Fractal Behavior Research Articles

Spatial Hurst–Kolmogorov Clustering

The stochastic analysis in the scale domain (instead of the traditional lag or frequency domains) is introduced as a robust means to identify, model and simulate the Hurst–Kolmogorov (HK) dynamics, ranging from small (fractal) to large scales exhibiting the clustering behavior (else known as the Hurst phenomenon or long-range dependence). The HK clustering is an attribute of a multidimensional (1D, 2D, etc.) spatio-temporal stationary stochastic process with an arbitrary marginal distribution function, and a fractal behavior on small spatio-temporal scales of the dependence structure and a power-type on large scales, yielding a high probability of low- or high-magnitude events to group together in space and time. This behavior is preferably analyzed through the second-order statistics, and in the scale domain, by the stochastic metric of the climacogram, i.e., the variance of the averaged spatio-temporal process vs. spatio-temporal scale.

Matrix Diffusion as a Mechanism Contributing to Fractal Stream Chemistry and Long‐Tailed Transit Time Distributions

AbstractSolute transit or travel time distributions (TTDs) in catchments are relevant to both hydrochemical response and inference of hydrologic mechanisms. Long‐tailed TTDs and fractal scaling behavior of stream concentration power spectra (∼1/frequency, or 1/frequency to a power <2) are widely observed in catchment studies. In several catchments, a significant fraction of streamflow is derived from groundwater in shallow fractured bedrock, where matrix diffusion significantly influences solute transport. I present frequency and time domain theoretical analyses of solute transport to quantify the influence of matrix diffusion on fractal scaling and long‐tailed TTDs. The theoretical concentration power spectra exhibit fractal scaling, and the corresponding TTDs resemble a gamma distribution. The tails of the TTDs are influenced by accessible matrix width, exhibiting a sustained power‐law (rather than exponential) decline for large matrix widths. Application to an experimental catchment shows that theoretical spectra match previously reported power spectral estimates derived from concentration measurements.

Monitoring a chemical reaction using pH measurements: An approach based on multiscale fractal analysis

The monitoring of (bio)-chemical reactions is mandatory to determine the operating conditions that maximize the performance of the processes and guarantee the process safety of chemical and biotechnological processes. However, it is not always possible to have monitoring systems that determine the reactive/product concentration in real-time. Therefore, offline analysis methods are commonly used, which are usually expensive and can generate considerable lag times. In this work, an indirect monitoring approach based on the fractal analysis of fluctuations of pH time series to monitor a chemical saponification reaction is presented. First, an experimental design was implemented to set the validation of the proposed monitoring system, evaluating the temperature and stirring speed effects on the conversion rate. Then, pH time series were analyzed using two multiscale methodologies (i.e., DFA and R/S analysis), which show that the pH series fluctuations exhibit fractal behavior. Also, two distinct regions can be identified, suggesting that the pH signals may contain information on the phenomena that intervene in the reaction process (i.e., transport phenomena). The dynamic scaling exponents show direct correlations with the reagent concentration, obtaining coefficients of determination R2> 0.98 in all experiments carried out. pH measurement is inexpensive, easy to implement, and can be obtained in real-time. The results suggest that multiscale analysis of pH series has a high potential for online monitoring of a class of reactive processes with low economic and computational costs.

Multifractal Behaviors of Stock Indices and Their Ability to Improve Forecasting in a Volatility Clustering Period

The financial market is a complex system, which has become more complicated due to the sudden impact of the COVID-19 pandemic in 2020. As a result there may be much higher degree of uncertainty and volatility clustering in stock markets. How does this “black swan” event affect the fractal behaviors of the stock market? How to improve the forecasting accuracy after that? Here we study the multifractal behaviors of 5-min time series of CSI300 and S&P500, which represents the two stock markets of China and United States. Using the Overlapped Sliding Window-based Multifractal Detrended Fluctuation Analysis (OSW-MF-DFA) method, we found that the two markets always have multifractal characteristics, and the degree of fractal intensified during the first panic period of pandemic. Based on the long and short-term memory which are described by fractal test results, we use the Gated Recurrent Unit (GRU) neural network model to forecast these indices. We found that during the large volatility clustering period, the prediction accuracy of the time series can be significantly improved by adding the time-varying Hurst index to the GRU neural network.

Fractal analysis of wildfire pattern dynamics using a Small World Network model

The dynamics of a spreading wildfire is modeled by using a variant of the Small World Network stochastic model (Zekri et al., 2005). The temporal evolution of the fire front width and its pattern’s fractal dimension exhibits an anomalous relaxation process before the extinction of the initial fire-line that is well described by a Kohlrausch–Williams–Watts-like relaxation function. The decrease of the fire pattern’s dimension corresponds to a transition from a structured fire front induced by the collective effects of burning fuels to a dispersed one caused by their individual effects. This anomalous relaxation with an exponent γ≈0.5 indicates the existence of a sub-diffusion process of the spreading fire regardless of the impact length and fuel occupation density. After the extinction of the initial fire-line, the fractal dimension of the fire pattern fluctuates around an average value corresponding to an average spread rate of fire and front intensity. The amplitude of these fluctuations decreases as the impact length of the burning fuel or the occupation density increases. The analysis of the local scale exponents shows that the fractal (self-similar) behavior of the fire pattern is observed in the spatial resolution range delimited by the mean free path and the pattern size. Further discussion on the competition effect between the heat release rate of the flame and the mean free path on fire behavior is provided.

The fractal behaviour of gravity field elements: case study in Egypt

ABSTRACT The fractal behaviour of gravity field elements was investigated. Particularly, the geoidal heights, gravity anomalies and vertical deflection components were considered. Such data were synthesized from the ultra-high degree SGG-UGM-1 geopotential model. The Egyptian Territory was selected as the study area. To supplement the results, parallel fractal analyses were performed on the GTM3A, Earth2012 and SRTM30 terrain data. The regularization, variogram and global Power Spectral Density (PSD) concepts were applied as different tools for fractal analysis. Consistent fractal dimensions resulted from the regularization and variogram algorithms. These two algorithms ensured the fractalness of the gravitational signals, including the geoidal height. The global PSD algorithm failed to give meaningful fractal dimensions for the topography and geoidal heights. However, it yielded realistic dimensions for the other elements. So, fractal dimensions could characterize the roughness and spatial precisions of different gravitational data. Also, multifractal modeling of the geopotential might be a promising task.

Gel Formation in Reversibly Cross-Linking Polymers

By means of Langevin dynamics simulations, we investigate the gel formation of randomly functionalized polymers in solution, with the ability to form both intra- and intermolecular reversible bonds. Under highly dilute conditions, these polymers form soft nano-objects (so-called single-chain nanoparticles, SCNPs), resulting from the purely intramolecular cross-linking of the reactive functional groups. Here we show that the competition between intra- and intermolecular bonds at finite concentration is governed by a delicate balance of various entropic contributions and leads to a density dependent effective valence. System-spanning networks are formed at relatively low monomer densities and their stability is mediated by just a small number of intermolecular connections per chain. The formation of intermolecular bonds furthermore can induce a non-monotonic dependence of the polymer size on the density for long bond lifetimes. Concomitantly, the polymers in the percolating cluster adopt an intramolecular structure characteristic for self-avoiding chains, which constitutes a strong contrast to the fractal globular behavior of irreversible SCNPs in crowded solutions with purely topological interactions (no intermolecular bonds). Finally, we study the dynamics of the system, which displays signatures expected for reversible gel-forming systems. An interesting behavior emerges in the reorganization dynamics of the percolating cluster. The relaxation is mostly mediated by the diffusion over long distances, through breaking and formation of bonds, of chains that do not leave the percolating cluster. Regarding the few chains that are transiently free, the time they spend until they reattach to the cluster is solely governed by the bond strength.

Alleviating Fractal and Ill-Conditioning Problems of the AC Power Flow Using a Polynomial Form

Power system resilience assessment, and mitigation, which deals with high-impact, low probability events, is gaining a great deal of attention among power researchers. To perform such an analysis, the availability of tools able to analyze modern power grids under normal and extreme disturbances is imperative. One such tool is the AC power flow algorithm. The latter should exhibit a numerical stability and an excellent convergence property under a broad range of operating conditions of a power system. One major difficulty that needs to be overcome stems from the nonlinearity and non-convexity of the AC power flow equations. As shown by Thorp and Naqavi, the Newton-Raphson (NR) method suffers from a fractal behavior, and the power flow Jacobian matrix becomes singular at points located on the fractal hypersurface, preventing that algorithm to converge. Furthermore, the Jacobian matrix becomes ill-conditioned for power systems that have relatively short lines or are heavy loaded. To address these problems, we propose a transformation of the AC power flow equations from a sinusoidal form to a polynomial form, which is solved using a modified Newton's method. The new polynomial form, whose order is equal to the number of solutions, encompasses node-based equations, Pythagorean equations, and loop-based equations. Furthermore, our approach is quadratic in three cases. Simulation results performed on several power systems reveal the good performance of the proposed method.

Using Fractals to Describe Ecologically Relevant Patterns in Distributions of Large Rocks in Streams

AbstractMeasuring the physical complexity of habitats or ecological resources is often achieved using system‐specific methods that make comparisons across ecosystems difficult. One measure that is applicable across multiple ecosystems and scales is the fractal dimension, which has the benefit of generality as well as potential scale independence. This study evaluated the use of box‐counting and entropy fractal dimensions for characterizing the complexity of emergent rock distributions in six streams across Scotland and Australia. Emergent rocks (ER) are important hydraulic features and ecological resources, including as oviposition sites for aquatic insects and cover for fish. We completed fractal analysis on counts of ER in 5‐m segments along longitudinal stretches of the six streams. All six streams exhibited fractal behavior (self‐similarity), suggesting that fractals can be used to measure the complexity of longitudinal ER distributions in a way that is scale independent. Entropy was a superior measure due to its ability to differentiate among the six streams whereas box‐counting could not. Together, field results and numerical simulations showed that fractal dimensions of emergent rock distributions were related to stream geomorphology. Well‐developed bedforms, like alternating pools and riffles had better organized emergent rocks because large bed materials were more likely to be emergent in topographic highs. Streams with coarser bed materials had more chaotic arrangements of emergent rocks because this increased the general abundance of emergent rocks, making differentiation between topographic highs and lows less distinctive. Fractal dimensions, therefore, can measure the complexity of river systems in a way that is relevant to geomorphological and ecological processes.

Fractal behavior of surface oxide crack patterns on AISI 4140 high-strength low-alloy steel exposed to the simulated offshore environment

Wear and corrosion are surface-dependent degradation phenomena pertinent to the hostile environment existing in deep-sea oil and gas wells containing aggressive chemical species such as chloride/bromide salts, organic acids, carbon dioxide, and hydrogen sulfide,etc. Apart from the identification of the potential failures of alloys that can occur on a large and devastating scale by its interaction with these deleterious species, it is also essential to comprehend the nature and character, particularly the cracking pattern of the surface oxides formed on alloys undergoing corrosion in a particular environment. This article uses fractal geometry to study the branching of the surface oxide crack patterns in AISI 4140 high-strength low-alloy steel after exposure in a simulated offshore corrosive environment containing aqueous 3.5 wt.% sodium chloride solution. The fractal geometry was evaluated using the box-count technique in three different electrolytic test conditions: Electrochemical Anodic Polarization, Electrochemical Cathodic Polarization, and Immersion Corrosion for 24 h. Our results showed that the magnitude of fractal dimension is highest during immersion corrosion for 24 h and lowest during anodic polarization. Fractals can provide complementary quantifiable information about the surface geometries and fracture crack patterns from macrostructural to microstructural length scales by image analysis. This approach can be leveraged by emerging technologies like Artificial Intelligence (AI), Machine Learning (ML) and Computer Vision (CV), in structural health monitoring of alloys used in exotic and challenging conditions.

Effects of Weakly Nonlinear Waves on Acoustic Scattering from the Ocean Surface

The elevation of ocean waves is always modeled in linear theory as a superposition of the sinusoidal components with crests and troughs of identical heights. However, under some circumstances, the wave amplitude is outside the linear range and presents as a weakly nonlinear asymmetrical waveform with sharper crests and shallower troughs. We studied the impact of the weakly nonlinear effect of ocean waves in deep and intermediate waters on acoustic scattering from the surface of the ocean using two rough surface models with fractal geometry and power law spectral behavior in the equilibrium range. The classic Weierstrass–Mandelbrot function was used to model the linear waves and a new fractal function, the fractional Weierstrass function developed in studies of electromagnetism, was used to model the weakly nonlinear waves. We evaluated these two models using the Pierson–Moskowitz spectrum and the incident wavelength. The bistatic scattering strength was obtained via a numerical method based on the “exact” solution of the integral equation. The weakly nonlinear phenomenon led to a very small reduction in the narrow area around the specular reflection angle and a small increase in the remaining wide area, including the backpropagation area with a scattering angle [Formula: see text]. The differences in backscattering strength between the two models were similar to the bistatic scattering strength in the backpropagation area and did not depend on the incident grazing angle.

Some engineering applications of newly constructed algorithms for one-dimensional non-linear equations and their fractal behavior

The aim of this paper is to develop some novel numerical algorithms for finding roots of one-dimensional non-linear equations. We derive these algorithms by utilizing the main and basic idea of the variational iteration technique. The convergence rate of the suggested algorithms is discussed. It is corroborated that the proposed numerical algorithms possess sixth-order convergence. To demonstrate the validity, applicability, and the performance of the proposed algorithms, we solved different test problems. These problems also include some real-life applications associated with the chemical engineering such as van der Wall’s equation, conversion of nitrogen-hydrogen feed to ammonia and the fractional-transformation in the chemical reactor problem. The numerical results of these problems show that the proposed algorithms are more effective against the other well-known similar nature existing methods. Finally, the dynamics of the suggested algorithms in the form of the polynomiographs of different complex polynomials have been analyzed that reveals the fractal nature and the other dynamical aspects of the suggested algorithms.

Multiple-scale incision-infill cycles in deep-water channels from the lacustrine Transylvanian Basin, Romania: Auto- or allogenic controls?

Many hierarchical schemes have been developed to describe multiple-scale cut-and-fill architecture of deep-water channel systems and analyse their evolution. A widely applicable genetical-based scheme is not on the horizon yet. Near seismic-scale exposures of two deep-water channel systems from the late Miocene of the Transylvanian Basin, Romania offer a fresh insight into the dynamics of erosional and depositional processes. Both channel systems are underlain by mudstone-prone successions; the channel-complex scale incisional surface can only be observed at Tău. Channel-form surfaces interpreted as the base of channel elements and storeys are present in both outcrops. At Tău, the shallow, base-of-slope channel complex consists of amalgamated sandstones separated by thin mudstones. The axis of the deep, slope channel at Daia is characterized by pebble lags, thick amalgamated sandstones that thin towards the margin, and thin mudstones. Backsets, possible cyclic steps, and erosional features, such as rip-up mud-boulders, detachment edges, and ghost bedding also occur. Cycles of incision, bypass with coarse lag accumulation, deposition from gravelly then sandy turbidity currents were repeated numerous times with differing amplitudes, which were followed by the abandonment of the channel complex. Novel incision-infill graphs were created to analyse and compare the cyclicity pattern of deep-water channels. Successive incision depth of channel-form surfaces, and infill height of channel-form bodies aids in distinguishing hierarchical levels, vertical and lateral offsets. Daia and Tău channel complexes show a remarkably similar cyclicity pattern, which might indicate an inherent character of this turbidite system. To investigate whether this pattern is universal, incision-infill graphs of two Laingsburg Karoo channel systems were constructed, based on published data. In all the four examples, a pattern of lateral-then-vertical offset of channel elements was present, and a similarity between channel element-complex and channel storey-element aggradation was found, suggesting a fractal behaviour. Two models are applicable to interpret the revealed pattern of incision-infill graphs, for the Tău and Daia channels: 1) three scales of waxing-waning flow magnitude cyclicity could create the architecture of channel complexes, or 2) the presence of autocyclic channel element incision-infill could eliminate one scale of flow magnitude changes, but two still need to be governed by allocyclic factors. The late Miocene turbidite system in the Transylvanian Basin developed in Lake Pannon, in close relationship with the uplifting Carpathians fringing the basin. As the duration of cycles is shorter than the time range of possible changes in uplift rate or fault activity, tectonically induced cyclicity is discarded. Climatically induced cycles are the most probable controls on varying the flow magnitude that created the complex architecture of channel systems in this lacustrine setting.

Investigation of fractal behavior, optical properties and electronic environments of carbon-doped of ZnO Thin Films

The undoped zinc oxide (ZnO) and carbon-doped zinc oxide (C:ZnO) thin films were synthesized with different process temperatures by using a chemical vapor deposition technique. The fractal dimension of AFM images was evaluated by power spectral density function, spatial function distribution, cube counting, and triangular counting by Gwyddion software. AFM results reveal undoped and C:ZnO indicated a larger particle size (~ 1.5 µm) and 200–50 nm, respectively, with increase in process temperature from 650 to 750 °C. The estimated lattice strain varied from 0.0033 to 0.0008 and 0.0038 to 0.0078 corresponding to the process temperature 650–750 °C for ZnO and C:ZnO thin films, respectively. The oxygen and zinc contents varied from 43.65 to 47.16 at.% and 47.13 to 33.83 at.%, respectively. The optical bandgap varied from 3.17 to 3.19 eV and 3.14 to 3.06 eV for undoped ZnO and C:ZnO thin films, respectively, and ID/IG varied 1.97–2.22 as the carbon content increased from 9.21 to 19 at.%. Photoluminescence of C:ZnO indicated deep level defect shifted from 481.56 to 515.1 nm with increase in process temperature from 650 to 750 °C. The possible bonding network of core orbital of Zn(2p), C(1s), and O(1s) in the C:ZnO thin films has been discussed by deconvolution with the Origin 2018.

Pore Structure Characterization of Eocene Low-Permeability Sandstones via Fractal Analysis and Machine Learning: An Example from the Dongying Depression, Bohai Bay Basin, China.

Poroperm analysis, mercury injection capillary pressure (MICP), and nuclear magnetic resonance (NMR) measurements were performed to delineate the pore structures and fractal behaviors of the Eocene low-permeability sandstones in the Dongying Depression, Bohai Bay Basin, China. Three types of pore structures (I, II, and III) have been classified by applying the self-organizing map (SOM) clustering model. Comparative analysis of three different fractal models indicates that the MICP tubular model and NMR model are quite effective for pore structure characterization. The results show that the reservoirs generally exhibit high fractal dimensions, indicative of complex pore structures. The presence of small pore throats is primarily responsible for the heterogeneities and complexities in the Eocene low-permeability sandstones. A modified Winland model was established for the permeability estimation using MICP data. Different from high-permeability reservoirs or unconventional (e.g., shale and tight formation) reservoirs, r10 is the best parameter for permeability estimation, indicating that the permeability of the Eocene low-permeability sandstones is largely controlled by the large pore systems. Additionally, a porosity model derived from movable fluids using NMR data has been established and provided better prediction effect compared with the classic Coates and Schlumberger Doll Research (SDR) models. Fractal analysis and permeability estimation are shown to be quite effective for investigating microscopic behaviors and in predicting the reservoir quality of low-permeability sandstone reservoirs.

Fractal behaviors of NMR saturated and centrifugal T2 spectra in oil shale reservoirs: The Paleogene Funing formation in Subei basin, China

Fluorescence thin section and scanning electron microscopy (SEM) analysis were performed to delineate pore spaces and pore network assemblages of oil shales of Member 2 of Paleogene Funing Formation (E1f2) in Subei Basin, China. Nuclear magnetic resonance (NMR) T2 (transverse relaxation time) spectra were used to gain insight into pore size distribution. Fractal analysis was performed on the NMR T2 spectra measured at saturated and centrifugal status, respectively, with the aim to reveal the different fractal behaviors of irreducible pore realms (centrifugal T2 spectrum) and the entire pore systems (saturated T2 spectrum). Relationships between NMR parameters and fractal dimensions were revealed. The results show that the pore spaces consist of 1) large pore realms including interparticle pores, and microfractures, 2) small pore realms including intragranular dissolution pores, intercrystal micropores and organic matter pores. The T2 spectra are mostly bimodal and left-skewed due to abundance in small pore realms. The right peak or tail distribution is associated with large pore realms. Fractal analysis performed on saturated and centrifugal T2 spectra show that fractal dimensions calculated from centrifugal T2 spectrum are higher. In addition, fractal dimensions for centrifugal T2 spectrum show negative correlation relationships with BVI and T2cutoff values. The different fractal behaviors of saturated and centrifugal T2 spectra reflect the complex pore assemblages in oil shales. The small pore realms determine the microscopic complexity, while large pore realms control the macroscopic reservoir quality of oil shales. However large pore realms (interparticle pores and micro-fractures) are not self-similar with the small pore realms, and can't be described by fractal dimension. Fractal analysis gives insights in the heterogeneous assemblage of pore systems and helps describe complexity of pore structure.

Fractal behaviors of hydraulic head and baseflow of the nested groundwater flow systems in response to rainfall

Fractal behaviors of hydraulic head and baseflow of the nested groundwater flow systems in response to rainfall

Coastlines and percolation in a model for hierarchical random deposition

We revisit a known model in which (conducting) blocks are hierarchically and randomly deposited on a D-dimensional substrate according to a hyperbolic size law with the block size decreasing by a factor λ>1 in each subsequent generation. In the first part of the paper the number of coastal points (in D=1) or coastline segments (in D=2) is calculated, which are points or lines that separate a region at “sea level” and an elevated region. We find that this number possesses a non-universal character, implying a Euclidean geometry below a threshold value Pc of the deposition probability P, and a fractal geometry above this value. Exactly at the threshold, the geometry is logarithmic fractal. The number of coastline segments in D=2 turns out to be exactly twice the number of coastal points in D=1. We comment briefly on the surface morphology and derive a roughness exponent α. In the second part, we study the percolation probability for a current in this model and two extensions of it, in which both the scale factor and the deposition probability can take on different values between generations. We find that the percolation threshold Pc is located at exactly the same value for the deposition probability as the threshold probability of the number of coastal points. This coincidence suggests that exactly at the onset of percolation for a conducting path, the number of coastal points exhibits logarithmic fractal behaviour.

A Global-Scale Investigation of Stochastic Similarities in Marginal Distribution and Dependence Structure of Key Hydrological-Cycle Processes

To seek stochastic analogies in key processes related to the hydrological cycle, an extended collection of several billions of data values from hundred thousands of worldwide stations is used in this work. The examined processes are the near-surface hourly temperature, dew point, relative humidity, sea level pressure, and atmospheric wind speed, as well as the hourly/daily streamflow and precipitation. Through the use of robust stochastic metrics such as the K-moments and a second-order climacogram (i.e., variance of the averaged process vs. scale), it is found that several stochastic similarities exist in both the marginal structure, in terms of the first four moments, and in the second-order dependence structure. Stochastic similarities are also detected among the examined processes, forming a specific hierarchy among their marginal and dependence structures, similar to the one in the hydrological cycle. Finally, similarities are also traced to the isotropic and nearly Gaussian turbulence, as analyzed through extensive lab recordings of grid turbulence and of turbulent buoyant jet along the axis, which resembles the turbulent shear and buoyant regime that dominates and drives the hydrological-cycle processes in the boundary layer. The results are found to be consistent with other studies in literature such as solar radiation, ocean waves, and evaporation, and they can be also justified by the principle of maximum entropy. Therefore, they allow for the development of a universal stochastic view of the hydrological-cycle under the Hurst–Kolmogorov dynamics, with marginal structures extending from nearly Gaussian to Pareto-type tail behavior, and with dependence structures exhibiting roughness (fractal) behavior at small scales, long-term persistence at large scales, and a transient behavior at intermediate scales.

Co-movement, fractal behaviour and forecasting of exchange rates

In this paper we analyse the fractal behaviour and correlations of exchange rates and develop a new approach to obtain better forecast results with small errors. Multifractal de-trended fluctuation analysis (MFDFA) is used to get fractal time series and their Hurst Exponents in specific time period and based on these results. We established Vector Autoregressive Fractionally Integrated Moving Average (VARFIMA) models of exchange rates on certain dates. Hurst Exponents allows us to detect small fluctuations (persistent characteristic) of time series and their multifractal behaviour. We consider, USD/CAD, USD/GBP, USD/EUR and USD/CHF exchange rates prices are as time series. Our theory is to show that if a time series has the highest multifractal degree then it has the most succesfull prediction results for the time interval which four time series have highest co-movement. We use Wavelet coherence analysis to understand co-movement of the exchange rates. The findings of Wavelet Coherence showed correlation between these exchange rates, the relationships among the two and four rates (with multiple wavelet coherence-MWC) and how these connections differ in the space of frequency-time. These relations and MFDFA results authorize us to construct VARFIMA models of exchange rate data which produce effective forecast outcomes with small errors.