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The fractal and chaos are bound tightly, and their relevant researches are well-established. Few of them, however, concentrates on the research of the possibility of combining the fractal and the chaotic systems to generate multi-scroll chaotic attractors. This paper presents a novel non-equilibrium point chaotic system, exhibiting extremely rich and complex hidden behaviors including chaos, hyper-chaos, multi-scroll attractors, extreme multi-stability and initial offset-boosting. The proposed system is combined with fractal transformation respectively, and a new class of multi-scroll attractors, such as multi-ring attractors and separated-scroll attractors, is observed. Particularly, swallow-shaped attractors for the first time is found. Moreover, another efficient method to generate a different class of chaotic attractors uses parabola transformation and triangle transformation. Additionally, the spectrum entropy ( SE ) complexity is employed to discuss the complexity of the proposed system before and after fractal, resulting in a chaotic sequences with fractal transformation that has higher complexity. Finally, we develop a hardware platform to implement the presented attractors before and after fractal in a way to confirm the accuracy of the numerical simulations, providing a theoretical basis for the next application in image encryption.

The environment around us naturally represents number of its components in fractal structures. Some fractal patterns are also artificially simulated using real life mathematical systems. In this paper, we use the fractal operator combined to the fractional operator with both exponential and Mittag-leffler laws to analyze and solve generalized three-dimensional systems related to real life phenomena. Numerical solutions are provided in each case and applications to some related systems are given. Numerical simulations show the existence of the models' initial three-dimensional structure followed by its self- replication in fractal structure mathematically produced. The whole dynamics are also impacted by the fractional part of the operator as the derivative order changes.

The concept of dimension is an important task in geometry. It permits a description of the growth process of objects. It may be seen as an invariant measure characterizing the object. Fractal dimensions are a kind of invariants permitting essentially to describe the irregularity hidden in irregular objects, by providing a suitable growth law. Among fractal geometrical objects, Moran’s types play an important role in explaining many situations, in pure mathematics as the general context of Cantor’s, and in applied physics as a suitable context for studying scaling laws. In the present paper, some non-regular homogeneous Moran measures are investigated, by establishing some new sufficient conditions permitting an explicit computation of the relative multifractal dimensions of the level sets for which the classical formulation does not hold. Besides, the mutual singularity of the relative multifractal measures for the homogeneous Moran case with different multifractal dimensions is investigated. This is very important, as in quasi-all existing situations, the validity of the multifractal formalism passes through the equality of the multifractal Hausdorff dimension with the packing one.

A general fractional calculus is described using fractional operators with respect to another function, and some often used propositions are presented in this framework. Together with the continuous time random walk (CTRW), a general time-fractional Fokker–Planck equation is derived and the governing equation meets the general fractional derivative. Finally, various new probability density functions are proposed in this paper.

Структурата на различни системи, целящи снабдяването на обем крайно вещество с определен флуид, каквато е например кръвта в съдовата система, се подчиняват на сходни закони, които могат да се изразят със съответните математически уравнения. Тези системи имат фрактална структура, т.е. всяка малка част на системата повтаря устройството на цялата система. Познаването на тези зависимости могат да ни позволят да изчислим определен параметър от сърдечно-съдовата система, например големината на кръвоснабдявания миокард, познавайки диаметъра на кръвоснабдяващата го коронарна артерия. Този подход е изключително подходящ при болните с коронарна болест на сърцето, където обемът на исхемичния миокард е от особено голямо значение за съдбата на болния. Целта на настоящия обзор е да представи основните взаимовръзки между показателите от анатомията и физиологията на сърдечно-съдовата система. The structure of different systems, aiming to supply a volume of definite tissue with a specific fluid, which is for example the blood in the vascular system, obeys on similar lows, which can be expressed by mathematical equations. Those systems have fractal structure, that means every small part of the system repeats the structure of entire system. Knowing those dependencies permits calculation of one particular parameter of cardiovascular system, for example, the amount of myocardium on the basis of diameter of coronary artery supplying it. This approach is extremely applicable to the patients with coronary artery disease, where the amount of ischemic myocardium is of paramount importance for the patient`s fate. The aim of the current review is to present the main interdependencies between anatomical and physiological parameters of cardiovascular system.

The degree of ordering of the structure of technologically important materials formed as a result of the evolution of complex physicochemical systems determines their physical properties, in particular optical. In this regard, the primary task for the theoretical study of methods for obtaining materials with predetermined physical properties is to develop approaches to describe the evolution of fractal (scale-invariant) objects in the formation of self-similar structures in systems exhibiting chaotic behavior. The paper forms an idea of the processes of evolution in materials formed as a result of stochastic processes. It is established that the conduct of ultrametrics in time space allows to characterize the time of the evolutionary process of fractal dimension, which is calculated either theoretically or model. The description of evolutionary processes in a condensed medium, accompanied by topological transformations, is significantly supplemented by the method of describing the stages of evolution of structures, which makes it possible to analyze a wide range of materials and can control their properties, primarily optical. It is shown that the most large-scale invariant structures, due to the investigated properties, can be used as information carriers. It is demonstrated that the presence in physical systems of fractal temporal dimension and generates a self-similar (consisting of parts in a sense similar to the whole object) evolutionary tree, which, in turn, generates spatial objects of non-integer dimension, observed in real situations. On the other hand, temporal fractality provides analysis of systems with dynamic chaos, leading to universal relaxation functions. In particular, in systems with a large-scale invariant distribution of relaxation characteristics, an algebraic law of relaxation is manifested, which leads to rheological models and equations of states, which are characterized by fractional derivatives. It is argued that the fractal dimension of time hierarchies stores information that determines the process of self-organization. Developed in the paper ideas about the processes of building the structure of materials, which lead to the fractal geometry of objects, can be used to predict their properties, in particular, optical.

The discovery of a simple numerical formula for the projection of all the atomic mass of life-sustaining CONHSP bioatoms leads to the emergence of a set of Nested CODES unifying all the biological, genetic and genomic components by unifying them from bioatoms up to 'to whole genomes. In particular, we demonstrate the existence of a digital meta-code common to the three languages ​​of biology that are RNA, DNA and amino acid sequences. Through this meta-code, genomic and proteomic images appear almost analogous and correlated. The analysis of the textures of these images then reveals a binary code as well as an undulatory code whose analysis on the human genome makes it possible to predict the alternating bands constituting the cariotypes of the chromosomes. The application of these codes to perspectives in astrobiology and the emergence of binary codes and regions of local stability (voting process), whose fractal nature we demonstrate, is illustrated. The fundamental discovery described here will undoubtedly one day constitute a new biomathematic approach to the emergence of living things.

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.

This paper reports a cellular automata (CA) study of the transient dynamic responses of anti-plane shear Lamb’s problems on random fields (RFs) with fractal and Hurst effects. Both Cauchy and Dagum random field models are employed to capture the combined effects of spatial randomness in both mass density and stiffness tensor fields. First, with a dyadic representation, we formulate a second-rank anti-plane stiffness tensor random field (TRF) model with full anisotropy. Its statistical, fractal, and Hurst properties are investigated, leading to introduction of a so-called MOSP model of TRF. Then, we generalize the CA approach to incorporate the inhomogeneity in mass density as well as stiffness fields. Through parametric studies for both Cauchy and Dagum TRFs, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. In general, the mean response amplitude is lowered by the presence of randomness, and the Hurst parameter (especially, for $$\beta < 0.5$$ ) is found to have a stronger influence than the fractal dimension on the response. The results are compared with two simpler random fields: (1) randomness is present only in the mass density field; (2) randomness is present in the mass density field and in a locally isotropic stiffness tensor field. Overall, the results show that a second-rank anti-plane stiffness TRF with full anisotropy leads to the strongest fluctuation in displacement responses followed by a locally isotropic RF model.

Cities are spatially evolving complex systems. The order and pattern beneath the apparent chaos and diverse physical forms of cities are still unclear. How the form and structure of a city evolve t...

This work addresses the role of rigid inclusions in determining the elastic modulus of particle-filled colloidal networks by modifying an established fractal scaling model. The approach acknowledges the heterogeneous nature of stress distribution at length scales beyond the colloidal aggregates, while maintaining structural information at the level of individual clusters. This was achieved by introducing a scaling factor to account for system heterogeneity which contains intrinsic information about the network’s capacity to form load-bearing links. Rigid fillers bound to the network induce stress concentration, but additionally serve as junction zones which introduce additional load-bearing pathways. This gives rise to the observed increase in the modulus with filler volume fraction. The proposed relationship between the load-bearing network connectivity and scaling behavior may have additional implications on the fractal dimension determined by rheological methods. Further, this model accommodates an experimentally observed correlation between the scaling behavior of the modulus associated with the addition of fillers and that arising from increasing structurant concentration. The modified fractal model thus provides an alternative view of how fillers contribute to the small- and large-deformation mechanical behavior of filled colloidal gels in a manner consistent with experimental observations.

The representative volume element (RVE) and/or the representative elementary volume (REV) and their existence and determination are a basic scientific issue in the research field of size effect. The similarities and differences between the RVE and REV are analyzed from the fields of composite micromechanics and geotechnical engineering. A three-level classification of the REV research on geomaterials is then proposed based on these fields, especially REV research in geotechnical engineering. The pore size distribution and the corresponding fractal dimension are introduced and selected as the constrained parameters for describing the complex pore structure of geomaterials. Accordingly, an approximate criterion is first presented via theoretical deduction and comparison with previous research results, through which the size of a P-REV of mesoscale porous geomaterials can be determined using their pore size distributions (i.e., the edge length of the cubic P-REV is approximately equal to five times of the maximum pore diameter). Finally, some statistical methods, such as low-order probability functions, are utilized to validate the performance of the proposed criterion. The verification results using two types of geomaterials, namely, sandstone and foamed concrete, reveal that both are in a reasonable agreement, indicating that the P-REV determined according to this approximate criterion can provide representative statistical results and find a balance between characterization accuracy and calculation efficiency.

The excluded volume effect is added to a fractional viscoelastic model for modeling fractal polymers. This reveals a physical connection between the fractional time derivative, fractal geometry, and excluded volume effect. This derivation is a general theoretical framework based on the Scott-Blair fractional model of viscoelasticity when the excluded volume and the hydrodynamic interaction are explicitly taken into account to derive the microscopic stress within the molecular theory of Rouse and Zimm. The methodology extends the generalized molecular theory of Zimm by adding the effect of excluded volume where the new relaxation formulation contains internal state variables that naturally depend on the fractional time derivative of deformation. The modified distribution of the end-to-end vector of a monomer contained within a polymer network is used for pre-averaging approximations of the mobility matrix in the Zimm model. The pre-averaging approximation is important since the mobility matrix is a nonlinear function and it is difficult to explicitly calculate. Through application of thermodynamic laws, we derive the linear fractional model of viscoelasticity based on its spectral dimension, fractal dimension, and the excluded volume parameter for fractal media. This derivation shows how the order of the fractional derivative in the linear fractional model of viscoelasticity is strongly correlated with fractal structure and excluded volume effects.

Atmosphere and ocean dynamics display many complex features and are characterized by a wide variety of processes and couplings across different timescales. Here we demonstrate the application of multivariate empirical mode decomposition (MEMD) to investigate the multivariate and multiscale properties of a reduced order model of the ocean–atmosphere coupled dynamics. MEMD provides a decomposition of the original multivariate time series into a series of oscillating patterns with time-dependent amplitude and phase by exploiting the local features of the data and without any a priori assumptions on the decomposition basis. Moreover, each oscillating pattern, usually named multivariate intrinsic mode function (MIMF), represents a local source of information that can be used to explore the behavior of fractal features at different scales by defining a sort of multiscale and multivariate generalized fractal dimensions. With these two complementary approaches, we show that the ocean–atmosphere dynamics presents a rich variety of features, with different multifractal properties for the ocean and the atmosphere at different timescales. For weak ocean–atmosphere coupling, the resulting dimensions of the two model components are very different, while for strong coupling for which coupled modes develop, the scaling properties are more similar especially at longer timescales. The latter result reflects the presence of a coherent coupled dynamics. Finally, we also compare our model results with those obtained from reanalysis data demonstrating that the latter exhibit a similar qualitative behavior in terms of multiscale dimensions and the existence of a scale dependency of the statistics of the phase-space density of points for different regions, which is related to the different drivers and processes occurring at different timescales in the coupled atmosphere–ocean system. Our approach can therefore be used to diagnose the strength of coupling in real applications.

It is shown that the classical equation describing the Yukawa diagram cannot satisfactorily describe a system of charged grains in a dusty plasma in a wide range of the structure factor values. A new approach to the diagrammatic analysis of Yukawa systems is proposed. Equations are obtained for the analytical description of the Yukawa diagram in a wide range of the structure factor values in the framework of the thermodynamic theory of similarity (scaling). Power-law dependences of the coupling parameter on the parameters of the structure are obtained. The relationships between the exponential functions describing Yukawa diagrams and the fractal dimension of the plasma system are revealed. The possibility of describing large-scale structures of dust particles (grains) such as Saturn’s E-ring using the same functions, but with different values of the fractal dimension, is discussed.

In the article we invoke a new treatment of the non-traditional elasticity with use of the fractal scaling-law vector calculus. The fundamental theorems for the fractal scaling-law vector calculus are given in detail. The result presents a new insight into the descriptions for the behaviors of the solids in the Mandelbrots scaling-law phenomena.

Present study is aimed at investigating the solar faculae area from 1990 to 2007 which partially covered the22nd and 23rd solar cycle. Rescaled Range Analysis (RRA) and Detrended Fluctuation Analysis (DFA) have beenadopted to evaluate the behaviour of nonlinear dynamics of solar faculae area. Results show that the value of Hurstexponent for solar faculae area from RRA and DFA is negatively correlated. It means it is non-persistent and longrange correlated. Obtained result is inaccurate so the only solution is to transform the data into stationary data by takingdifferencing. RRA is applied on residuals and RRA to evaluate the fractal property of the time series. Solar faculae areainvestigated in this study is fractal in nature and predictable ass well. Moreover, the time series of solar faculae area isnon-linear as established by the Brock – Dechert – Scheinkman (BDS) test results

In this study, we have used the concept of product-like fractal measure to analyze the fractal heat transfer in anisotropic media. This concept was introduced by Li and Ostoja-Starzewski in order t...

Surface roughness of shales has a key influence on the petroleum resources they are able to store and the fraction of them that can be recovered. The fractal dimension quantifies the degree of roughness and is influenced primarily by the pore surfaces within the shale that typically include micro-, meso-and macro-pores. Isotherms generated by gas adsorption experiments are the common data source used to derive estimates of fractal dimension. The Frenkel-Halsey-Hill fractal technique is the most widely applied fractal dimension estimation method. Other methods can derive fractal dimension from isotherm data but typically the values they generate are different from the Frenkel-Halsey-Hill derived fractal dimension values. Moreover, those differences can vary significantly depending on the type of shales involved. Those shales displaying more complex pore-scale distributions including extensive micro-porosity components tend to be associated with the greatest discrepancies. A comparison of three fractal dimension calculation methods applied to shales reveals aspects of their calculation and interpretation methods that explain the differences in the fractal dimension values they generate. This study identifies the uncertainties that should be taken into account when applying the methods and the appropriate curve fitting optimization configurations that should be evaluated. Taking these factors into account leads to more realistic selections of appropriate fractal dimension values from gas adsorption isotherms of organic-rich shales. Cited as : Wood, D. A. Techniques used to calculate shale fractal dimensions involve uncertainties and imprecisions that require more careful consideration. Advances in Geo-Energy Research, 2021, 5(2): 153-165, doi: 10.46690/ager.2021.02.05

The properties of network traffic of multiservice networks, which are observed in practice today, are quite dif-ficult to interpret using the methods of statistical analysis, characteristic of the established modes of operation of network applications. Typically, backbone traffic focused on the simultaneous maintenance of information flows is often fractal in nature. This makes it impossible to obtain traffic characteristics by conventional approaches. In particular, problems arise in determining the statistical characteristics of the second order. The article proposes an approach to the calculation of the correlation coefficient of fractal traffic, which is based on the analysis of fractal properties of network processes. The model of generalized Brownian motion is used for calculation. This uses the operation of fractional integration of Brownian motion with the kernel, which has the form of a power function with a fractional exponent, and the resulting expression shows the dependence of the power tail of the distribution on the Hearst parameter. The proposed method allows to perform an analytical assessment of the correlation coefficient, as well as to determine the degree of fractal properties of this random process.

It is widely known that there are several interdependent phenomena. The zones of influence (ZoIs) of water bodies (WBs) can be considered as typical examples. Over 289000 ZoIs are computed for the corresponding WBs mapped from remotely sensed satellite data of Indian peninsular. The size of WBs ranges between the smallest size of 5.1594 m2 to 628.1642 km2, and the range of the ZoIs is in between 416.2497 m2 to 8862.3362 km2. The basic geometric measures were computed for all WBs and their ZoIs. These measures were employed to derive power laws that are found scale-invariant, supporting the fact that these interdependent phenomena belong to two different universality classes, and explaining the commonly shared physical mechanisms. It is also inferred that power laws derived for this very large set of WBs and their ZoIs are similar to the ones derived for a smaller data set, that is, they are indeed fractals. The study shows the importance of deriving power-law relationships for natural phenomena mapped from remotely sensed satellite data across very fine to coarse spatial, spectral, and temporal resolutions.

In this paper, the structures of spatial self-organization of steels after thermos-cyclic exposure were studied, using the fractal formalism method. It is shown that the structures, observed after cyclic heat exposure, can be characterized by the presence of the invariance property of their structure at different scale levels.

Studies on entomopter’s performance under the influence of gust have received impetus in the past decade. There exists a dire need to ascertain the threshold of the frontal gusty conditions which would destabilize these anthropogenic flyers. This would help to devise methods to mitigate the detrimental effects of gust. In light of this aspect, the present study aims at analyzing the onsets of instability in a flapping wing system subjected to temporal gust by employing recurrence period density entropy (RPDE) and detrended fluctuation analysis (DFA). Simulation of the flapping wing along inclined stroke is carried out for a Reynolds number of 150. This Reynolds number lies in the typical operating regime of fruit flies and entomopters like the Pico aerial vehicle. Numerical simulations are carried out to solve the laminar, unsteady, and incompressible Navier–Stokes equations. The dynamic meshing technique is employed to model flapping kinematics. Nine gusts with a combination of frequency and velocity ratios of 0.1, 0.5, and 1.0 are considered. Instantaneous horizontal and vertical forces are estimated. Time series of these forces are analyzed using RPDE and DFA paradigms. These analyses indicate that gust frequency of an order of magnitude higher than flapping frequency and gust amplitude of the order of magnitude as the wing’s root mean square velocity induces a possible onset of instability.

In this contribution, we present for the first time, explicit expressions for time-dependent entropy production, in the classical context. Here, we pursue the understanding of the time dependence of entropy by the approach of nonlocal fractional derivatives and also with local deformed derivatives. We claim that the entropy production depends on the dynamics and the geometry of the complex system considered. Some closed expressions for the entropy growth rate were obtained. We also give a possible explanation for the generalized Pesin relations here obtained, by considering the connection between the fractal geometry of phase-space with the types of particles interactions.

Methods of digital image analysis find wide application both in scientific research and in many branches of industry. During the last decades, interest has grown in images with multifractal structure which are obtained in biology, medicine, chemistry and studying the soil. The mathematics of fractals and fractal geometry are well known and studied. However, despite this, a common approach to designing the methods of practical investigation of such images has not been developed until now. The main purpose of this work is to propose the using of multifractal formalism as the mathematical tool for the statistical description of multifractal sets. Such a description adequately depicts the chaotic behavior of the majority of real systems. The method for calculation of Rényi and singularity spectra based on using parametrized spectra, which are obtained from escort (zooming) distributions of an initial measure, is considered. The method for comparing images based on using vectors of divergences calculated for the sequence of escort distributions is proposed. The role of parametrized spectra as the tool for the approximation of any part of the singularity spectrum is substantiated. An estimation of the rate of growth of the divergence vector is obtained. Main theoretical results are confirmed by numerical experiments with images of biomedical preparations. These show the ability of the implemented methods to find subtle distinctions in image structure for a simple choice of an initial measure.

Identifying urban boundaries involves analysing both the functional and the morphological aspects of urban systems. In this paper, we adopt a purely morphological approach and compare three methods...

The allometric scaling laws of metabolism in 447 animal and 1200 plant species showed convex and concave curvatures between mass and metabolic rate, respectively. The objective of the study is to explain the difference of curvatures between animals and plants based on fractal models. Several intraspecific scaling laws were derived from an asymmetric vascular tree with the fractal dimension (i.e., a in $${k}_{1}^{a}+{k}_{2}^{a}+\dots =1$$ , where $${k}_{i}$$ refers to the ratio of daughter to mother diameters). Based on the intraspecific scaling laws, the allometric scaling exponent of metabolism (i.e., an interspecific scaling law) was shown to be equal to one-third of fractal dimension. Moreover, a novel piecewise-defined function in conjunction with the intraspecific scaling laws was proposed to explain the diverse metabolic scaling in animals and plants. The intraspecific and interspecific scaling laws showed good agreement with morphometric measurements. The experimentally-validated scaling models predict the diversity of intraspecific and interspecific scaling seen in nature. To our knowledge, this is the first study to use fractal models to explain the convex and concave forms of metabolic scaling in animals and plants. The study resolves the long-term controversies to use the resource-transport network models for explanation of the allometric scaling law of metabolism. This study validated intraspecific scaling laws of asymmetric vascular trees instead of idealized symmetric trees. The fractal dimension was found to greatly affect the scaling exponents. Based on the intraspecific scaling laws, the allometric scaling exponent of metabolism was shown to be equal to one-third of fractal dimension. The piecewise-defined asymmetric vascular tree was further proposed to accurately explain convex and concave curvatures in metabolic scaling of animals and plants, respectively.

Background: Allometric scaling is a well-known research tool used for the metabolic rates of organisms. It measures the living systems with fractal physiology. The metabolic rate versus the mass of the living species has a definite scaling and behaves like a four-dimensional phenomenon. The extended investigations focus on the mass-dependence of the various physiological parameters. Objective: Proving the length of vascularization is the scaling parameter instead of mass in allometric relation. Method: The description of the energy balance of the ontogenic growth of the tumor is an extended cell-death parameter for studying the mass balance at the cellular level. Results: It is shown that when a malignant cellular cluster tries to maximize its metabolic rate, it changes its allometric scaling exponent. A growth description could follow the heterogenic development of the tumor. The mass in the allometric scaling could be replaced by the average length of the circulatory system in each case. Conclusion: According to this concept, the dependence of the mass in allometric scaling is replaced with a more fundamental parameter, the length character of the circulatory system. The introduced scaling parameter has primary importance in cancer development, where the elongation of the circulatory length by angiogenesis is in significant demand.

Fractal calculus is an excellent tool to explaining natural phenomena in porous media. In this paper, we first give a simple introduction on He?s fractal derivative, and then it is used to establish a model for thermal conduction of silkworm cocoon reveal its biomechanism. The theoretical results obtained in this paper are helpful for the biomimetic design.

This work analyzes the $$SePaC$$ method of fractals and various background events (mixed, random, and distributed according to Gauss’s law, as well as by power and exponential laws) in order to search and reconstruct fractal structures. It is shown that some of the background events are reconstucted as fractals. Dimensional distributions of various events are considered. It is found that these distributions for fractals and background events are different. Characteristics are proposed that describe the features of fractal structures. They are compared for fractals and background events. Criteria for background suppression are proposed and the results of their application are presented. It is found that the application of the criteria changes the distribution of background events by dimension, and the use of a combination of criteria suppresses the background.

In this article, complex random polar media (CRPM) are synthesized by a forward numerical modeling based on the fractal statistical properties of the marginal ice zone (MIZ) in terms of its energy balance under sea state conditions. The proposed modeling was carried out by the parallel processing of a modified JONSWAP directional spectrum function in a two-dimensional hybrid domain. As a novel approach, the scattering formulation has been modified in the form of a two-scale implementation, which is optimized by the product of cosine-power and Gamma probability distribution function (PDF). As a result, a composite model of CRPM under different sea state conditions, including wind speed and direction, has been made possible. Due to the high dynamics of CRPM under sea state conditions, the presence of high entropy samples in the form of random roughness fluctuations is inevitable. These highly textured areas not only reduce the ability to resolve fine details but also make the interpretation difficult from a remote sensing point of view. According to the importance of synthetic aperture radar in the observation of MIZ and analysis of scattering properties, the spectral distribution modeling of these roughness anomalies and their electromagnetic interactions, along with surface tension synthesis, are presented here. Meanwhile, several objective quality metrics under different sea state conditions have been derived. The results show that the sea state not only changes the spectral components, but also affects their contribution to the roughness fluctuations and texture compositions. In short, this is for the first time that the composite modeling of CRPM based on directional spectrum function under sea state conditions along with electromagnetic interactions investigation and its pertinent texture analysis has been done.

A first report of unveiling the fractality and fractal nature of severe acute respiratory syndrome coronavirus (SARS CoV-2) responsible for the pandemic disease widely known as coronavirus disease 2019 (COVID 19) is presented. The fractal analysis of the electron microscopic and atomic force microscopic images of 40 coronaviruses (CoV), by the normal and differential box-counting method, reveals its fractal structure. The generalised dimension indicates the multifractal nature of the CoV. The higher value of fractal dimension and lower value of Hurst exponent (H) suggest higher complexity and greater roughness. The statistical analysis of generalised dimension and H is understood through the notched box plot. The study on CoV clusters also confirms its fractal nature. The scale-invariant value of the box-counting fractal dimension of CoV yields a value of 1.820. The study opens the possibility of exploring the potential of fractal analysis in the medical diagnosis of SARS CoV-2.

The phase structure of synthetic isoprene rubber samples subjected to stretching was studied by transmission electron microscopy. A procedure for determining the fractal dimension of extended one-dimensional structures arising during deformation was proposed. Image parametrization, being the extraction of substantial acting parameters, their description, and quantitative estimation of the obtained parameters, made it possible to characterize each stage of the deformation. A combination of standard methods of studying electron microscopic images using the fractal theory to describe the supramolecular and phase structures of elastomers allows one to understand the mechanism of the deformation process and the nature of the structures arising in structure rearrangement.

In this work, combined experimental and fractal nature analysis procedures are proposed in order to both model and design mechanical properties of porous ceramics. Several porous ceramics samples have been considered both from an in-situ experimental campaign and from the literature. Microstructural information concerning pore size distribution has been approximated by the Intermingled Fractal units (IFU) approach and effective mechanical properties are derived by a simple discrete model. The capability of the proposed methodology to reproduce high-scattered mechanical properties is fully shown and a comparison with classical bounds and estimates is also reported. Finally, the combined experimental, fractal nature analysis and homogenisation scheme is implemented as a design procedure for the technological production of advanced porous ceramics.

Concepts of formation and propagation of radiation with a fractal amplitude–phase profile are analyzed and systematized on a single methodological basis using main points of fractal physics. Motion of a plane wave through fractal phase and amplitude screens is considered. Characteristics of beams with a stochastic and a regular self-similar structure are discussed. A range of questions on direct formation of waves with a fractal amplitude and phase distribution in laser systems is examined. The analysis shows that scaling characteristics complement the space–time parameters traditionally used for describing light beams. In particular, mutually complementing are statistical and fractal data on propagation of beams in the turbulent atmosphere. To date, the mathematical and physical apparatus has been developed that allows self-similar and scaling properties of radiation to be adequately described. With this apparatus, it is possible to develop efficient methods for optical diagnostics of fractal structures using probing light beams. A high degree of stability of Fourier images of fractals is an important factor for increasing stability and noise immunity of optical information transmission channels based on fractal beams.

The paper aims to obtain the cushion curves of a polymer foam commonly used in the protective packaging of fragile products and its fractal cushioning performance. First, the impact curves are obtained to evaluate the dropping damage of a critical component considering a fractal nonlinear dynamic packaging system model, more specifically a fractal cubic-quintic-heptic Duffing’s equation. Then, the two-scale fractal derivative transform is introduced to determine the packaging material fractal behavior due to the material molecular structure. The findings explain the dynamic response behavior of the porous cushion material, and the proposed fractal dynamic model sheds new light on the design and analysis of cushioning packaging systems.

In dynamical systems, basins of attraction connect a given set of initial conditions in phase space to their asymptotic states. The basin entropy and related tools quantify the unpredictability in the final state of a system when there is an initial perturbation or uncertainty in the initial state. Based on the basin entropy, the ln 2 criterion allows for efficient testing of fractal basin boundaries at a fixed resolution. Here, we extend this criterion into a new test with improved sensitivity that we call the S b b fractality test. Using the same single scale information, the S b b fractality test allows for the detection of fractal boundaries in many more cases than the ln 2 criterion. The new test is illustrated with the paradigmatic driven Duffing oscillator, and the results are compared with the classical approach given by the uncertainty exponent. We believe that this work can prove particularly useful to study both high-dimensional systems and experimental basins of attraction.

Why fractal distribution is so frequent? It is true that fractal dimension is always less than 3? Why fractal dimension of 2.5 to 2.9 seems to be steady-state or stable? Why the fractal distributions are the limit distributions of the degradation path? Is there an ultimate distribution? It is shown that the finite fractal grain size distributions occurring in the nature are identical to the optimal grading curves of the grading entropy theory and, the fractal dimension n varies between –¥ and ¥. It is shown that the fractal dimensions 2.2–2.9 may be situated in the transitional stability zone, verifying the internal stability criterion of the grading entropy theory. Micro computed tomography (μCT) images and DEM (distinct element method) studies are presented to show the link between stable microstructure and internal stability. On the other hand, it is shown that the optimal grading curves are mean position grading curves that can be used to represent all possible grading curves.