Fractal Behavior Research Articles

Fractal Continuum Maxwell Creep Model

In this work, the fractal continuum Maxwell law for the creep phenomenon is introduced. By mapping standard integer space-time into fractal continuum space-time using the well-known Balankin’s approach to variable-order fractal calculus, the fractal version of Maxwell model is developed. This methodology employs local fractional differential operators on discontinuous properties of fractal sets embedded in the integer space-time so that they behave as analytic envelopes of non-analytic functions in the fractal continuum space-time. Then, creep strain ε(t), creep modulus J(t), and relaxation compliance G(t) in materials with fractal linear viscoelasticity can be described by their generalized forms, εβ(t),Jβ(t) and Gβ(t), where β=dimS/dimH represents the time fractal dimension, and it implies the variable-order of fractality of the self-similar domain under study, which are dimS and dimH for their spectral and Hausdorff dimensions, respectively. The creep behavior depends on beta, which is characterized by its geometry and fractal topology: as beta approaches one, the fractal creep behavior approaches its standard behavior. To illustrate some physical implications of the suggested fractal Maxwell creep model, graphs that showcase the specific details and outcomes of our results are included in this study.

A model characterising the fractal supercritical adsorption behaviour of methane in shale: Incorporating principles, methods, and applications

The pore structure of shale is patently heterogeneous, with a conventional adsorption model being difficult to characterise the complex adsorption behavior of methane in shale. Therefore, a model is being proposed to characterise the fractal supercritical adsorption behavior of methane in shale. The principles and derivation of the model have been described in detail. The results showed that the adsorption characteristic parameters relating to different scale pores were affected by temperature and fractal dimension to different degrees. In addition, the relationship between isosteric heat of adsorption (qst) and adsorption capacity of different samples revealed different trends. The absolute values of ΔS of FC-66 and FC-72 decreased linearly. This might have been because of the fact that the adsorption saturation’s inflection point, which caused a fall in the absolute value of the entropy change (ΔS), was not reached. Compared with traditional adsorption models, the proposed model could reflect menthane’s adsorption in shale more accurately. The research results provide theoretical support for the evaluation of deep shale gas reserves, including the optimisation of exploitation strategies.

Adaptive synchronization and anti-synchronization of Julia sets generated by the competitive model

In this paper, we study the fractal behaviour of a competitive model that describes the interaction of plankton allelopathy. This paper aims to establish synchronization and anti-synchronization of Julia sets of two competitive systems with some different parameters by using an adaptive control strategy. Firstly, a discrete version of the competitive model is obtained, and then the Julia set of the discrete version is generated by using the escape-time algorithm. Adaptive controllers and parameter update laws for unknown parameters are designed to achieve synchronization and anti-synchronization of Julia sets. Furthermore, we can determine unknown parameters of the competitive system by using this adaptive control technique. Here, the adaptive synchronization and anti-synchronization of Julia sets are accomplished by its trajectories synchronization and anti-synchronization due to the close relation of trajectories of the system with the Julia set of the system. Numerical simulations are carried out to validate several key theoretical results as well as the efficacy and accuracy of the applied methodologies. Moreover, with the help of this analysis, we can study other models of a similar type.

Crack characteristic and fractal analysis of SFRC shear wall with CFST columns under repeated low cycle load

In this study, we conducted a comprehensive analysis of the crack characteristics and fractal behavior in steel fiber reinforced concrete shear walls (SFRCSW) reinforced with concrete-filled steel tube columns (CFSTCs) under repeated low-cycle loading conditions after failure. Notably, the incorporation of steel fibers and CFSTCs did not alter the fundamental shear failure mode but effectively constrained crack widths to 2.0 mm and 1.6 mm, respectively, from an initial width of 2.4 mm. The use of steel fibers or CFSTCs led to significant improvements in the surface crack fractal dimension (SCFD) of shear walls, with increase ratios of 9.7 % and 7.88 %, respectively. Conversely, an increase in SCFD was associated with improvements in yield load, peak load, failure load, and cumulative hysteretic energy consumption in SFRCSWs. SCFD is suitable for comparative evaluation of the bearing capacity and energy dissipation capacity of different shear walls, but it is not suitable for comparative evaluation of displacement and ductility. For shear walls, it seems more reasonable to evaluate the damage degree in loading process by displacement - fractal dimension relation, rather than load - fractal dimension relation, with the critical values of fractal dimension are about 1.12 and 1.17.

Fractal Behavior of Nanostructured Pt/TiO2 Catalysts: Synthesis, Characterization and Evaluation of Photocatalytic Hydrogen Generation

Fractal Behavior of Nanostructured Pt/TiO2 Catalysts: Synthesis, Characterization and Evaluation of Photocatalytic Hydrogen Generation

FRACTAL DIFFUSION IN AN ISOTROPOUS MEDIUM

This paper provides a comprehensive examination of fractal diffusion in an isotropic medium. This process has a fascinating temporal memory and an extremely high initial response, which are unquestionably characterized by the Caputo fractional derivative. Its spatio-symmetric property is described by the Riesz fractional derivative. To illustrate the diffusion mechanism, we present an example of a toxic gas leak in a subway station. The numerical results are extremely promising and challenging. They clearly indicate that a MEMS-based gas sensor can track a toxic gas leak immediately due to its extremely fast diffusion at the initial time. This is of great importance for public safety. Furthermore, the diffusion process is primarily contingent upon the density of the air, which we can easily control. The diffusion pattern in an isotropic medium exhibits fractal behavior, which is extremely helpful for designing an emergency system for evacuating people from the center of toxic gas leaks by changing air density or temperature. This paper presents a rigorous mathematical concept for monitoring systems and early warning systems for toxic gas leaks.

Generation of fractal curves using new binary 8-point interpolatory subdivision scheme

The subdivision scheme is a valuable tool for designing shapes and representing geometry in computer-aided geometric design. It has excellent geometric properties, such as fractals and adjustable shape. In this research paper, we explore the generation of fractal curves using a novel binary 8-point interpolatory subdivision scheme with two parameters. We analyse different properties of the proposed scheme, including convergence, special cases, and fractals. Additionally, we demonstrate through various examples the relationship between the shape parameters and the fractal behaviour of the resulting curve. Our research also identifies a specific range of shape parameters that can effectively produce fractal curves. The findings of this study provide a fast and efficient method for generating fractals, as demonstrated by numerous examples. Modelling examples show that the 8-point interpolatory scheme can enhance the efficiency of computer design for complex models.

Dynamical complexity in microscale disk-wind systems

Context. Powerful winds at accretion-disk scales have been observed in the past 20 years in many active galactic nuclei (AGN). These are the so-called ultrafast outflows (UFOs). Outflows are intimately related to mass accretion through the conservation of angular momentum, and they are therefore a key ingredient of most accretion disk models around black holes (BHs). At the same time, nuclear winds and outflows can provide the feedback that regulates the joint BH and galaxy growth. Aims. We reconsidered UFO observations in the framework of disk-wind scenarios, both magnetohydrodynamic disk winds and radiatively driven winds. Methods. We studied the statistical properties of observed UFOs from the literature and derived the distribution functions of the ratio ω̄ of the mass-outflow and -inflow rates and the ratio λw of the mass-outflow and the Eddington accretion rates. We studied the links between ω̄ and λw and the Eddington ratio λ = Lbol/LEdd. We derived the typical wind-activity history in our sources by assuming that it can be statistically described by population functions. Results. We find that the distribution functions of ω̄ and λw can be described as power laws above some thresholds, suggesting that there may be many wind subevents for each major wind event in each AGN activity cycle, which is a fractal behavior. We then introduced a simple cellular automaton to investigate how the dynamical properties of an idealized disk-wind system change following the introduction of simple feedback rules. We find that without feedback, the system is overcritical. Conversely, when feedback is present, regardless of whether it is magnetic or radiation driven, the system can be driven toward a self-organized critical state. Conclusions. Our results corroborate the hypothesis that AGN feedback is a necessary key ingredient in disk-wind systems, and following this, in shaping the coevolution of galaxies and supermassive BHs.

RETRACTED: Zhong et al. Fractal Behavior of Particle Size Distribution in the Rare Earth Tailings Crushing Process under High Stress Condition. Appl. Sci. 2018, 8, 1058

The Applied Sciences Editorial office retracts the article “Fractal Behavior of Particle Size Distribution in the Rare Earth Tailings Crushing Process under High Stress Condition” [...]

A maximum-likelihood-based approach to estimate the long memory parameter using fractional spline wavelets

Fractional spline wavelets act as fractional differentiators for essentially low-pass signals, which is a useful property to analyze time series with fractal behavior. Moreover, under suitable scaling of wavelet coefficients, one can whiten a fractional Gaussian noise inputted in the wavelet channels. In this article, we demonstrate the two former statements and, by employing them, propose a method based on maximum likelihood to estimate the long-memory parameter using fractional spline wavelets. The proposed estimator exhibits competitive behavior in terms of mean squared error in the simulation studies we conducted, dominating all other methods as the sample size increases. In empirical applications, the proposed method outperformed other approaches in 10 of 12 evaluations for three different time series. Moreover, the proposed method allows for estimation in a continuous search grid, even when the Hurst coefficient lies outside of stationarity boundaries, which is a clear advantage over many algorithms: in such cases, the proposed method displays a stable behavior and an increasing outperformance margin. We also propose an identification procedure for the persistence or antipersistence of a given signal, based on the energy concentration of the coefficients of wavelet decomposition and its theoretical properties.

Predator–Prey Interaction with Fear Effects: Stability, Bifurcation and Two-Parameter Analysis Incorporating Complex and Fractal Behavior

This study investigates the dynamics of predator–prey interactions with non-overlapping generations under the influence of fear effects, a crucial factor in ecological research. We propose a novel discrete-time model that addresses limitations of previous models by explicitly incorporating fear. Our primary question is: How does fear influence the stability of predator–prey populations and the potential for chaotic dynamics? We analyze the model to identify biologically relevant equilibria (fixed points) and determine the conditions for their stability. Bifurcation analysis reveals how changes in fear levels and predation rates can lead to population crashes (transcritical bifurcation) and complex population fluctuations (period-doubling and Neimark–Sacker bifurcations). Furthermore, we explore the potential for controlling chaotic behavior using established methods. Finally, two-parameter analysis employing Lyapunov exponents, spectrum, and Kaplan–Yorke dimension quantifies the chaotic dynamics of the proposed system across a range of fear and predation levels. Numerical simulations support the theoretical findings. This study offers valuable insights into the impact of fear on predator–prey dynamics and paves the way for further exploration of chaos control in ecological models.

Duality in the Directed Landscape and Its Applications to Fractal Geometry

AbstractGeodesic coalescence, or the tendency of geodesics to merge together, is a hallmark phenomenon observed in a variety of planar random geometries involving a random distortion of the Euclidean metric. As a result of this, the union of interiors of all geodesics going to a fixed point tends to form a tree-like structure that is supported on a vanishing fraction of the space. Such geodesic trees exhibit intricate fractal behaviour; for instance, while almost every point in the space has only one geodesic going to the fixed point, there exist atypical points that admit two such geodesics. In this paper, we consider the directed landscape, the recently constructed [ 18] scaling limit of exponential last passage percolation (LPP), with the aim of developing tools to analyse the fractal aspects of the tree of semi-infinite geodesics in a given direction. We use the duality [ 39] between the geodesic tree and the interleaving competition interfaces in exponential LPP to obtain a duality between the geodesic tree and the corresponding dual tree in the landscape. Using this, we show that problems concerning the fractal behaviour of sets of atypical points for the geodesic tree can be transformed into corresponding problems for the dual tree, which might turn out to be easier. In particular, we use this method to show that the set of points admitting two semi-infinite geodesics in a fixed direction a.s. has Hausdorff dimension $4/3$, thereby answering a question posed in [ 12]. We also show that the set of points admitting three semi-infinite geodesics in a fixed direction is a.s. countable.

Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain

The soliton propagation of the fractional-in-space nonlinear Schrodinger equation (NLSE) is much more complicated than that of the corresponding integer NLSE. The aim of this paper is to discover some novel fractal soliton propagation behaviors (FSPBs) of this fractional-in-space NLSE. Firstly, the exact solution is compared with the present numerical solution, and the validity and accuracy of the present numerical method are verified. Secondly, the effect of fractional derivatives on soliton propagation is explored through the present numerical simulation results. At the same time, the present method is extended to the three-dimensional fractional-order NLSE. Finally, some novel FSPBs of the fractional-in-space NLSE are given.

Statistical Study of the Bias and Precision for Six Estimation Methods for the Fractal Dimension of Randomly Rough Surfaces

The simulation and characterisation of randomly rough surfaces is an important topic in surface science, tribology, geo- and planetary sciences, image analysis and optics. Extensions to general random processes with two continuous variables are straightforward. Several surface generation algorithms are available, and preference for one or another method often depends on the specific scientific field. The same holds for the methods to estimate the fractal dimension D. This work analyses six algorithms for the determination of D as a function of the size of the domain, variance, and the input value for D, using surfaces generated by Fourier filtering techniques and the random midpoint displacement algorithm. Several of the methods to determine fractal dimension are needlessly complex and severely biased, whereas simple and computationally efficient methods produce better results. A fine-tuned analysis of the power spectral density is very precise and shows how the different surface generation algorithms deviate from ideal fractal behaviour. For large datasets defined on equidistant two-dimensional grids, it is clearly the most sensitive and precise method to determine fractal dimension.

Influence of swift heavy ion irradiation on structure and morphology of La0.25Pr0.375Ca0.375MnO3 film

The effects of Ag15+ (120 MeV) swift heavy ion (SHI) irradiation on the structural and morphological properties of epitaxial La0.25Pr0.375Ca0.375MnO3 (LPCMO) thin films were investigated by x-ray scattering and atomic force microscopy (AFM) techniques. LPCMO films of thickness ∼ 280 Å were irradiated with an Ag15+ ion beam at different fluences of 1 × 1011, 5 × 1011, and 1 × 1012 ions cm−2. XRD results suggested the development of the tensile stress along the out-of-plane direction of the LPCMO film upon ion irradiation, which increases on increasing the ion fluence. The morphology of the film was also modified with the irradiation and an increase in the fluence of the ion beam enhanced the in-plane height-height correlation length scale (grain size) with a loss of the fractal behaviours. The linear variation of microstrain with ion irradiation fluence in thin LPCMO film can be considered for a possible strain-driven application in modifying functional properties of such a phase separated complex oxide.

Theoretical analysis and numerical simulation of methane adsorption behavior on rough surfaces featuring fractal property

This work focuses on the control mechanism of adsorption behavior from rough surfaces featuring fractal property numerically. For that, we develop an equivalent characterization method for rough fractal surfaces, reproduce the adsorption process of methane on the surface by combining molecular dynamics and Grand canonical Monte Carlo (GCMC) simulations, analyze the control mechanisms of complex types of surface geometry on the methane adsorption process, and clarity the ratio of vertical roughness to horizontal complexity as key parameter to describe the adsorption capability. Our numerical results indicate that the adsorption density of methane in fractal and non-fractal slit pores varies with the fractal behavior of surface geometry, while the density of gas phase methane remains unaffected (all distributed around 0.06 g/cm3). And, the complexities of surface geometry process various effects on occurrence mode, adsorption potential energy (ɛ), and adsorption amounts. The surface original complexity will influence the force field, and alter the occurrence mode from monolayer adsorption to pore filling. Meanwhile, the width-to-depth ratio of surface original complexity has a negatively effects the ɛ of methane. The behavioral complexity of the surface reflects the number of small pores on it. For fractal slit pores, the maximum adsorption amounts are 2405.77, 2634.90, and 2749.46 mol/m3, respectively. They are smaller than that of the non-fractal slit pore, which is due to the increase of non-adsorption pores for the small diameter of pores. Above results give a clear explanation to the effects of fractal property on methane adsorption, which will provide a reference to study the adsorption process of methane in Coalbed Methane reservoir.

Micro- and Nano-Roughness Separation Based on Fractal Analysis.

When describing the tribological behaviour of technical surfaces, the need for full-length scale microtopographic characterization often arises. The self-affine of surfaces and the characterisation of self-affine using a fractal dimension and its implantation into tribological models are commonly used. The goal of our present work was to determine the frequency range of fractal behaviour of surfaces by analysing the microtopographic measurements of an anodised aluminium brake plunger. We also wanted to know if bifractal and multifractal behaviour can be detected in real machine parts. As a result, we developed a new methodology for determining the fractal range boundaries to separate the nano- and micro-roughness. To reach our goals, we used an atomic force microscope (AFM) and a stylus instrument to obtain measurements in a wide frequency range (19 nm-3 mm). Power spectral density (PSD)-based fractal evaluation found that the examined surface could not be characterised by a single fractal dimension. A new method capable of separating nano- and micro-roughness has been developed for investigating multifractal behaviour. The presented procedure separates nano- and micro-roughness based on the geometric characteristics of surfaces. In this way, it becomes possible to specifically examine the relationship between the micro-geometry that can be measured in each wavelength range and the effects of cutting technology and the material structure that creates them.

THE FRACTAL RHEOLOGY OF MAGNETORHEOLOGICAL ELASTOMERS DESCRIBED THROUGH THE MODIFIED ZENER MODEL AND THE COLE–COLE PLOT

The main objective of this study focuses on using, for the first time, the Cole–Cole semilogarithmic fractal plot to study the fractal rheological behavior of magnetorheological elastomer materials using the fractional Zener model. Experimental data and theoretical results show that the Cole–Cole semilogarithmic fractal plot predicts the Zener mathematical model’s material parameters. This paper elucidates the benefits of using Cole–Cole semilogarithmic fractal plots to accurately predict material rheological properties of composite elastomer reinforced with nano or microiron particles observed during experimental tests.

PREFACE — SPECIAL ISSUE ON FRACTALS AND LOCAL FRACTIONAL CALCULUS: RECENT ADVANCES AND FUTURE CHALLENGES

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems.

INTELLIGENT EXTRACTION OF COMPLEXITY TYPES IN FRACTAL RESERVOIR AND ITS SIGNIFICANCE TO ESTIMATE TRANSPORT PROPERTY

Fractal pore structure exists widely in natural reservoir and dominates its transport property. For that, more and more effort is devoted to investigate the control mechanism on mass transfer in such a complex and multi-scale system. Apparently, effective characterization of the fractal structure is of fundamental importance. Although the newly emerged concept of complexity assembly clarified the complexity types and their assembly mechanism in a fractal system, equivalent extraction of the complexity types is the key for effective characterization. For these, we proposed a deep learning-based method to extract the original and behavioral complexity assembled in bed-packing fractal porous media for simplification and without loss of generality. In detail, the UNeXt network model was trained to obtain the independent connected regions of scaling objects with different scales, the edge detection and clustering analysis algorithms were employed to extract the number-size relationship between two successive scaling objects, and the unique inversion of fractal behavior was realized by taking the number-size model and fractal topography together. Consequently, an equivalent characterization method for fractal complex pore structure was developed based on the concept of complexity assembly. Our investigation provides a theoretical guidance and method reference for the quantitative characterization of fractal porous media that will guarantee the fundamental requirement for the accurate evaluation of the transport properties of natural reservoir.