Fractal Dynamics Research Articles

Mercury Bonding to Xerogel: The Interface Fractal Dynamics of the Interaction between Two Complex Systems.

This study describes novel solid substances founded on chitosan and TEGylated phenothiazine that have a high ability for hydrargyrum recovery from watery liquid solutions. These compounds were taken into account, consisting of two distinct entity interactions inside of the classic fractal dynamics conjecture of an "interface". They were assimilated through fractal-type mathematical objects and judged as such. The bi-stable behavior of two fractally connected objects was demonstrated both numerically and graphically. The fractal character was demonstrated by the fractal analysis made using SEM images of the xerogel compounds with interstitial fixed hydrargyrum. For the first time, SEM helped to verify such samples from two distinct bodies, with the multifractal parameter values being listed in continuation. The fractal dimension of the rectangular mask is D1 = 1.604 ± 0.2798, the fractal dimension of the square mask is D2 = 1.596 ± 0.0460, and the lacunarity is 0.0402.

Resting-state fractal brain connectivity is associated with impaired cognitive performance in healthy aging

Aging affects cognitive functions even in the absence of ongoing pathologies. The neurophysiological basis of age-related cognitive decline (CD), however, is not completely understood. Alterations in both functional brain connectivity and in the fractal scaling of neuronal dynamics have been linked to aging and cognitive performance. Recently, fractal connectivity (FrC) has been proposed — combining the two concepts — for capturing long-term interactions among brain regions. FrC was shown to be influenced by increased mental workload; however, no prior studies investigated how resting-state FrC relates to cognitive performance and plausible CD in healthy aging. We recruited 19 healthy elderly (HE) and 24 young control (YC) participants, who underwent resting-state electroencephalography (EEG) measurements and comprehensive cognitive evaluation using 7 tests of the Cambridge Neurophysiological Test Automated Battery. FrC networks were reconstructed from EEG data using the recently introduced multiple-resampling cross-spectral analysis (MRCSA). Elderly individuals could be characterized with increased response latency and reduced performance in 4–4 tasks, respectively, with both reaction time and accuracy being affected in two tasks. Auto- and cross-spectral exponents — characterizing regional fractal dynamics and FrC, respectively, — were found reduced in HE when compared to YC over most of the cortex. Additionally, fractal scaling of frontoparietal connections expressed an inverse relationship with task performance in visual memory and sustained attention domains in elderly, but not in young individuals. Our results confirm that the fractal nature of brain connectivity — as captured by MRCSA — is affected in healthy aging. Furthermore, FrC appears as a sensitive neurophysiological marker of age-related CD.

On fractal thermodynamics of the superconducting transition and the roles of specific heat, entropy and critical magnetic field in disordered superconductors

In this study, we have introduced the concept of ``fractal temperature derivative'' motivated from recent works of Li and Ostoja-Starzewski in their analysis of nonlinear fractal dynamics in porous and anisotropic media. We have introduced various analytical forms of the critical magnetic field and we have discussed their thermodynamical properties, mainly the specific heat and entropy. These forms have been compared with the exact analytical form obtained based on the ``fractal temperature derivative'' approach. It was observed that these analytical forms hold a number of advantageous features which are practical to various types of superconductors including strongly disordered superconductors. Additional properties have been obtained and analyzed accordingly.

Universal System of Tripling Periods for the Eye Is the Way to the “Logarithmic” Treatment of Neurological Diseases with Modern Lighting Technologies

Previously, the pattern of evolutionary adjustment of photodetectors of the human retina to the frequencies of the UPTS (the so-called universal period-tripling system) was phenomenologically established. The period tripling effect is a new direction in the study of complex systems. In these papers, it was established that the UPTS is a characteristic of a wide class of natural fractal systems described, in particular, by complex cubic mappings. In this work, taking into account the fact that both vision and the processing of visual information by the brain work based on the principles of UPTS, a fundamental frequency, dynamic model of the mechanism of the eye-vision system is constructed, generalizing experimental and theoretical data on the fractal dynamics of the brain in normal and neurological diseases. Data on the Luscher test, on MRI and slow EEG waves, neuro-hormones, photooxy-therapy of brain diseases allow us to establish a new paradigm of lighting technology for health-friendly lighting and ways of a new successful therapeutic effect on Pacrkinson’s disease, effects of traumatic brain injury, stroke, schizophrenia, psychopathy, neurosis. In this case, the lighting scheme requires the simultaneous observance of five rows of rankings of lighting parameters: by frequencies (1), according to the combination of frequencies, and the dynamics of the presentation of combinations (2), by colour perception errors (3), according to the patient’s refusals determine the colours (4), according to the fractal laws of the frequencies of vision and the brain (5). The universality of period-tripling system gives new ways to the principles of lighting engineering and photo-medicine.

Time–frequency analysis of cryptocurrency attention

ABSTRACT We present a wavelet analysis of retail investor attention and the daily returns of Bitcoin, Ethereum, and Litecoin at five selected crypto exchanges that identifies the fractal dynamics of the short- and long-term persistent processes. The investors’ attention is proxied by the Search Volume Index provided by Google at daily frequency. We detect significant temporal cyclical movements and coherence between cryptocurrency returns and retail investor attention at long investment horizons: from the beginning of 2017 to the middle of 2018 and, to a lesser degree, in 2019. Investment horizons that dominated in 2017 and 2018 were mainly driven by retail investor attention rather than by uncertainty, risk, or stock markets. Therefore, we do not confirm that cryptocurrencies can be considered a safe-haven asset in times of crisis because there is no significant negative comovement between the returns of cryptocurrencies and stock returns or economic uncertainty. Furthermore, the phase shift analysis indicates that attention can serve as a leading indicator for the cryptocurrency returns, particularly in 2017 and 2018. Therefore, retail investors are encouraged to use the Search Volume Index as an early warning indicator in case of sudden changes in the cryptocurrency returns to maximise profits or minimise losses.

Fractal dynamics of individual mitochondrial oscillators measure local inter-mitochondrial coupling

Mitochondrial inner membrane potentials in cardiomyocytes may oscillate in cycles of depolarization/repolarization when the mitochondrial network is exposed to metabolic or oxidative stress. The frequencies of such oscillations are dynamically changing while clusters of weakly coupled mitochondrial oscillators adjust to a common phase and frequency. Across the cardiac myocyte, the averaged signal of the mitochondrial population follows self-similar or fractal dynamics; however, fractal properties of individual mitochondrial oscillators have not yet been examined. We show that the largest synchronously oscillating cluster exhibits a fractal dimension, D, that is indicative of self-similar behavior with D=1.27±0.11, in contrast to the remaining network mitochondria whose fractal dimension is close to that of Brownian noise, D=1.58±0.10. We further demonstrate that fractal behavior is correlated with local coupling mechanisms, whereas it is only weakly linked to measures of functional connections between mitochondria. Our findings suggest that individual mitochondrial fractal dimensions may serve as a simple measure of local mitochondrial coupling.

Propagation of waves in fractal spaces

In this study, we study waves propagation in fractal spaces based on two independent variational approaches: the first one is based on the ‘product-like fractal measure’ introduced by Li and Ostoja-Starzewski in their analysis of nonlinear fractal dynamics in anisotropic porous media whereas the second and another is based on fractal calculus, which includes analytical functions with fractal support. The first approach is motivating since it describes physics in anisotropic media characterized by fractal dimensions. The second one is also of particular importance since it offers a new framework to reformulate the laws of physics in spaces with fractional dimensions. In both approaches, the fractional Laplace equation is derived and solved using plausible boundary conditions. This study proves the importance of fractals in wave motion We show that these models are able to describe the propagation of waves in a fractal Hausdorff dimensional space.

Simulation of Nonlinear Viscous Fingering in a Reactive Flow Displacement: A Multifractal Approach

fractal analysis of viscous fingering of a reactive miscible flow displacement in homogeneous porous media is investigated and multifractal spectrum, and fractal dimension are introduced as two essential features to characterize the irregularity of finger patterns. The Reaction of the two reactant fluids generates a miscible chemical product C in the contact zone. Considering the similarity between chemical products and coastline, monofractal and multifractal analyzes are performed. In monofractal analysis, the box-counting method is implemented on binary images and in multifractal analysis, due to the image processing; the fractal characteristics of viscous fingering instability are analyzed by means of fractal quantities such as Holder exponent, multifractal spectrum, f (α)-image and fractal dimension dynamics. Fractal analysis shows that the fractal dimension increases with time. Also, by considering five different nonlinear simulations, the results show that in the case both sides of the chemical product C are unstable, the multifractal spectrum curve has the highest peak, which means the more complex finger patterns lead to more values of fractal dimension. In addition, a comparison between different values of Ar is conducted and the results show similar behavior. However, small value of aspect ratio leads to a broader width of the multifractal spectrum curve. Furthermore, f (α)-images of concentration contour were investigated for different precisions and some undetectable finger patterns were observed in these images. It can be concluded that the use of f (α)-image represents more detailed image than concentration contours.

Analysis and Fractal Dynamics of Local Fractional Partial Differential Equations Occurring in Physical Sciences

Abstract In this paper, we implement the local fractional natural homotopy perturbation method (LFNHPM) to solve certain local fractional partial differential equations (LFPDEs) with fractal initial conditions occurring in physical sciences in a fractal domain. LFPDEs successfully exhibit the important properties of physical models occurring in a fractal medium. The working methodology depicts the feasibility and accuracy of the implemented approach for given LFPDEs. Moreover, the solutions for LFPDEs are obtained in a closed form and are in good agreement with the previously determined results. The numerical simulations are also investigated for each of the LFPDE on Cantor set. The implementation of the method in view of numerical simulations authenticates that the applied method is precise, and useful to investigate the solutions of partial differential equations with local fractional derivatives.

A Generalization of Poiseuille’s Law for the Flow of a Self-Similar (Fractal) Fluid through a Tube Having a Fractal Rough Surface

In this paper, a generalization of Poiseuille’s law for a self-similar fluid flow through a tube having a rough surface is proposed. The originality of this work is to consider, simultaneously, the self-similarity structure of the fluid and the roughness of the tube surface. This study can have a wide range of applications, for example, for fractal fluid dynamics in hydrology. The roughness of the tube surface presents a fractal structure that can be described by the surface fractal noninteger dimensions. Complex fluids that are invariant to changes in scale (self-similar) are modeled as a continuous medium in noninteger dimensional spaces. In this work, the analytical solution of the Navier–Stokes equations for the case of a self-similar fluid flow through a rough “fractal” tube is presented. New expressions of the velocity profiles, the fluid discharge, and the friction factor are determined analytically and plotted numerically. These expressions contain fractal dimensions describing the effects of the fractal aspect of the fluid and of that of the tube surface. This approach reveals some very important results. For the velocity profile to represent a physical solution, the fractal dimension of the fluid ranges between 0.5 and 1. This study also qualitatively demonstrates that self-similar fluids have shear thickening-like behavior. The fractal (self-similarity) nature of the fluid and the roughness of the surface both have a huge impact on the dynamics of the flow. The fractal dimension of the fluid affects the amplitude and the shape of the velocity profile, which loses its parabolic shape for some values of the fluid fractal dimension. By contrast, the roughness of the surface affects only the amplitude of the velocity profile. Nevertheless, both the fluid’s fractal dimension and the surface roughness have a major influence on the behavior of the fluid, and should not be neglected.

Experimental observations of fractal landscape dynamics in a dense emulsion.

Many soft and biological materials display so-called 'soft glassy' dynamics; their constituents undergo anomalous random motions and complex cooperative rearrangements. A recent simulation model of one soft glassy material, a coarsening foam, suggested that the random motions of its bubbles are due to the system configuration moving over a fractal energy landscape in high-dimensional space. Here we show that the salient geometrical features of such high-dimensional fractal landscapes can be explored and reliably quantified, using empirical trajectory data from many degrees of freedom, in a model-free manner. For a mayonnaise-like dense emulsion, analysis of the observed trajectories of oil droplets quantitatively reproduces the high-dimensional fractal geometry of the configuration path and its associated local energy minima generated using a computational model. That geometry in turn drives the droplets' complex random motion observed in real space. Our results indicate that experimental studies can elucidate whether the similar dynamics in different soft and biological materials may also be due to fractal landscape dynamics.

Machine learning control of complex nonlinear dynamics in fibre lasers - INVITED

We review our recent work on the use of genetic algorithms to control non-stationary nonlinear wave dynamics in ultrafast fibre lasers, including the generation of breathing-soliton dynamics with controlled characteristics, the disclosure of the fractal dynamics of breathers, and the generation of rogue waves with controlled intensity.

ARTIFICIAL INTELLIGENCE TO DEAL WITH THE POST COVID-19 FRACTAL DYNAMICS LINKED WITH ECONOMY

Data-based studies have always provided useful insights of the research problem, provided that correct statistical modeling and inference strategies are adopted. For bi-directional studies or longitudinal datasets, it is always difficult to analyze the dependence of variables and their impact on the key features. With the advancement in the fields of data science and the applied mathematical modeling, these difficulties are well resolved. The machine learning algorithms can help to streamline the data-based studies in a robust manner. During this research, the COVID-19 data are analyzed with the aid of machine learning classification tools to identify the predictors, directly influenced by the pandemic. The impact of COVID-19 on the world’s economy can be better interpreted with the aid of data-based research. The data linked to the unemployment rates, during the frequent waves of COVID-19, are extracted from different sources and analyzed during this research for better forecasting measures. The nonlinear dynamical analysis can be visualized with the aid of the 2D fractal pattern generation approach. Thus the current research is an attempt to connect the outcomes of the classification analysis to the 2D fractal generation for better visual interpretation of lengthy large datasets.

Novel analytical approach to modified fractal gas dynamics model with the variable coefficients

AbstractIn this study, the modified fractal gas dynamics model (GDM) with variable coefficients is successfully represented using the fractal derivative. We obtain the fractal variational principle of the modified fractal GDM by employing the fractal semi‐inverse method. Based on the established fractal variational principle, a new and fascinating algorithm is presented to solve the fractal model, which is called fractal two‐scale variational method (FTSVM). Finally, two numerical examples are given to indicate the efficiency and accuracy of the proposed algorithm. The FTSVM sheds a new light on the fractal differential equations.

Analysis and modeling of fractal evolutionary dynamics of industrial systems

The industry is the key to economic development. This paper proposed a fractal evolutionary dynamic model of industry based on the propagation dynamics theory from the perspective of fractal theory. The model quantitatively portrays the evolutionary trend of industries and provides reasonable solutions to the problems faced by industrial structure optimization. First, the fractal characteristics of the industrial system are analyzed, and the proposed model is used to study the evolution of industry and enterprise impact values at different granularities. Subsequently, the impact of micro firms on the macro industrial development was further investigated. Finally, the correctness of the model simulation results was verified by real data. The results showed that the evolution of industries under different granularities shows self-similarity and that micro industries with small and medium-sized enterprises (SMEs) were more influential in the development of industries. Finally, the proposed fractal evolutionary dynamics model of the industry has proven to be reasonable and effective through empirical comparison. This study provided new insights for solving industrial development problems with high theoreticality and reliability.

Marks-Tarlow, Terry, & Shapiro, Yakov. (2021). 'Synchronicity, acausal connection, and the fractal dynamics of clinical practice'. Psychoanalytic Dialogues, 31, 4, 468-86. https://doi.org/10.1080/10481885.2021.1925283. Harris, Adrienne. (2021). 'Discussion: "Synchronicity, acausal connection, and the fractal dynamics of clinical practice"'. Psychoanalytic Dialogues, 31, 4, 487-92. https://doi.org/10.1080/10481885.2021.1926789. Cartwright, Duncan. (2021). 'Can we sing potatoes? Primary substrates, synchronicity, and fractal dynamics in clinical practice'. Psychoanalytic Dialogues,

Marks-Tarlow, Terry, & Shapiro, Yakov. (2021). 'Synchronicity, acausal connection, and the fractal dynamics of clinical practice'. Psychoanalytic Dialogues, 31, 4, 468-86. https://doi.org/10.1080/10481885.2021.1925283. Harris, Adrienne. (2021). 'Discussion: "Synchronicity, acausal connection, and the fractal dynamics of clinical practice"'. Psychoanalytic Dialogues, 31, 4, 487-92. https://doi.org/10.1080/10481885.2021.1926789. Cartwright, Duncan. (2021). 'Can we sing potatoes? Primary substrates, synchronicity, and fractal dynamics in clinical practice'. Psychoanalytic Dialogues,

Measuring fractal dynamics of FECG signals to determine the complexity of fetal heart rate

In this research, we study the fetal heart rate from abdominal signals using multi-fractal spectra and fractal analysis. We use the Abdominal and Direct Fetal Electrocardiogram Database contains multichannel fetal electrocardiogram (FECG) recordings obtained from 5 different women in labor, between 38 and 41 weeks of gestation. We apply autocorrelation or power spectral densities (PSD) analysis on these five FECG recordings to estimate the exponent from realizations of these processes and to find out if the signal of interest exhibits a power-law PSD. We perform multi-fractal analysis to discover whether some type of power-law scaling exists for various statistical moments at different scales of these FECG signals. We plot the multi-fractal spectra of this database to compare the width of the scaling exponent for each spectrum. A quantitative analysis commonly known as the Fractal Dimension (FD) using the Higuchi algorithm has been carried out to illustrate the fractal complexity of input signals. Our finding shows that the fractal geometry can be used as a mathematical model and computational framework to further analysis and classification of clinical database. Moreover, it can be considered as a framework to compare the complexity of FECG signals and a useful tool to differentiate between their patterns.

A brief note on fractal dynamics of fractional Mandelbrot sets

This paper preliminary examines a kind of Mandelbrot set generated by a fractional difference quadratic map involving Caputo-like fractional h-difference operators. A connectivity index is proposed based on numerical methods, which avoids difficulties in discussion at the topological level. The dynamics of those sets in two kinds of noise environments are considered involving connectivity, symmetry and dimension. Several typical cases are visualized to illustrate the main conclusions.

Fractal Operators and Fractional Dynamics with 1/2 Order in Biological Systems

This paper reports the new advances in biological fractal dynamics. The following contents are included: (1) physical (or functional) fractal spaces abstracted from biological materials, biological structures and biological motions; (2) fractal operators on fractal spaces; (3) 1/2-order fractional dynamics controlled by fractal operators; and (4) the origin of 1/2-order. Based on the new progress, we can make a judgment that all the two-bifurcation physical functional fractal motions in the living body can be attributed to the fractional dynamics with 1/2-order.

Fractal-like random pore model applied to CO2 capture by CaO sorbent

• The application of fractal dynamics concepts to the random pore model is proposed. • The case-study is CO 2 capture by CaO carbonation in fluidised bed calcium looping. • Results were discussed as a function of the carbonator operating conditions. • The building-up of a solid product layer calls for a time-decreasing diffusivity. • A correlation between carbonation degree and product layer diffusivity is proposed. CO 2 capture from industrial flue gas by CaO in fluidised bed calcium looping processes is one of the means to reduce anthropogenic greenhouse gas emissions. The heterogeneous carbonation reaction has been widely investigated in literature, but in almost all cases the product layer (CaCO 3 -rich) CO 2 diffusivity has been considered as a constant parameter. In this work, the application of fractal-like dynamics concepts to the present case has led to the formulation of an expression for the product layer diffusivity that is function of the carbonation time. This expression has been used in the canonical random pore model, to propose an original fractal model, whose superior accuracy has been demonstrated by validation with experimental data obtained under realistic operating conditions (dual interconnected fluidised beds system, use of a CaCO 3 -rich limestone sorbent, carbonation carried out at 650 °C treating a flue gas with 15% CO 2 and in absence/presence of steam - 10% when present - and SO 2 - 1500 ppmv when present-). Main model parameters were the following: fractal heterogeneity parameter, average value for product layer diffusivity, characteristic time for CO 2 diffusion into the sorbent pores, intrinsic kinetic constant, average value for a modified Biot number. Their values were discussed in the light of the reaction mechanism, and of the effect of the absence/presence of steam and SO 2 in the flue gas to be treated. It has been observed that the presence of steam in the carbonator favours both the intrinsic carbonation kinetics and the CO 2 product layer diffusion, while the presence of SO 2 limits the product layer diffusion.