Electrostatics in fractal geometry: Fractional calculus approach

An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume V_d ～ r^d with the fractal dimension d and a complementary host volume V_h = V₃ ? V_d. Integrations over these fractal volumes correspond to the convolution integrals that eventually lead to the employment of the fractional integro-differentiation.     

A set of formulae on fractal dimension relations and its application to urban form

The area-perimeter scaling can be employed to evaluate the fractal dimension of urban boundaries. However, the formula in common use seems to be not correct. By means of mathematical method, a new formula of calculating the boundary dimension of cities is derived from the idea of box-counting measurement and the principle of dimensional consistency in this paper. Thus, several practical results are obtained as follows. First, I derive the hyperbolic relation between the boundary dimension and form dimension of cities. Using the relation, we can estimate the form dimension through the boundary dimension and vice versa. Second, I derive the proper scales of fractal dimension: the form dimension comes between 1.5 and 2, and the boundary dimension comes between 1 and 1.5. Third, I derive three form dimension values with special geometric meanings. The first is 4/3, the second is 3/2, and the third is 1 + 2^1/2/2 ≈ 1.7071. The fractal dimension relation formulae are applied to China’s cities and the cities of the United Kingdom, and the computations are consistent with the theoretical expectation. The formulae are useful in the fractal dimension estimation of urban form, and the findings about the fractal parameters are revealing for future city planning and the spatial optimization of cities.     

Surface investigation by electrochemical methods and application of chaos theory and fractal geometry

The present study has considered the application of the noise analysis and fractal geometry as a promising dynamic method for exploiting the corrosion mechanism of the stainless steel 304 that is immersed in different concentrations of FeCl₃. The fractal dimension calculated from the electrochemical noise technique has a good correlation with the surface fractal dimension obtained by electrochemical impedance spectroscopy and scanning electron microscopy results. The complexity of system increases by divergence of Electrochemical Potential noise fractal dimension from 1.5 value and also the roughness of surface increases by an increase in surface fractal dimension. As the concentration of FeCl₃ increases (0.001 M, 0.01 M and 0.1 M) the value of Electrochemical Potential noise fractal dimension diverges from 1.5 value (1.57, 1.33 and 1.01 respectively) and the value of surface fractal dimension increases (2.107, 2.425 and 2.756 for impedance results and 2.073, 2.425 and 2.672 for scanning electron microscopy images). These results show that the complexity of system and roughness of the surface increases by an increase in concentration of FeCl₃. The present study has shown that chaos and noise analysis are effective methods for the study of the mechanism of the corrosion process.     

Calculation of a static potential created by plane fractal cluster

In this paper we demonstrate new approach that can help in calculation of electrostatic potential of a fractal (self-similar) cluster that is created by a system of charged particles. For this purpose we used the simplified model of a plane dendrite cluster [1] that is generated by a system of the concentric charged rings located in some horizontal plane (see Fig. 2). The radiuses and charges of the system of concentric rings satisfy correspondingly to relationships: r_n = r₀ξⁿ and e_n = e₀bⁿ, where n determines the number of a current ring. The self-similar structure of the system considered allows to reduce the problem to consideration of the functional equation that similar to the conventional scaling equation. Its solution represents itself the sum of power-low terms of integer order and non-integer power-law term multiplied to a log-periodic function [5], [6]. The appearance of this term was confirmed numerically for internal region of the self-similar cluster (r₀ ≪ r ≪ r_N−1), where r₀, r_N−1 determine the smallest and the largest radiuses of the limiting rings correspondingly. The results were obtained for homogeneously (b > 0) and heterogeneously (b < 0) charged rings. We expect that this approach allows to consider more complex self-similar structures with different geometries of charge distributions.     

The concept of fractal experiments: New possibilities in quantitative description of quasi-reproducible measurements

In this paper the authors suggest a new conception of the so-called fractal (self-similar) experiment. Under the fractal experiment (FE) one can imply a cycle of measurements that are subjected by the scaling transformations F(z) → F(zξ^m) in contrast with conventional scheme F(z) → F(z + mT) (m = 0,1,…, M–1), where z defines the controllable (input) variable and can be associated with time, complex frequency, wavelength and etc., T – mean period of time between successive measurements and m defines a number of successive measurements. One can connect a fractal experiment with specific memory effect that arises between successive measurements. The general theory of experiment for quasi-periodic measurements proposed in [1] after some transformations can be applied for the set of the FE, as well. But attentive analysis shown in this paper allows generalizing the previous results for the case when the influence of uncontrollable factors becomes significant. The theory developed for this case allows to consider more real cases when the influence of dynamic (unstable) processes taking place during the cycle of measurements corresponding to some FE is becoming essential. These experiments we define as quasi-reproducible (QR) fractal experiments.The proposed concept opens new possibilities in theory of measurements and numerous applications, especially in different nanotechnologies, when the influence of the scaling factor plays the essential role. This concept allows also to introduce the so-called intermediate model (IM) which can serve as an unified platform for reconciliation of the proposed microscopic theory with reliable experiments “refined” from the influence of the random noise and apparatus function. We forced to consider a modified model experiment in order to demonstrate some common peculiarities that can be appeared in real cases. We know only couple of similar examples of experiments that are close to the proposed concept. Mechanical relaxation and dielectric spectroscopy (based on measurements of the complex susceptibility ε(jω)) represent the branches of physics related to consideration of mechanical and electric relaxation phenomena in different heterogeneous materials. The dielectric spectroscopy can be considered as an instructive example for better understanding of the proposed concept.In cases, when the microscopic model is absent the results of measurements can be expressed in terms of the fitting parameters associated with the generalized Prony spectrum (GPS) belonging to the IM. The authors do hope that this new approach will find an interesting continuation in various applications of different nanotechnologies.     

The second law of thermodynamics and multifractal distribution functions: Bin counting,pair correlations,and the Kaplan–Yorke conjecture

We explore and compare numerical methods for the determination of multifractal dimensions for a doubly-thermostatted harmonic oscillator. The equations of motion are continuous and time-reversible. At equilibrium the distribution is a four-dimensional Gaussian, so that all the dimension calculations can be carried out analytically. Away from equilibrium the distribution is a surprisingly isotropic multifractal strange attractor, with the various fractal dimensionalities in the range 1 < D < 4. The attractor is relatively homogeneous, with projected two-dimensional information and correlation dimensions which are nearly independent of direction. Our data indicate that the Kaplan–Yorke conjecture (for the information dimension) fails in the full four-dimensional phase space. We also find no plausible extension of this conjecture to the projected fractal dimensions of the oscillator. The projected growth rate associated with the largest Lyapunov exponent is negative in the one-dimensional coordinate space.     

Multifractal scaling of crack images from pyroligneous acid dried sludge

Tannery effluent (sludge, wastewater) is treated by natural way. The waste sludge has been taken for two treatment process. The alkali chemicals are neutralized by pyroligneous acid which is obtained by pyrolysis process of wood. This sludge is sent out for drying. The dried sludge contains some crack pattern formation. Photographs were used to record two sludge cracking surfaces. Experiment has been performed to study the fracture pattern formation in thin film sludge. We studied changes of crack surface of a sludge by image analysis and also assessed applicability of fractal scaling to crack surfaces. The calculated crack surface dimension shows that the fracture surface exhibit fractal structure. Image size was 256 × 256 pixels. MFA (multifractal analysis) was carried out to the method of moments, i.e., the probability distribution was estimated for moments ranging from ?150 < q < 150 and the generalized dimension were calculated from the log/log slope of the probability distribution for the respective moments over box sizes. Generalized dimension D(q) were attained for this box size range, which are capable of characterizing heterogeneous spatial crack structure. Multifractal spectra analyzed two fracture surface of the image and results were indicated that the width of spectra increases due to pyroligneous acid. Multifractal method is sensitive enough to measure the fracture distribution and can be seen as a different approach for changing research of crack images of manure sludge drying.     

Dynamics of landslide model with time delay and periodic parameter perturbations

In present paper, we analyze the dynamics of a single-block model on an inclined slope with Dieterich–Ruina friction law under the variation of two new introduced parameters: time delay T_d and initial shear stress μ. It is assumed that this phenomenological model qualitatively simulates the motion along the infinite creeping slope. The introduction of time delay is proposed to mimic the memory effect of the sliding surface and it is generally considered as a function of history of sliding. On the other hand, periodic perturbation of initial shear stress emulates external triggering effect of long-distant earthquakes or some non-natural vibration source. The effects of variation of a single observed parameter, T_d or μ, as well as their co-action, are estimated for three different sliding regimes: β < 1, β = 1 and β > 1, where β stands for the ratio of long-term to short-term stress changes. The results of standard local bifurcation analysis indicate the onset of complex dynamics for very low values of time delay. On the other side, numerical approach confirms an additional complexity that was not observed by local analysis, due to the possible effect of global bifurcations. The most complex dynamics is detected for β < 1, with a complete Ruelle–Takens–Newhouse route to chaos under the variation of T_d, or the co-action of both parameters T_d and μ. These results correspond well with the previous experimental observations on clay and siltstone with low clay fraction. In the same regime, the perturbation of only a single parameter, μ, renders the oscillatory motion of the block. Within the velocity-independent regime, β = 1, the inclusion and variation of T_d generates a transition to equilibrium state, whereas the small oscillations of μ induce oscillatory motion with decreasing amplitude. The co-action of both parameters, in the same regime, causes the decrease of block’s velocity. As for β > 1, highly-frequent, limit-amplitude oscillations of initial stress give rise to oscillatory motion. Also for β > 1, in case of perturbing only the initial shear stress, with smaller amplitude, velocity of the block changes exponentially fast. If the time delay is introduced, besides the stress perturbation, within the same regime, the co-action of T_d (T_d < 0.1) and small oscillations of μ induce the onset of deterministic chaos.     

Formation patterns at the air-grain interfaces in spinning granular films at high rotation rates

Experimental and numerical studies are described in which a thin film of air-immersed grains is spun in vertical and tilted containers about their axis. At high rotation rates a steep depression appears around the axis of rotation. Interesting fractal type patterns with dimension D = 1.7 ± 0.05 are observed at the air-grain interfaces in the depression. By utilizing computer simulations, it is shown that the fractal-like patterns may be associated with a sharp deformation of the volume occupied by the particles within the depression hole due to turbulent diffusion.     

Bounds of the hyper-chaotic Lorenz–Stenflo system

To estimate the ultimate bound and positively invariant set for a dynamical system is an important but quite challenging task in general. This paper attempts to investigate the ultimate bounds and positively invariant sets of the hyper-chaotic Lorenz–Stenflo (L–S) system, which is based on the optimization method and the comparison principle. A family of ellipsoidal bounds for all the positive parameters values a, b, c, dand a cylindrical bound for a > 0, b > 1, c > 0, d > 0 are derived. Numerical results show the effectiveness and advantage of our methods.     

Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach

In the present paper, the wave propagation in one-dimensional elastic continua, characterized by nonlocal interactions modeled by fractional calculus, is investigated. Spatial derivatives of non-integer order 1 < α < 2 are involved in the governing equation, which is solved by fractional finite differences. The influence of long-range interactions is then analyzed as α varies: the resonant frequencies and the standing waves of a nonlocal bar are evaluated and the deviations from the classical (local) ones, recovered by imposing α = 2, are discussed.     

Multifractal characterization of a dental restorative composite after air-polishing

Roughness is of critical importance for the surface of dental materials. Air-polishing is a procedure commonly used on dental surfaces to remove the biofilm, however it can also damage the material surface. As a result its roughness is increased, and the possible fractal dimension, if any, may change. This study reviews atomic force microscope images of a reference dental restorative composite, treated with abrasive powders of either sodium bicarbonate or glycine, for times of 5, 10 or 30 s, and from distances of 2 or 7 mm. To fully characterize the structural complexity of surface damage, the images were analyzed according to both statistical parameters and multifractal approach. The singularity spectrum f(α) provided quantitative values data to characterize the local scale properties of 3D surface geometry at nanometer scale. The lowest roughening of the surfaces was obtained by air-polishing with glycine for 5 s, independent of the used distance. This observation of least damage and thus best treatment was confirmed by the multifractal analysis. Multifractal analysis provides quantitative information complementary to that offered by traditional surface statistical parameters, with great potential for use also in the field of examination of quality of dental material surfaces.     

Fractional dynamics of systems with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction defined by a term proportional to 1/∣n − m∣^α+1. Continuous medium equation for this system can be obtained in the so-called infrared limit when the wave number tends to zero. We construct a transform operator that maps the system of large number of ordinary differential equations of motion of the particles into a partial differential equation with the Riesz fractional derivative of order α, when 0 < α < 2. Few models of coupled oscillators are considered and their synchronized states and localized structures are discussed in details. Particularly, we discuss some solutions of time-dependent fractional Ginzburg–Landau (or nonlinear Schrodinger) equation.     

Visibility graph approach to the analysis of ocean tidal records

By using the recent method of the visibility graph, three time series of oceanic tide level in central Argentina were investigated. The degree distributions show a rich structure; in particular the maximum is due to the main periodic oscillations at 24 hours and 12 hours and higher harmonics. The degree distributions of the residuals (obtained removing from the original signals the cyclic components) suggest that the local effects, linked with the particular coastal conditions of the sites, are discernible for the degree k < 20, while the global effects, linked with linked with the more general and common atmospheric forcing and ocean current conditions, are visible for k > 100. Although a relationship between the spectral exponent α and the exponent of the degree distribution γ of tidal signals can be recognized, this cannot be simply stated due to the very rich and complex structure of time dynamics of tides. The present study, even if still preliminary, show the importance of the visibility graph method in investigating the complex time dynamics of observational and experimental signals.     

Existence and asymptotics of traveling waves for nonlocal diffusion systems

In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder’s fixed point theorem for c > c₁(τ) and using a limiting argument for c = c₁(τ). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara’s Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for ?₂f(0, 0) > 0 and the delay is independent of the minimal wave speed for ?₂f(0, 0) = 0.     

Structurally unstable regular dynamics in 1D piecewise smooth maps,and circle maps

In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X′ = aX if ∣X∣ < z, X′ = bX if ∣X∣ > z, where a and b can take any real value, and may have several applications. We show that its dynamic behaviors are those of a linear rotation: either periodic or quasiperiodic, and always structurally unstable. A generalization to piecewise monotone functions X′ = F(X) if ∣X∣ < z, X′ = G(X) if ∣X∣ > z is also given, proving the conditions leading to a homeomorphism of the circle.     

The non-traveling wave solutions and novel fractal soliton for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients

A method is proposed by extending the linear traveling wave transformation into the nonlinear transformation with the (G′/G)-expansion method. The non-traveling wave solutions with variable separation can be constructed for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients via the method. A novel class of fractal soliton, namely, the cross-like fractal soliton is observed by selecting appropriately the arbitrary functions in the solutions.     

An approximate solution of nonlinear fractional reaction–diffusion equation

The article presents a mathematical model of nonlinear reaction diffusion equation with fractional time derivative α (0 < α ? 1) in the form of a rapidly convergent series with easily computable components. Fractional reaction diffusion equation is used for modeling of merging travel solutions in nonlinear system for popular dynamics. The fractional derivatives are described in the Caputo sense. The anomalous behaviors of the nonlinear problems in the form of sub- and super-diffusion due to the presence of reaction term are shown graphically for different particular cases.     

Some properties for the intersection of Moran sets with their translates

Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for the classification purpose. In this paper, we pursue this study on a class of Moran sets with their rational translates. We also get the fractal structure of intersection I(x, y) of a class of Moran sets with their rational translates, and the formula of the box-counting dimension. We find that the Hausdorff measures of these sets form a discrete spectrum whose non-zero values come only from shifting vector with the expansion in fraction of (x, y). Concretely, when (x, y) has a finite expansion in fraction, a very brief calculation formula of the measure is given.     

Fractal discrimination of random fractal aggregates and its application in biomarker analysis for blood coagulation

A recent rheological study has established that the fractal dimension, d_f, of an incipient clot, formed at the Gel Point (sol–gel transition) of coagulating blood is a significant new biomarker of haemostasis. In whole healthy blood, incipient clots show a clearly defined value of d_f = 1.7 within a narrow range, which represents a new ‘healthy index’ for normal clotting. The addition of unfractionated heparin significantly prolongs the onset of clot formation with a corresponding reduction of d_f as a function of heparin dose. However, as clots mature they exhibit (i) an expected increase in d_f and (ii) a significant increase to spread of these values, i.e. d_f’s in the range 2.0–2.5, limiting the use of d_f as a discriminant of clot microstructure.The present study, details how and why the spectral dimension, d_s, can be used to accommodate this shortcoming and allow discrimination of mature forms of clot microstructure in indistinguishable in terms of their fractal dimension. To elucidate why d_s permits discrimination a numerical experiment was conducted on computationally generated random fractal aggregates (RFAs) with a priori set value of d_f. Starting from RFAs with a d_f of 1.7, mature RFAs are evolved from these incipient templates by two differing growth processes achieving a final d_f of 2.1. Fractal and statistical analysis of the mature RFAs reveals, for the first time, that their differing internal structure is manifest in the magnitude of d_s. The potential clinical significance of these findings is discussed in terms of the possibility of exploiting the incipient clot’s ability to template the internal arrangement of the mature clot to better predict long term clot susceptibility to lysis.