Showing papers in "Fractals in 1995"
TL;DR: In this paper, various methods for estimating the self-similarity parameter and/or the intensity of long-range dependence in a time series are available. But some of these methods are more reliable than others.
Abstract: Various methods for estimating the self-similarity parameter and/or the intensity of long-range dependence in a time series are available. Some are more reliable than others. To discover the ones t...
1,105 citations
TL;DR: In this article, it was shown that fractional calculus can be used to precisely change the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time.
Abstract: The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-differentiation are developed. Based on an understanding of fractional calculus applied to random fractal functions, consideration is given to an analogous application to deterministic or nonrandom fractals. The concept of expressing each coordinate of a deterministic fractal curve as a “pseudo-time” series is investigated. Fractional integro-differentiation of such series is numerically carried out for the case of quadric Koch curves. The resulting time series produces self-similar patterns with fractal dimensions which are linear functions of the order of the fractional integro-differentiation. These curves are assigned the name, fractional Koch curves. The general conclusion is reached that fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time (or pseudo-time).
194 citations
TL;DR: In this paper, the authors investigate the properties of fractal stochastic point processes (FSPPs) and develop several mathematical formulations for these processes, showing that over a broad range of conditions they converge to a particular form of FSPP.
Abstract: We investigate the properties of fractal stochastic point processes (FSPPs). First, we define FSPPs and develop several mathematical formulations for these processes, showing that over a broad range of conditions they converge to a particular form of FSPP. We then provide examples of a wide variety of phenomena for which they serve as suitable models. We proceed to examine the analytical properties of two useful fractal dimension estimators for FSPPs, based on the second-order properties of the points. Finally, we simulate several FSPPs, each with three specified values of the fractal dimension. Analysis and simulation reveal that a variety of factors confound the estimate of the fractal dimension, including the finite length of the simulation, structure or type of FSPP employed, and fluctuations inherent in any FSPP. We conclude that for segments of FSPPs with as many as 106 points, the fractal dimension can be estimated only to within ±0.1.
149 citations
TL;DR: In this article, the U.S. stock market price index was examined using wavelet transform localized in time to indicate how the power of the projection of the signal onto the kernel varies with the scale of observation.
Abstract: Using wavelets we re-examine the U.S. stock market price index for any evidence of self-similarity or order that might be revealed at different scales. The wavelet transform localized in time can be used to indicate how the power of the projection of the signal onto the kernel varies with the scale of observation. By comparing how the local power scales vary over time much information about the structure of the data can be obtained. Such evidence is not at all evident from standard analyses of untransformed data, including projections onto a Fourier basis. Wavelets can detect structures in data that are highly localized in time and therefore non-detectable by Fourier transforms. The main conclusion is that while the data are clearly complex, there seems to be some evidence of non-randomness in the data. There is also some limited evidence of quasi-periodicity in the occurrence of large amplitude shocks to the system.
115 citations
TL;DR: In this paper, a fractional diffusion equation was investigated and the corresponding probability density function for the location of a random walker on a fractal object was presented, where Fox-functions played a dominant part.
Abstract: When Benoit Mandelbrot discussed the problem of fractional Brownian motion in his classic book The Fractal Geometry of Nature, he already pointed out some strong relations to the Riemann-Liouville fractional integral and differential calculus Over the last decade several papers have appeared in which integer-order, standard differential equations modeling processes of relaxation, oscillation, diffusion and wave propagation are generalized to fractional order differential equations The basic idea behind all that is that the order of differentiation need not be an integer but a fractional number (ie dq/dtq with 0
114 citations
TL;DR: Fractal time random walks with generalized Mittag-Leffler functions as waiting time densities are studied in this article, where exact Greens functions corresponding to fractional diffusion are obtained using Mellin transform techniques.
Abstract: Fractal time random walks with generalized Mittag-Leffler functions as waiting time densities are studied. This class of fractal time processes is characterized by a dynamical critical exponent 0<ω≤1, and is equivalently described by a fractional master equation with time derivative of noninteger order ω. Exact Greens functions corresponding to fractional diffusion are obtained using Mellin transform techniques. The Greens functions are expressible in terms of general H-functions. For ω<1 they are singular at the origin and exhibit a stretched Gaussian form at infinity. Changing the order ω interpolates smoothly between ordinary diffusion ω=1 and completely localized behavior in the ω→0 limit.
97 citations
TL;DR: In this article, it was shown that induced flows are given generically by stable convolution semigroups and not by the conventional translation groups, and that the induced semigroup are generated by fractional time derivatives of orders less than unity.
Abstract: Time flow in dynamical systems is reconsidered in the ultralong time limit. The ultralong time limit is a limit in which a discretized time flow is iterated infinitely often and the discretization time step is infinite. The new limit is used to study induced flows in ergodic theory, in particular for subsets of measure zero. Induced flows on subsets of measure zero require an infinite renormalization of time in the ultralong time limit. It is found that induced flows are given generically by stable convolution semigroups and not by the conventional translation groups. This could give new insight into the origin of macroscopic irreversibility. Moreover, the induced semigroups are generated by fractional time derivatives of orders less than unity, and not by a first order time derivative. Invariance under the induced semiflows therefore leads to a new form of stationarity, called fractional stationarity. Fractionally stationary states are dissipative. Fractional stationarity also provides the dynamical foundation for a previously proposed generalized equilibrium concept.
83 citations
TL;DR: In this paper, the authors present some results dealing with the multifractal analysis of sequences of Choquet capacities, and the possibility of constructing such capacities with prescribed spectrum, and related results concerning the pointwise irregularity of a continuous function at each point are given in the frame of iterated functions systems.
Abstract: Some recent advances in the application of fractal tools for studying complex signals are presented. The first part of the paper is devoted to a brief description of the theoretical methods used. These essentially consist of generalizations of previous techniques that allow us to efficiently handle real signals. We present some results dealing with the multifractal analysis of sequences of Choquet capacities, and the possibility of constructing such capacities with prescribed spectrum. Related results concerning the pointwise irregularity of a continuous function at each point are given in the frame of iterated functions systems. Finally, some results on a particular stochastic process are sketched: the multifractional Brownian motion, which is a generalization of the classical fractional Brownian motion, where the parameter H is replaced by a function. The second part consists of the description of selected applications of current interest, in the fields of image analysis, speech synthesis and road traffic modeling. In each case we try to show how a fractal approach provides new means to solve specific problems in signal processing, sometimes with greater success than classical methods.
77 citations
TL;DR: In this paper, the generalized entropic form (q∈ℜq=1 corresponds to the standard entropy S1≡−kB ∑i pi ln pi) was used to identify Sq=0 is equivalent to q=df/d, where df and d respectively are fractal and Euclidean dimensions.
Abstract: We discuss the modifications of standard, extensive, thermodynamics that long-range interactions are expected to introduce In addition to that, by using the generalized entropic form (q∈ℜq=1 corresponds to the standard entropy S1≡−kB ∑i pi ln pi) we establish, through a simple identification, that Sq=0 is equivalent to q=df/d, where df and d respectively are fractal and Euclidean dimensions
77 citations
TL;DR: In this article, the authors analyzed the self-similarity of high-frequency DEM-USD exchange rate records and 30 main German stock price records and found that the full distributions of lag-k jumps have a scaling behavior characterized by the above Hurst exponent.
Abstract: A simple quantitative measure of the self-similarity in time-series in general and in the stock market in particular is the scaling behavior of the absolute size of the jumps across lags of size k. A stronger form of self-similarity entails that not only this mean absolute value, but also the full distributions of lag-k jumps have a scaling behavior characterized by the above Hurst exponent. In 1963, Benoit Mandelbrot showed that cotton prices have such a strong form of (distributional) self-similarity, and for the first time introduced Levy’s stable random variables in the modeling of price records. This paper discusses the analysis of the self-similarity of high-frequency DEM-USD exchange rate records and the 30 main German stock price records. Distributional self-similarity is found in both cases and some of its consequences are discussed.
76 citations
TL;DR: In this article, the authors concentrate on the fractal behavior of dynamical systems, focusing on the sol-gel transition and the patterns of motion displayed by polyampholytes (polymers containing positive and negative charges).
Abstract: Scaling aspects are of common occurrence in polymer science. Here we concentrate on the fractal behavior of dynamical systems. As examples we focus on the sol-gel transition and on the patterns of motion displayed by polyampholytes (polymers containing positive and negative charges).
TL;DR: Evidence is presented supporting the idea that the DNA sequence in genes containing noncoding regions is correlated, and that the correlation is remarkably long range--indeed, base pairs thousands of base pairs distant are correlated.
Abstract: We present evidence supporting the idea that the DNA sequence in genes containing noncoding regions is correlated, and that the correlation is remarkably long range--indeed, base pairs thousands of base pairs distant are correlated. We do not find such a long-range correlation in the coding regions of the gene. We resolve the problem of the "non-stationary" feature of the sequence of base pairs by applying a new algorithm called Detrended Fluctuation Analysis (DFA). We address the claim of Voss that there is no difference in the statistical properties of coding and noncoding regions of DNA by systematically applying the DFA algorithm, as well as standard FFT analysis, to all eukaryotic DNA sequences (33 301 coding and 29 453 noncoding) in the entire GenBank database. We describe a simple model to account for the presence of long-range power-law correlations which is based upon a generalization of the classic Levy walk. Finally, we describe briefly some recent work showing that the noncoding sequences have certain statistical features in common with natural languages. Specifically, we adapt to DNA the Zipf approach to analyzing linguistic texts, and the Shannon approach to quantifying the "redundancy" of a linguistic text in terms of a measurable entropy function. We suggest that noncoding regions in plants and invertebrates may display a smaller entropy and larger redundancy than coding regions, further supporting the possibility that noncoding regions of DNA may carry biological information.
TL;DR: In this paper, the Barkhausen signal ν, as well as the size Δx and the duration Δu of jumps follow distributions of the form ν−α, Δx−β, Δu−γ, with α=1−c, β=3/2−c/2, γ=2-c, where c is a dimensionless parameter proportional to the applied field rate.
Abstract: The main physical aspects and the theoretical description of stochastic domain wall dynamics in soft magnetic materials are reviewed. The intrinsically random nature of domain wall motion results in the Barkhausen effect, which exibits scaling properties at low magnetization rates and 1/f power spectra. It is shown that the Barkhausen signal ν, as well as the size Δx and the duration Δu of jumps follow distributions of the form ν−α, Δx−β, Δu−γ, with α=1−c, β=3/2−c/2, γ=2–c, where c is a dimensionless parameter proportional to the applied field rate. These results are analytically calculated by means of a stochastic differential equation for the domain wall dynamics in a random perturbed medium with brownian properties and then compared to experiments. The Barkhausen signal is found to be related to a random Cantor dust with fractal dimension D=1−c, from which the scaling exponents are calculated using simple properties of fractal geometry. Fractal dimension Δ of the signal v is also studied using four different methods of calculation, giving Δ≈1.5, independent of the method used and of the parameter c. The stochastic model is analyzed in detail in order to clarify if the shown properties can be interpreted as manifestations of self-organized criticality in magnetic systems.
TL;DR: In this article, the influence of fractal surface morphology on the Knudsen diffusivity was investigated, and a momentum transfer technique was used to obtain a smooth field approximation for the KF.
Abstract: Porous amorphous catalysts often have a fractal internal surface down to molecular scales. The movement of gas molecules inside the narrow channels that constitute all or most of the pore space, is mainly hindered by collisions with the surface, rather than with each other: Knudsen diffusion, rather than bulk molecular diffusion is then the main diffusion mechanism. The influence of the fractal surface morphology on the Knudsen diffusivity is investigated. A momentum transfer technique leads to a smooth field approximation for the Knudsen diffusivity. The dependence of the surface accessibility on the size of the molecules causes the Knudsen diffusivity of a molecule to depend on its effective diameter. Because the previous method assumes a uniform surface accessibility, a first-passage time technique is developed, which accounts for the true accessibility distribution over the surface. The problem is solved analytically and the result is a simple expression of the Knudsen diffusivity. Extensions of this technique to other problems in catalysis and to other fields are discussed. Once a property, like the Knudsen diffusivity, is known for a medium with smooth walls, the method developed here allows us to calculate the same property for a medium with fractal walls.
TL;DR: In this paper, the authors show that the experiments described in the literature all have avalanche-size distributions that can be very well described by stretched-exponential distributions, since the stretchedexponential distribution has a characteristic size.
Abstract: Self-organized criticality (SOC) is thought to describe avalanche dynamics in sandpiles. Experiments on the dynamics of “sandpiles” fall in two broad categories. Sandpiles in rotating drums exhibit periodic large avalanches, which is inconsistent with SOC. Other experiments study sandpiles on a circular support driven by the addition of grains of sand at a low rate from above and centered with respect to the support. The mass of avalanches that drop sand off the support is typically measured by a balance. It has been claimed that the observed avalanche statistics is consistent with SOC. However, I find that experiments described in the literature all have avalanche-size distributions that can be very well described by stretched-exponential distributions. Since the stretched-exponential distribution has a characteristic size, I conclude that the experiments described in the literature are inconsistent with SOC.
TL;DR: In this paper, the authors discuss the difficulties in describing chaotic systems in general relativity and investigate the motion of particles in two and three black hole spacetimes, and show that the dynamics is chaotic by exhibiting the basins of attraction of the black holes which have fractal boundaries.
Abstract: Fractal basin boundaries provide an important means of characterizing chaotic systems. We apply these ideas to general relativity, where other properties such as Lyapunov exponents are difficult to define in an observer independent manner. Here we discuss the difficulties in describing chaotic systems in general relativity and investigate the motion of particles in two and three black hole spacetimes. We show that the dynamics is chaotic by exhibiting the basins of attraction of the black holes which have fractal boundaries. Overcoming problems of principle as well as numerical difficulties, we evaluate Lyapunov exponents numerically and find that some trajectories have a positive exponent.
TL;DR: In this paper, various aspects of the question "Can one hear the shape of a fractal drum?" were studied, both for "drums with fractal boundary" (or "surface fractals") and for drums with fract fractal membrane (or ''mass fractals'').
Abstract: We study various aspects of the question “Can one hear the shape of a fractal drum?”, both for “drums with fractal boundary” (or “surface fractals”) and for “drums with fractal membrane” (or “mass fractals”).
TL;DR: A quarter century has passed since the bulk of Benoit Mandelbrot's articles on economics were published as mentioned in this paper, and there are two separate literatures which have responded skeptically to his innovations: that which consider the empirical possibility that economic time series are best described as stochastic processes based upon Levy stable distributions; and that which considers the possible evidence for chaotic dynamics in economic time-series.
Abstract: A quarter century has passed since the bulk of Benoit Mandelbrot’s articles on economics were published. This paper outlines the broad reactions to Mandelbrot’s theses within the economics profession. There are two separate literatures which have responded skeptically to his innovations: that which considers the empirical possibility that economic time series are best described as stochastic processes based upon Levy stable distributions; and that which considers the possible evidence for chaotic dynamics in economic time series. The paper concludes that much of the skepticism in orthodox economics is founded upon inadequate empirical premises, especially when compared to the approach of physicists to similar issues.
TL;DR: In this paper, the authors discuss geometric optimization as a mathematical device with which to model several natural processes involving surface free energy, and discuss the use of geometric optimization to model a number of natural processes.
Abstract: We discuss geometric optimization as a mathematical device with which to model several natural processes involving surface free energy.
TL;DR: In a previous paper as mentioned in this paper, we showed how one may read off dynamical information about orbits of Fc(x)=x2+c from the geometry of the Mandelbrot set.
Abstract: In a previous paper in this series,1 we showed how one may read off dynamical information about orbits of Fc(x)=x2+c from the geometry of the Mandelbrot set. In this paper we extend these ideas to show further relations between the dynamics of x2+c and the Mandelbrot set. In particular, we show that Mandelbrot set may be used as a vehicle to show students how to “count” and how to “add” geometrically. The ideas in this paper arose from a series of experiments conducted by high school students in “chaos clubs” organized in the Boston public schools by Jonathan Choate, Mary Corkery, Beverly Mawn, and the author. The goal of these clubs was to expose young students to contemporary ideas in mathematics. Students discovered various facts about the Mandelbrot set using a combination of computer experiments and group projects. This paper presents a summary of some of the students’ findings.
TL;DR: This work presents a non-local communicating walkers model to study the effect of local bacterium-bacterium interaction and communication via chemotaxis signaling and shows that seemingly unrelated patterns can result from the employment of the same generic strategies.
Abstract: In nature, bacterial colonies often must cope with hostile environmental conditions. To do so they have developed sophisticated cooperative behavior and intricate communication channels on all levels. The result is that a profusion of complex patterns are formed during growth of various bacterial strains and for different environmental conditions. Some qualitative features of the complex morphologies may be accounted for by invoking ideas from pattern formation in non-living systems together with a simplified model of chemotactic “feedback”. We present a non-local communicating walkers model to study the effect of local bacterium-bacterium interaction and communication via chemotaxis signaling. The model is an hybridization of the continuous approach (to handle chemicals’ diffusion) and the atomistic approach (each “atom” or “walker” represents 104–105 bacteria). Using the model we demonstrate how communication enables the colony to develop complex patterns in response to adverse growth conditions. Efficient response of the colony requires self-organization on all levels, which can be achieved only via cooperative behavior of the bacteria. It can be viewed as the action of an interplay between the micro-level (the individual bacterium) and the macro-level (the colony) in the determination of the emerging pattern. We show that seemingly unrelated patterns can result from the employment of the same generic strategies.
TL;DR: In this article, the inverse fractal problem for self-affine functions in R2 is solved by means of testing the invariance of the wavelet transform of the function.
Abstract: The inverse fractal problem for self-affine functions in R2 is solved by means of testing the invariance of the wavelet transform of the function. The wavelet maxima bifurcation representation used is derived from the continuous wavelet decomposition and possesses translational and scale invariance necessary to reveal the invariance of the self-affine fractal. Algorithms are presented which give satisfactory results for the self-affine fractal and which potentially can be applied to a variety of fractal types in order to solve the related inverse fractal problem.
TL;DR: In this article, the relationship between geometric roughness and energetic heterogeneity is discussed by considering a thermodynamically consistent model of adsorption isotherms (Keller model) which encompasses fractal scaling and the dependence of the adsorship energies on the coverage.
Abstract: The relationships between geometric roughness and energetic heterogeneity are discussed by considering a thermodynamically consistent model of adsorption isotherms (Keller model) which encompasses fractal scaling and the dependence of the adsorption energies on the coverage Experimental results validate this model and indicate that it can be used not only to interpolate experimental data but also to predict adsorption equilibria of multicomponent rnixtures The peculiar non-Henry behaviour of the Keller model at low pressure is discussed by considering a simple model of preferential adsorption on a rough energy landscape and including the effect of surface diffusion
TL;DR: In this article, the arterial system is characterized as a non-homogeneous fractal, and the spatial resolution of local scaling and dispersion of the fractal dimension within the organs are analyzed.
Abstract: Three-dimensional data sets of kidney arterial vessels were obtained from resin casts by serial sectioning and by micro-NMR-tomography, and were analyzed by the mass-radius-relation both for global and local scaling properties. We present for the first time the spatial resolution of local scaling and thus the dispersion of the fractal dimension within the organs. The arterial system is characterized as a non-homogeneous fractal. We discuss and relate the fractal structure to the scaling and allometry of metabolic rates in living organisms.
TL;DR: In this paper, the authors use both the global Mandelbrot-Hurst exponent and the distribution of local MHP exponents, in combination with dynamical entropies, to quantitate the property of nonuniform persistence which they treat as both deterministically expansive and statistically diffusive.
Abstract: Intermittency, in which the normalized weight of large fluctuations grows for increasingly longer statistical samples, is seen as irregular bursting activity in time and is characteristic of the behavior of many brain and behavioral systems. This pattern has been related to the brain-stabilizing interplay of the general mechanisms of silence-evoked sensitization and activity-evoked desensitization, which can be found at most levels of neurobiological function and which vary more smoothly and at much longer times than the phasic observables. We use both the global Mandelbrot-Hurst exponent and the distribution of local Mandelbrot-Hurst exponents, in combination with dynamical entropies, to quantitate the property of nonuniform persistence which we treat as both deterministically expansive and statistically diffusive. For example, varying the parameter of the one-dimensional, Manneville-Pomeau intermittency map generated time series which demonstrated systematic changes in these statistical indices of persistence. Relatively small doses of cocaine administered to pregnant rats increased statistical indices of expansiveness and persistence in fetal motor behavior. These techniques also model and characterize a breakdown of statistical scaling in 72-hour time series of the amount of motor activity in some hospitalized manic-depressive patients.
TL;DR: For intersecting circles, a modification of the limit set generated by inversion in circles is proposed in this article, which leads to a graphical representation of the grammatical complexity of the restricted limit set.
Abstract: For intersecting circles, we propose a modification of the limit set generated by inversion in circles. This restricted limit set is always a subset of the discs bounded by the generating circles. We give examples of restricted limit sets and show arrangements of generating circles for which the restricted limit set equals the limit set, and also arrangements for which they differ. In addition, we give a visual presentation, based on Iterated Function Systems, of the excluded strings in the restricted limit set. This leads to a graphical representation of the grammatical complexity of the restricted limit set.
TL;DR: In this article, a geometric and very general relation between the size distribution and the fractal dimensions of a set of objects is presented, which can be used to understand the fragment-size distribution of fragmenting gypsum.
Abstract: A geometric and very general relation between the size distribution and the fractal dimensions of a set of objects is presented. The applications are numerous, ranging from fragmentation experiments to time series. For example, it may be used to understand the fragment-size distribution of fragmenting gypsum. The formalism also generalizes to self-affine fractals, and here it is applied to the scaling properties of self-interactions in (1+1)-d directed percolation.
TL;DR: In this article, the autocorrelation functions obey a power-law behavior, implying a power spectrum of the kind 1/f, and amplitude distribution N(V) of ultrasonic acoustic emission signals follows a power law.
Abstract: Relaxation processes taking place after microfracturing of laboratory samples give rise to ultrasonic acoustic emission signals. Statistical analysis of the resulting time series has revealed many features which are characteristic of critical phenomena. In particular, the autocorrelation functions obey a power-law behavior, implying a power spectrum of the kind 1/f. Also the amplitude distribution N(V) of such signals follows a power law, and the obtained exponents are consistent with those found in other experiments: N(V) dV≃V–γ dV, with γ=1.7±0.2. We also analyzed the distribution N(τ) of the delay time τ between two consecutive acoustic emission events. We found that a N(τ) distribution rather close to a power law constitutes a common feature of all the recorded signals. These experimental results can be considered as a striking evidence for a critical dynamics underlying the microfracturing processes.
TL;DR: In this paper, the authors introduced the concept of local average dimension of a measure µ, at x∈ℝn as the unique exponent where the lower average density of µ at x jumps from zero to infinity.
Abstract: In this note we introduce the concept of local average dimension of a measure µ, at x∈ℝn as the unique exponent where the lower average density of µ, at x jumps from zero to infinity. Taking the essential infimum or supremum over x we obtain the lower and upper average dimensions of µ, respectively. The average dimension of an analytic set E is defined as the supremum over the upper average dimensions of all measures supported by E. These average dimensions lie between the corresponding Hausdorff and packing dimensions and the inequalities can be strict. We prove that the local Hausdorff dimensions and the local average dimensions of µ at almost all x are invariant under orthogonal projections onto almost all m- dimensional linear subspaces of higher dimension. The corresponding global results for µ and E (which are known for Hausdorff dimension) follow immediately.