Showing papers on "Fractal dimension published in 1983"
TL;DR: In this paper, the correlation exponent v is introduced as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise, and algorithms for extracting v from the time series of a single variable are proposed.
5,239 citations
TL;DR: In this article, a measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable, and the relation of this measure to fractal dimension and information-theoretic entropy is discussed.
Abstract: A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and information-theoretic entropy is discussed.
4,323 citations
TL;DR: In this article, it was shown that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions Dq, q > 0, which correspond to exponents associated with ternary, quaternary and higher correlation functions.
1,577 citations
TL;DR: Fractal dimension D as discussed by the authors is a global measure of surface irregularity, which labels an extremely heterogeneous surface by a value far from two, and it implies that any monolayer on such a surface resembles three-dimensional bulk rather than a two-dimensional film because the number of adsorption sites within distance l from any fixed site, grows as lD.
Abstract: In this, the first of a series of papers, we lay the foundations for appreciation of chemical surfaces as D‐dimensional objects where 2≤D<3. Being a global measure of surface irregularity, this dimension labels an extremely heterogeneous surface by a value far from two. It implies, e.g., that any monolayer on such a surface resembles three‐dimensional bulk rather than a two‐dimensional film because the number of adsorption sites within distance l from any fixed site, grows as lD. Generally, a particular value of D means that any typical piece of the surface unfolds into mD similar pieces upon m‐fold magnification (self‐similarity). The underlying concept of fractal dimension D is reviewed and illustrated in a form adapted to surface‐chemical problems. From this, we derive three major methods to determine D of a given solid surface which establish powerful connections between several surface properties: (1) The surface area A depends on the cross‐section area σ of different molecules used for monolayer cov...
976 citations
TL;DR: In this paper, a scaling exponent for the final-state-uncertain phase space volume dependence on uncertainty in initial conditions is defined and related to the fractal dimension of basin boundaries.
292 citations
TL;DR: In this paper, a generalization to the d-fractal geometry of the relation which controls the interfacial energy and mass exchange in the linear approximation of the TEISI model was proposed.
279 citations
TL;DR: In this paper, it was shown numerically that the stability intervals for limit cycles of the circle map form a complete devil's staircase at the onset of chaos and the complementary set to the stability interval is a Cantor set of fractal dimension $D=0.87.
Abstract: It is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil's staircase at the onset of chaos. The complementary set to the stability intervals is a Cantor set of fractal dimension $D=0.87$. This exponent is found to be universal for a large class of functions.
238 citations
TL;DR: Fractal analysis has been used for morphological characterization of irregular and crenellate sedimentary particles as discussed by the authors, where the fractal dimension (D) equals 1.0 and 2.0.
Abstract: For several decades, sedimentologists have had difficulty in obtaining an efficient index of particle form that can be used to specify adequately irregular morphology of sedimentary particles. Mandelbrot has suggested the use of the fractal dimension as a single value estimate of form, in order to characterize morphologically closed loops of an irregular nature. The concept of fractal dimension derives from Richardson's unpublished suggestion that a stable linear relationship appears when the logarithm of the perimeter estimate of an irregular outline is plotted against the logarithm of the unit of measurement (step length). Decreases in step length result in an increase in perimeter by a constant weight (b) for particles whose morphological variations are the same at all measurement scales (self-similarity). The fractal dimension (D) equals 1.0-(b), where b is the slope coefficient of the best-fitting linear regression of the plot. The value of D lies between 1.0 and 2.0, with increasing values of D correlating with increasing irregularity of the outline. In practice, particle outline morphology is not always self-similar, such that two or possibly more fractal elements can occur for many outlines. Two fractal elements reflect the morphological difference between micro-scale edge textural effects (D1) and macro-scale particle structural effects (D2) generated by the presence of crenellate-edge morphology (re-entrants). Fractal calibration on a range of regular/irregular particle outline morphologies, plus examination of carbonate beach, pyroclastic and weathered quartz particles indicates that this type of analysis is best suited for morphological characterization of irregular and crenellate particles. In this respect, fractal analysis appears as the complementary analytical technique to harmonic form analysis in order to achieve an adequate specification of all types of particles on a continuum of irregular to regular morphology.
168 citations
TL;DR: On donne une estimation de la dimension fractale D F'des attracteurs etranges as discussed by the authors, des amas de n points plus proches voisins sont echantillonnes a partir d'une serie temporelle.
Abstract: On donne une estimation de la dimension fractale D F ' des attracteurs etranges. Des amas de n points plus proches voisins sont echantillonnes a partir d'une serie temporelle; on trouve D F ' a partir de R DF' ∼n ou R est le rayon d'amas moyen
89 citations
TL;DR: In this article, the first renormalization-group approach for irreversible growth models of randomly branched aggregates is presented, and the main result is that the Witten-Sander diffusion-limited aggregation model, a discrete version of a dendritic growth model, is in a different universality class than "equilibrium" lattice animals.
Abstract: The first renormalization-group approach for irreversible growth models of randomly branched aggregates is presented. The main result is that the Witten-Sander diffusionlimited aggregation model, a discrete version of a dendritic growth model, is in a different universality class than "equilibrium" lattice animals. Also calculated is the fractal dimension for the Witten-Sander model and the Eden model (a model developed for the study of biological structures).
83 citations
TL;DR: It is speculated that in common with the human lung, another living surface with a high D value, the coral reef slope is maximizing its contact with the surrounding medium.
Abstract: Conventional analytical techniques do not cope adequately with multiscale processes. Mandelbrot's concept of fractal dimension, a novel approach to multiscale phenomena, is applied to the problem of coral reef topography. A fractal dimension D = 1.9-2.0 for a contour on the reef slope is obtained. This contrasts strongly with the well established value of D = 1.2-1.3 for coastlines. We speculate that in common with the human lung, another living surface with a high D value, the coral reef slope is maximizing its contact with the surrounding medium.
TL;DR: In this paper, the authors re-measured the fractal dimension of the Henon attractor by direct box-counting and observed that the number of boxes visited after n iterations obeys a scaling law.
16 Oct 1983
TL;DR: In this article, the fractal dimension of a surface is defined as a measure of its geometric complexity and can take on any non-integer value between 2 and 3; however, their fractal dimensions increase with greater amounts of complexity or roughness.
Abstract: : The fractal dimension of a surface is a measure of its geometric complexity and can take on any non-integer value between 2 and 3. Normally, the topological dimension of surfaces is 2; however, their fractal dimensions increase with greater amounts of complexity or roughness. For example, a fractal dimension of 2.3 is found to be a common value in describing the relief on the earth. This paper discusses and presents examples of an algorithm designed to measure the fracticality of surfaces. The algorithm was developed at The Ohio State University and is shown to be reliable and robust. It is placed in an interactive setting and is based on the premise that the complexity of isarithm lines may be used to approximate the complexity of a surface. The algorithm operates with the following scenario: Starting with a matrix of Z-heights, an isarithm interval is selected and isarithm lines are constructed on the surface. A fractal dimension is computed for each isarithm line by calculating their lengths over a number of sampling intervals. The surface's fractal dimension is the result of averaging the fractal dimensions of all the isarithm lines and adding 1. Potential applications for this technique include a new means for data compression, a quantitative measure of surface roughness, and be used for generalization and filtering.
IBM1
TL;DR: The one-dimensional Ising model with long-range antiferromagnetic interaction in an applied field is known to exhibit a complete devil's staircase in its $T=0$ phase diagram as discussed by the authors.
Abstract: The one-dimensional Ising model with long-range antiferromagnetic interaction in an applied field is known to exhibit a complete devil's staircase in its $T=0$ phase diagram. In this Comment we discuss its self-similar properties and determine the fractal dimension.
TL;DR: In this article, an exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold, and the authors use this approach to calculate exactly the first low-density expansion of LBB(p) for arbitrary system dimensionality d, where LBBp is the mean of backbone bonds and p is the bond occupation probability.
Abstract: An exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold. The authors use this approach to calculate exactly the first low-density expansion of LBB(p) for arbitrary system dimensionality d, where LBB(p) is the mean of backbone bonds and p is the bond occupation probability. Standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d; these disagree with the Sierpinski gasket model of the backbone. They also calculate the first low-density expansions of Lmin(p) and Lred(p) which are, respectively, the mean number of bonds in the minimum path between i and j and the mean number of singly connected ('red') bonds.
TL;DR: Fractal properties and the concept of fractal dimension have been studied in this article and the applicability in structure analysis has been given to apply fractal analysis in the field of structure analysis.
Abstract: SUMMARY
Fractal properties and the concept of fractal dimension has been studied. Emphasis is given to the applicability in structure analysis. Comparison between different measurement procedures, analyses of mathematically defined lines and surfaces as well as measurements on real surfaces have been performed. The stereological consequences have been considered.
A restrictive use of the fractal analysis results as an indicator of size, shape and self-similarity is recommended. If results obtained by quantitative microscopy at different magnification-resolution levels are to be compared, fractal analysis may be of advantage. The actual choice of resolution should yet be determined from the physical relevance of the geometrical details.
TL;DR: The concept of helicity-fluctuation hierarchy as generating the fractal structure of turbulence in conjunction with renormalization-group theory are utilized to renormalize the iterative solution of the Navier-Stokes equation in all orders of perturbation theory as mentioned in this paper.
IBM1
TL;DR: In this article, the authors analyzed transmission electron micrographs of thin evaporated gold films with thickness varying from 6 to 10 nm and found that Au-Au and Au-substrate interactions set a natural correlation length of order 10nm.
Abstract: Transmission electron micrographs of thin evaporated gold films with thickness varying from 6 to 10 nm were analyzed by computer. The films cover the range from electrically insulating to conducting and thus span the 2D percolation threshold. The computer analysis allows the direct comparison of actual geometric cluster statistics with both the scaling theory of percolation and Mandelbrot's fractal geometry. We find that Au-Au and Au-substrate interactions set a natural correlation length of order 10nm. Small clusters are dominated by these effects and have simple almost-circular shapes. At larger scales, however, the irregular connected clusters are ramified with a perimeter linearly proportional to area. Near the percolation threshold the large scale power-law correlations and area distributions are consistent with the scaling theory of 2nd order phase transitions. In the fractal interpretation, we demonstrate that the boundary of all clusters is a fractal of dimension D=2 while the largest cluster boundary has a fractal dimension Dc≈1.9. Moreover, many of the usual analytic scaling relations between universal exponents are shown to have fractal geometric basis.
TL;DR: In this article, the fractal dimension of the strange attractor in the map x = ax - sgn (x ) + bz, z = x. The method is based on the construction of the unstable manifolds of period-two points.