Fat Fractals Research Articles

Structural investigations of fat fractals using small-angle scattering

Experimental small-angle scattering (SAS) data characterized, on a double logarithmic scale, by a succession of power-law decays with decreasing values of scattering exponents, can be described in terms of fractal structures with positive Lebesgue measure (fat fractals). Here we present a theoretical model for fat fractals and show how one can extract structural information about the underlying fractal using SAS method, for the well known fractals existing in the literature: Vicsek and Menger sponge. We calculate analytically the fractal structure factor and study its properties in momentum space. The models allow us to obtain the fractal dimension at each structural level inside the fractal, the number of particles inside the fractal and about the most common distances between the center of mass of the particles.

Small-angle scattering from fat fractals

A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.

Study of the island morphology at the early stages of Fe/Mo(110) MBE growth

Abstract We present theoretical study of morphology of Fe islands grown at Mo(110) surface in submonolayer MBE mode. We utilize atomistic SOS model with bond counting, and interactions of Fe adatom up to third nearest neighbors. We performed KMC simulations for different values of adatom interactions and varying temperatures. We have found that, while for the low temperature islands are fat fractals, for the temperature 500 K islands have faceted rhombic-like shape. For the higher temperature, islands acquire a rounded shape. In order to evaluate qualitatively morphological changes, we measured average aspect ratio of islands. We calculated dependence of the average aspect ratio on the temperature, and on the strength of interactions of an adatom with neighbors.

The VAK of vacuum fluctuation,: Spontaneous self-organization and complexity theory interpretation of high energy particle physics and the mass spectrum

The paper is a rather informal introduction to the concepts and results of the E-infinity Cantorian theory of quantum physics. The fundamental tools of complexity theory and non-linear dynamics (Hausdorff dimensions, fat fractals, etc.) are used to give what we think to be a new interpretation of high energy physics and to determine the corresponding mass-spectrum. Particular attention is paid to the role played by the VAK, KAM theorem, Arnold diffusion, Newhaus sinks and knot theory in determining the stability of an elementary “particle-wave” which emerges in self-organizatory manner out of sizzling vacuum fluctuation.

Restructuring in step decorations grown by simulated epitaxy of Cu on a bcc W(110) surface

The submonolayer growth of Cu deposit at an ascending step oriented in the [001] direction on a W(110) surface is studied by kinetic Monte Carlo simulations at temperatures ranging from 60 to 280 K and fluxes from 10 - 1 9 to 10 - 3 ML/s. A lattice-gas model that simulates deposition of adatoms, diffusion of monomers, diffusion along deposit edges, and reversible aggregation of adatoms is employed. The growth conditions are chosen in such a way that the growing deposit accumulates only at the step edge. Randomly ramified step decorations are found to grow in extended temperature regimes. The deposit morphology changes from an open branched aggregate with fractal dimension equal to 1.71′0.04 at low temperatures to a stripe of a compact shape at the highest temperature considered. With increasing temperature of deposition the partial restructuring processes yield widening of aggregate branches. The onset of the restructuring processes takes place at 90 K when single-coordinated adatoms become mobile. Twofold- and threefold-coordinated adatoms become mobile at 180 and at 250 K; respectively. There is a temperature window, 150 K<T<190 K, where the deposit equilibration slows down because the average number of hops performed by single-coordinated adatoms is nearly constant and the hops from twofold-coordinated sites do not start yet. At these temperatures, step decorations with a nearly constant average branch width of 3-4 atoms are obtained. The high-temperature region of crossover into the compact growth, existing above 200 K, results in aggregates called fat fractals with branches widened up to 10 or more atoms. These fat fractals resemble closely step decorations observed in molecular beam epitaxy experiments.

Intermingled fractal Arnold tongues

We present a pattern of multiply interwoven Arnold tongues in the case of the single-well Duffing oscillator at low dissipation and weak forcing: Strips of $\frac{2}{2}$ Arnold tongues form a truncated fractal structure and the tonguelike regions in between are filled by finely intermingled fractal-like $\frac{1}{1}$ and $\frac{3}{3}$ Arnold tongues, which are fat fractals characterized by the uncertainty exponent $\ensuremath{\alpha}\ensuremath{\approx}0.7$. The truncated fractal Arnold tongues are present in the case of high dissipation as well, while the intermingled fractal pattern gradually disappears with increasing dissipation.

Scaling relationships of channel networks at large scales: Examples from two large-magnitude watersheds in Brittany, France

We present a statistical analysis on two watersheds in French Brittany whose drainage areas are about 10,000 and 2000 km2. The channel system was analysed from the digitised blue lines of the 1:100,000 map and from a 250-m DEM. Link lengths follow an exponential distribution, consistent with the Markovian model of channel branching proposed by Smart (1968). The departure from the exponential distribution for small lengths, that has been extensively discussed before, results from a statistical effect due to the finite number of channels and junctions. The Strahler topology applied on channels defines a self-similar organisation whose similarity dimension is about 1.7, that is clearly smaller than the value of 2 expected for a random organisation. The similarity dimension is consistent with an independent measurement of the Horton ratios of stream numbers and lengths. The variables defined by an upstream integral (drainage area, mainstream length, upstream length) follow power-law distributions limited at large scales by a finite size effect, due to the finite area of the watersheds. A special emphasis is given to the exponent of the drainage area, aA, that has been previously discussed in the context of different aggregation models relevant to channel network growth. We show that aA is consistent with 4/3, a value that was obtained and analytically demonstrated from directed random walk aggregating models, inspired by the model of Scheidegger (1967). The drainage density and mainstream length present no simple scaling with area, except at large areas where they tend to trivial values: constant density and square root of drainage area, respectively. These asymptotic limits necessarily imply that the space dimension of channel networks is 2, equal to the embedding space. The limits are reached for drainage areas larger than 100 km2. For smaller areas, the asymptotic limit represents either a lower bound (drainage density) or an upper bound (mainstream length) of the distributions. Because the fluctuations of the drainage density slowly converge to a finite limit, the system could be adequately described as a fat fractal, where the average drainage density is the sum of a constant plus a fluctuation decreasing as a power law with integrating area. A fat fractal hypothesis could explain why the similarity dimension is not equal to the fractal capacity dimension, as it is for thin fractals. The physical consequences are not yet really understood, but we draw an analogy with a directed aggregating system where the growth process involves both stochastic and deterministic growth. These models are known to be fat fractals, and the deterministic growth, which constitutes a fundamental ingredient of these models, could be attributed in river systems to the role of terrestrial gravity.

Multifractal approach to inhomogeneous fractals

Vicsek et al. have shown that DLA can be described as a geometrical multifractal by defining the mass M i within box i normalized to the object's total mass M 0 as the measure μ i = M i / M 0. This measure shows its multifractal property by the dependence of the generalized dimensions D q on q. Recently, we have shown that the arterial blood vessels, which are fat fractals, are also geometrically multifractal. We have examined the origin of multifractality of thin and fat fractals and give a new classification of thin and fat monofractals and multifractals.

RELATING THIN AND FAT-FRACTAL SCALING OF RIVER-NETWORK MODELS

The fractal scaling of river networks has been described in the context of both thin and fat fractals. Whereas the thin-fractal characterization is presented as a fractal dimension, D, derived from the length properties of the river channels, a fat-fractal characterization is given as a scaling exponent, β, derived from the behavior of river-channel area. Several authors have related D to the bifurcation ratio, Rb, and length ratio, Rl, of idealized Hortonian-network trees. Here, for these types of trees, we asymptotically relate β to Rb, Rl, and to a diameter exponent, Δ, which governs the downstream channel-widening process. Using this result, we present a linkage of D to β and discuss the implications of the relative values of Rb, Rl, and Δ on this linkage and on network channel behavior. Finally, we illustrate bias in the estimation of β from preasymptotic trees.

Peculiarities of wave scattering by fat fractals.

The stationary problem of a one-dimensional wave transition through a fat-fractal slab is considered in the small-wavelength limit. The convergence of the numerical approach is discussed. An analysis of the accuracy of the calculation yields the conditions that allow one to approximate an authentic fractal layer by a structure obtained with a finite number of steps. This number decays exponentially with the incident wavelength.

Extension of the fat-fractal exponent β to arbitrary sets in D dimensions

Abstract Fat fractals, which are fundamental to nonlinear dynamical systems, are characterized by a fat-fractal exponent, rather than a dimension. In one dimension, the most discriminating exponent is β, but, for most sets in D >1 dimensions, β is not defined. Here, we extend the definition of β to all sets.

Measure of fat fractals in one-dimensional maps.

An efficient method to determine the measure of fat-fractal sets of parameter values giving rise to chaos in a wide class of nonlinear mappings on finite intervals is introduced. The method is based on perturbing a given map by a piecewise linear term -mchemically bondxchemically bond with mgreater than or equal to0. For nonzero values of the perturbation strength m the maps under study are chaotic on intervals in the nonlinearity parameter. As m diminishes, the number of these intervals increases, whereas their average length decreases. Accordingly, the measure of the studied fat fractals is defined as a limit (for m..-->..0 of the product of the number of chaotic intervals and their average size.

Relating the various scaling exponents used to characterize fat fractals in nonlinear dynamical systems

Fat fractals (sets with fractal structure, but nonzero measure) are becoming increasingly important in nonlinear dynamical systems. These sets have allowed quantitative analyses of sensitivity to parameters, final-state sensitivity, and quantum chaos, and have played important roles in discussions of nonlinear twist maps, one-dimensional maps with quadratic maxima, Arnol'd tongues in circle maps, and practal basin boundaries. Unfortunately, since fat fractals have nonzero measure, they must have the same (integer) dimension as the underlying space. Therefore, their dimension is insensitive to the fractal structure and does not provide an appropriate means of characterization. Hence, several scaling exponents have been suggested for this purpose, and there has been some debate over which is best. Also, there has been considerable confusion over how these exponents differ and whether they are simply related. We shed light on this issue by examining these exponents and their properties and deriving relationships between them.

Characterization of Fat Fractals in Nonlinear Dynamical Systems

Characterization of Fat Fractals in Nonlinear Dynamical Systems

Characterization of fat fractals in nonlinear dynamical systems.

Fat fractals (sets with fractal structure, but nonzero measure) are becoming increasingly important in the study of nonlinear dynamical systems. Since fat fractals are not well characterized by their (integer) dimensions, several scaling exponents have been proposed to characterize these sets, and there has been some debate over which exponent is best. We shed light on this issue by examining these exponents and finding relationships between them. Finally, these results lead naturally to quantitative definitions of the notions of sensitivity to parameters and final-state sensitivity.

Comment on "Sensitive dependence on parameters in nonlinear dynamics" and on "Fat fractals on the energy surface"

Comment on "Sensitive dependence on parameters in nonlinear dynamics" and on "Fat fractals on the energy surface"

Farmer and Umberger respond.

A reply to the comment on sensitive dependence on parameters in nonlinear dynamics and on fat fractals on the energy surface is presented.

Fat fractals in the lung? A comment on a paper by grebogi et al.

An exponent characterising the scaling behaviour of a branching structure occupying volume (an example of a “fat” fractal), introduced recently by Grebogi et al., is examined using data on the arteries, veins and airways of mammalian lungs. The results cast doubt on the usefulness of the exponent in this context.

Fat fractals on the energy surface.

For a closed system of two coupled nonlinear oscillators, chaotic orbits are punctuated by holes associated with stable periodic orbits. For the corresponding class of Hamiltonian maps we demonstrate that the combined area for all holes of size epsilon or greater scales as a power law with exponent ..beta.. and asymptotic area 0 < ..mu.. < 1. In contrast to previous results, this is a global scaling property, valid for a set of positive Lebesgue measure. It suggests that these chaotic orbits are fat fractals, i.e., Cantor setlike objects of positive area. We numerically compute lower bounds on their area.