during collective flight behavior. During the flight of the starling flocks, a real-time phase transition process for evading attacks and effective advancement has been identified. Experimental data confirms the effectiveness of controlling the phase transition. Full article

To study the creep compression characteristics and evolution mechanism of pressure-bearing graded crushed rock under constant load. Creep compression tests of crushed rock were conducted using the self-developed confined compression test system under different Talbot indexes and axial stresses. The axial displacement, void ratio, mass distribution, fractal dimension, and fragmentation of crushed rock during creep compression were analyzed. And the void ratio-fractal dimension model of crushed rock under pressure was established. The results reveal three-stage characteristics in axial displacement and void change, which correspond to rapid, attenuation, and stable change processes. The axial displacement and fragmentation amount are positively correlated with the axial stress and Talbot index, while the porosity is negatively correlated with them. The fractal dimension shows a positive correlation with axial stress and a negative correlation with the Talbot index. Additionally, a theoretical model was established to characterize the dynamic correlation between void ratio and fractal dimension during compression process, and its accuracy was verified, with a maximum error of only 0.0819. The research findings can provide insights for stability prediction and deformation control of crushed rock in engineering applications such as building foundation pits, ground treatment, and coal mine goafs. Full article

The pore architecture of the Nahr Umr Formation sandstone reservoirs is highly complex and heterogeneous, severely limiting efficient oilfield development. Conventional methods often fail to adequately characterize such intricate pore systems, necessitating the application of fractal theory. Focusing on sandstone samples from the Nahr Umr-B Member, this study integrates thin section identification, XRD, HPMI, and NMR to characterize the fractal features of the reservoir pore structure and evaluate fluid mobility. The results indicate that from Type I to Type III reservoirs, displacement pressure and median pressure gradually increase, whereas the average and median pore-throat radius gradually decrease, and the pore-throat sorting coefficient decreases. For instance, Type I reservoirs exhibit an average displacement pressure of 0.15 MPa, a median pressure of 0.81 MPa, an average pore-throat radius of 1.96 μm, and a median pore-throat radius of 2.85 μm; in contrast, Type III reservoirs show averages of 14.43 MPa, 45.32 MPa, 0.02 μm, and 0.03 μm, respectively. These trends reflect a gradual deterioration in pore connectivity, increased resistance to fluid flow, and a reduction in the development of larger pore throats. From Type I to Type III reservoirs, both the total fractal dimension (D_H) and the movable fluid pore fractal dimension (D_N2) show a gradual increasing trend. This indicates that the pore structure becomes increasingly complex and heterogeneous, the complexity of the movable fluid pore space increases, and fluid mobility progressively weakens. Furthermore, higher quartz content and lower cement and clay mineral contents correspond to smaller reservoir pore fractal dimensions and stronger fluid mobility. For example, Sample No. 3 (Type I) has a quartz content of 91.97%, a cement content of 1.64%, and a clay mineral content of 6.4%, with a D_H of 2.4385 and D_N2 of 2.9323. Conversely, Sample No. 4 (Type III) has a quartz content of 49.72%, a cement content of 11.21%, and a clay mineral content of 39.07%, with a D_H of 3.9099 and D_N2 of 2.9762. Compared to D_H, D_N2 reduces the prediction error for dynamic quality by over 70% on average, offering a more reliable prediction of fluid mobility and providing a more precise scale for evaluating reservoir development potential. Full article

Background: Glioblastoma (GBM) is among the most aggressive and morphologically heterogeneous brain tumors. Beyond static imaging biomarkers, its structural organization can be viewed as a nonlinear dynamical system. Characterizing morphodynamic attractors within such a system may reveal latent stability patterns of morphological change and potential indicators of morphodynamic organization. Methods: We analyzed 494 subjects from the multi-institutional BraTS 2020 dataset using a fully automated computational pipeline. Each multimodal MRI volume was encoded into a 16-dimensional latent space using a 3D convolutional autoencoder. Synthetic morphological trajectories, generated through bidirectional growth–shrinkage transformations of tumor masks, enabled training of a contraction-regularized Neural Ordinary Differential Equation (Neural ODE) to model continuous-time latent morphodynamics. Morphological complexity was quantified using fractal dimension (DF), and local dynamical stability was measured via a Lyapunov-like exponent (λ). Robustness analyses assessed the stability of DF–λ regimes under multi-scale perturbations, synthetic-order reversal (directionality; sign-aware comparison) and stochastic noise, including cross-generator generalization against a time-shuffled negative control. Results: The DF–λ morphodynamic phase map revealed three characteristic regimes: (1) stable morphodynamics (λ < 0), associated with compact, smoother boundaries; (2) metastable dynamics (λ ≈ 0), reflecting weakly stable or transitional behavior; and (3) unstable or chaotic dynamics (λ > 0), associated with divergent latent trajectories. Latent-space flow fields exhibited contraction-induced attractor-like basins and smoothly diverging directions. Kernel-density estimation of DF–λ distributions revealed a prominent population cluster within the metastable regime, characterized by moderate-to-high geometric irregularity (DF ≈ 1.85–2.00) and near-neutral dynamical stability (λ ≈ −0.02 to +0.01). Exploratory clinical overlays showed that fractal dimension exhibited a modest negative association with survival, whereas λ did not correlate with clinical outcome, suggesting that the two descriptors capture complementary and clinically distinct aspects of tumor morphology. Conclusions: Glioblastoma morphology can be represented as a continuous dynamical process within a learned latent manifold. Combining Neural ODE–based dynamics, fractal morphometry, and Lyapunov stability provides a principled framework for dynamic radiomics, offering interpretable morphodynamic descriptors that bridge fractal geometry, nonlinear dynamics, and deep learning. Because BraTS is cross-sectional and the synthetic step index does not represent biological time, any clinical interpretation is hypothesis-generating; validation in longitudinal and covariate-rich cohorts is required before prognostic or treatment-monitoring use. The resulting DF–λ morphodynamic map provides a hypothesis-generating morphodynamic representation that should be evaluated in covariate-rich and longitudinal cohorts before any prognostic or treatment-monitoring use. Full article

During drill-and-blast construction, complex and variable rock masses are frequently encountered. Owing to the transient nature of the explosion process and the randomness of crack propagation, the response of different rock masses to explosive loading is highly intricate. This study primarily investigates the dynamic response of rock masses with varying strengths under two different charge configurations. First, four cement mortar specimens of differing strengths were prepared then subjected to general blasting and slit charge blasting, respectively. High-speed cameras and digital image correlation techniques were employed to capture and analyse stress wave propagation and crack propagation during detonation. Fractal dimension analysis was subsequently employed to quantify and compare the extent of damage in the specimens. Findings indicate that rock strength influences stress wave attenuation patterns: lower-strength rocks exhibit higher peak strains but faster decay rates. Crack propagation velocity was calculated by deploying monitoring points along fracture paths and defining fracture initiation thresholds. Higher rock strength correlates with both peak and average crack propagation velocities. Slit charge blasting effectively optimizes damage distribution, concentrating it within the intended directions while reducing chaotic fracturing. These findings provide scientific justification for blasting operations in complex rock formations. Full article

River discharge scaling is fundamental to the global hydrological cycle and to water resource assessment. This study investigates the existence of multiple scaling regimes and introduces a novel framework for clustering river discharge using multiscale fractal characteristics. We analyzed daily discharge data from 38 stations across continental Canada over an 80-year period. Multifractal characterization was performed at decadal and long-term scales using three key parameters: the singularity exponent (${\alpha }_0$), multifractal strength ($\alpha$), and asymmetry index (r). K-means clustering in the $\alpha -r$, ${\alpha }_0-r$, and $\alpha -{\alpha }_0$ planes revealed distinct clusters, with the asymmetric parameter (r) emerging as the strongest distinguishing factor. These clusters represent groups of rivers with similar dynamical structures: the $\alpha -r$ clusters categorize discharge based on scaling strength and fluctuation influence. Analysis of the generalized Hurst exponent revealed anti-persistent behavior at most stations, with exceptions at five specific locations. This multifractal clustering approach provides a powerful method for classifying river regimes based on intrinsic characteristics and identifying the physical drivers of discharge fluctuations. Full article

Chaotic dynamical systems are ubiquitous in nature and modern technology, with applications ranging from secure communications and cryptography to the design of chaos-based sensors and modeling biological phenomena such as arrhythmias and neuronal behavior. Given their complexity, precise analysis of these systems is crucial for both theoretical understanding and practical implementation. The characterization of chaotic dynamical systems typically relies on conventional measures such as Lyapunov exponents and fractal dimensions. While these metrics are fundamental for describing dynamical behavior, they are often computationally expensive and may fail to capture subtle changes in the overall geometry of the attractor, limiting comparisons between systems with topologically similar structures and similar values in common chaos metrics such as the Lyapunov exponent. To address this limitation, this work proposes a geometric framework that treats chaotic attractors as spatial objects, using topological tools—specifically the $\alpha$-sphere—to quantify their shape and spatial extent. The proposed method was validated using Chua’s system (including two reported variations), the Rössler system (standard and piecewise-linear), and a fractional-order multi-scroll system. A parametric characterization of the Rössler system was also performed by varying parameter b. Experimental results show that this geometric approach successfully distinguishes between attractors where classical metrics reveal no perceptible differences, in addition to being computationally simpler. Notably, we observed geometric variations of up to 80% among attractors with similar dynamics and introduced a specific index to quantify these global discrepancies. Although this geometric analysis serves as a complement rather than a substitute for chaos detection, it provides a reliable and interpretable metric for differentiating systems and selecting attractors based on their spatial properties. Full article

To investigate the mechanical behavior and energy evolution characteristics of various rock materials under impact loading, dynamic impact tests were conducted on five representative rock types using a split Hopkinson pressure bar (SHPB) apparatus, combined with X-ray diffraction (XRD) and scanning electron microscopy (SEM) techniques. The dynamic mechanical response, energy characteristics, mineral composition, and associated microstructural features of these typical rocks were systematically analyzed. The results show that basalt exhibits the highest peak strength, followed by blue sandstone and granite; all three display typical brittle failure characteristics, whereas red sandstone and green sandstone demonstrate greater ductility and plastic deformation capacity. By introducing the energy-time density index, the energy-time density of the rocks ranks from strongest to weakest as follows: green sandstone, red sandstone, granite, blue sandstone, and basalt. An innovative dynamic strength–energy-time density mapping model was established to elucidate the clustering and distinguishing characteristics of these rock materials. Assay results and mesoscopic images confirm the relationship between mineral composition and the fine structure of rock fragmentation mechanisms, highlighting that the critical transition from intergranular to transgranular fracture is the key mechanism governing impact pulverization. Furthermore, fractal analysis reveals that higher fractal dimensions are associated with more complex microcrack structures and may correlate with the corresponding energy dissipation intensity. These findings provide profound insight into the failure mechanisms of rocks under dynamic loading, offering significant theoretical value and engineering application prospects, particularly in fields such as mining excavation and rock mass stability assessment. Full article

Prediction of coal and rock dynamic disasters is essential for ensuring the safety, efficiency, and long-term sustainability of deep mining operations. To improve the accuracy of acoustic methods for forecasting coal instability, acoustic emission (AE) source localization experiments are conducted on coal samples under uniaxial compression, and the multidimensional correlations among AE events together with the evolution characteristics of the corresponding complex network are investigated. The results show that the temporal correlations of AE events exhibit nonlinear decay with increasing time intervals, the spatial correlations display fractal clustering that transcends Euclidean geometry, and the energetic correlations reveal hierarchical transitions controlled by intrinsic material properties. To capture these interactions, a multidimensional correlation calculation method is developed to quantitatively characterize these multidimensional coupled relationships of AE events, and a complex network of AE events is constructed. The network evolution from sparse to highly interconnected is quantified using three parameters: average degree, clustering coefficient, and modularity. A rapid rise in the first two metrics, accompanied by a sharp decline in the latter, indicates the rapid strengthening of AE event correlations, the aggregation of local microcrack clusters, and their transition into a global fracture network, thereby providing a clear early warning of impending compressive failure of the coal sample. The study establishes a mechanistic link between microcrack evolution and macroscopic failure, offering a robust real-time monitoring tool that supports sustainable mining by reducing disaster risk, improving resource extraction stability, and minimizing socio-economic and environmental losses associated with dynamic failures in deep underground coal operations. Full article

In control systems, chaos is a natural dualistic phenomenon that can be both a beneficial resource to be used and a negative phenomenon to be avoided. The study examines two opposing paradigms: positive chaotic control, which aims to enhance performance, and negative chaos management, which aims to stabilize a system. More sophisticated suppression methods, including adaptive neural networks, sliding mode control, and model predictive control, can decrease convergence times. Controlled chaotic dynamics have significantly impacted the domain of embedded control systems. Specialized controller designs include fractal-based systems and hybrid switching systems that offer better control of chaotic behavior in many situations. The paper highlights the key issues that are related to chaos-based systems, such as the need to implement them in real time, parameter sensitivity, and safety. Recent research suggests an increased interdependence between artificial intelligence, quantum computing, and sustainable technology. The synthesis shows that chaos control has evolved into an engineering field, significantly impacting the industry, which was initially a theoretical concept. It also offers exclusive ideas in the design and improvement of complex control systems. Full article