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Fractal dimension: Limit capacity or Hausdorff dimension?
Abstract
It is shown that for often encountered nonpathological sets the limit capacity and the Hausdorff dimension differ. This raises the question in the context of physics as to which of these should be called the ``fractal'' dimension.
... Katz and Thompson (1985) used scanning electron microscopy (SEM) and optical microscope to study the sandstone pore, and their results showed that sandstone pore had obvious fractal phenomena. However, a single fractal dimension cannot perfectly describe the characteristics of the fractal signal (Stanley and Meakin, 1988;Essex and Nerenberg, 1990;. Many examples suggest that many images exhibit significant visual difference but have very similar fractal dimension. ...
... Real complex systems have self-similar characteristics, both in geometry and in mass distribution of space. Multifractal and generalized fractal provides a good way to characterize this distribution and weakens single fractal's deficiencies (Essex and Nerenberg, 1990). The details of theory have been discussed in earlier publications by researchers (e.g., Martín and Montero, 2002;Grau et al., 2006;Ke et al., 2015;Ge et al., 2015). ...
... Real complex fracture systems not only have surface-area-based self-similar characteristics but also have characteristics based on mass or height proportion distribution. Multifractals and generalised fractals are powerful in characterising this mass or height proportion [60]. The generalised entropy H q (δ) and generalised dimension D q are defined as [61]: ...
... In order to unify the measure standard of fractal figures, fractal dimension is proposed by Hausdorff. At present, Hausdorff dimension is commonly used to describe the fractal figure (Essex and Nerenberg 1990). In the field of mathematics, the definition of Hausdorff dimension has very strict criteria. ...
Fractal theory, with its novel architectures inspired by nature, provides some novel concepts for smart reactor design. Here, researches on the applications of fractal theory to micro-reactor design are reviewed, in term of its high surface area-to-volume ratio, rapid and direct numbering-up, safety, and precise control. In addition, two designs of fractal micro-reactor are introduced as typical examples. First, the H-type fractal structure is considered in the context of the design of a double-plate micro-reactor, which is used for photocatalytic reactions of CO 2 . Second, applications of fractal Hilbert curves are considered in the design of channel structures for gas-liquid reactions. These two fractal micro-reactors can be fabricated via 3D printing technology and used for CO 2 conversion under mild conditions.
... Referente a la geometría fractal de los atractores extraños, la noción de autosimilitud que presentan dichos atractores puede ser cuantificada por la llamada dimensión fractal [Essex & Nerenberg 1990;Farmer et al. 1983;Grassberger 1981;Grebogi et al. 1984]. En efecto, la dimensionalidad del espacio de estados está estrechamente relacionada con la dinámica del sistema. ...
This thesis contributes to a better understanding of the syncrhronization phenomena in chaotic dynamical systems and reports a mathematical formalism to estimate the energy cost involved in the synchronization process.
... However, fractal geometry has its drawback since it cannot depict the heterogeneity and local scale properties of the object (Essex and Nerenberg, 1990;Hargis et al., 1998;Tang et al., 2003;Brewer and Girolamo, 2006). Recent studies indicate that multiple fractal dimensions were needed to describe statistical scaling behavior in many fields such as geosciences, soil and material sciences (Bittelli et al., 1999;Posadas et al., 2001;Vázquez et al., 2008;Ferreiro and Vázquez, 2010;Martínez et al., 2010;Kwasny and Mikuła, 2012). ...
... The fractal dimension as defined by Eq. (3) is usually identified with the Hausdorff-Besicovitch dimension and is known as the capacity dimension D O (see, e.g., Tsonis 1992). Nevertheless, there is a fine distinction between the Hausdorff-Besicovitch dimension and the capacity dimension: while the former is obtained by covering the set minimally with hypercubes that may be different in size, the latter is obtained with the same process except that the hypercubes are the same size (Essex and Nerenberg 1990). ...
Mesoscale cloud patterns are analyzed through the application of fractal
box dimensions. Verification of fractal properties in satellite infrared
images is carried out by computing box dimensions with two different
methods and by computing the fraction of cloudy pixels for two sets of
images: 174 are considered the `control series,' and 178 (for
verification) are considered the `test series.' The main instabilities
in the behavior of such dimensions are investigated from the simulation
of circles filling space in several spatial distributions. It is shown
that the box dimensions are sensitive to the increase of the area
covered and to the spatial organization-that is, the number of cells,
the spatial clustering, and the isotropy of the distribution of pixels.
From a principal components analysis, the authors find six main patterns
in the cloudiness for the control series. The three main patterns
related to enhanced convection are the massive noncircular spread
cloudiness, the highly isotropic distribution of cloud in several cells,
and the most circular pattern associated with mesoscale convective
complexes. The six patterns are separated into a cluster analysis, and
the properties of each cluster are averaged and verified for the test
series. This method is a simple and skillful procedure to recognize
mesoscale cloud patterns in satellite infrared images.
... Referente a la geometría fractal de los atractores extraños, la noción de autosimilitud que presentan dichos atractores puede ser cuantificada por la llamada dimensión fractal [Essex & Nerenberg 1990;Farmer et al. 1983;Grassberger 1981;Grebogi et al. 1984]. En efecto, la dimensionalidad del espacio de estados está estrechamente relacionada con la dinámica del sistema. ...
Sand-conglomerate reservoir has been scarcely studied, and there is no effective method available for quantitative characterization of pore structure of such reservoir. In this paper, a multifractal study was made on the Triassic Karamay Formation sand-conglomerate reservoir in the Mahu rim region, the Junggar Basin, by using a variety of high-resolution analysis methods, such as Micro-CT, QEMSCAN and MAPS, in order to quantitatively characterize the heterogeneity of pore size distribution, relative differentiation of large and small pores, and mineral composition. The results reveal that the multifractal parameters have more influence on permeability than on porosity. The smaller the Δα (the multifractal spectral width) and the larger the Δƒ (the difference in fractal dimension of the maximum and minimum probability subsets), the better the reservoir physical property. To some extent, the relationship between multifractal parameter and mineral composition provides an opportunity to reflect the diagenesis. There is a positive correlation between the clay mineral content and the heterogeneity of the microscopic pore structure of reservoir. Kaolinite and chlorite cementations are the most significant factors that damage the reservoir pore space. This understanding matches well with the MAPS and QEMSCAN results. With outstanding advantage in quantitatively evaluating the heterogeneity of pore structure of sand-conglomerate reservoir, multifractal provides a new idea and method for quantitative characterization of pore structure of other heterogeneous oil reservoirs.
The nonlinear and nonstationary nature of structural damage (bridges, buildings, etc.) brings a great challenge to structural health monitoring (SHM). The correlation dimension is an effective index to capture the process nonstationarity. But the traditional algorithms for computation of correlation dimension have fairly high complexity O(N2)and thus are not suitable for online monitoring. To tackle this challenge, this paper presents a novel quasi-recursive correlation dimension algorithm (QRCD) for online damage detection of structures. It can significantly alleviate the complexity of computation for correlation dimension to approximate O(N), and thus, make the online monitoring of nonlinear/nonstationary processes using correlation dimension much more applicable and efficient. The case studies show that for detection of process nonstationarity (occurrence of damages), the EWMA control charts using correlation dimension have shorter average run length (ARL) than the control charts using wavelet coefficients. And the proposed method significantly reduces the computational complexity in terms of computation time by approximately 90% (for different levels of damage) as opposed to the traditional methods. Moreover, the developed methodology is less influenced by process noise compared to the wavelet analysis based approaches. All these results demonstrate that our proposed method is both effective and efficient for online SHM.
It is shown that under simple restrictions a series converges provided it’s set of terms has Kolmogorov dimension strictly smaller than .
This Resource Letter provides an introductory guide to the literature on nonlinear dynamics. Journal articles and books are cited for the following topics: general aspects of nonlinear dynamics and applications of nonlinear dynamics to various fields of physics, other sciences, and a few areas outside the sciences. Software and Internet resources are given also. (C) 1997 American Association of Physics Teachers.
A technique, called forecast entropy, is proposed to measure the difficulty of forecasting data from an observed time series. When the series is chaotic, this technique can also determine the delay and embedding dimension used in reconstructing an attractor. An ideal random system is defined. An observed time series from the Lorenz system is used to show the results.
Oscillations in the chattering region in plots of final action and collision time as a function of the initial vibrational phase of the diatom in collinear He+H+2(ni=0) collisions are shown to have characteristics of fractals with a capacity dimension 1.38–1.68 over a wide range of translational energies. For energies above the reaction threshold, the fractal zones are shown to occur between reactive and nonreactive bands and are related to known quantal reactive scattering resonances.
For a model diatomic molecule-surface collision, we show that the extent of chaoticity, as evidenced in action-angle and lifetime plots, decreases dramatically with an increase in the translational energy of the molecule. In addition to the existence of a threshold for the dissociative chemisorption, there is also an antithreshold at higher energies. The implications for the existence (or lack thereof) of chaos/fractals in molecule-surface collisions are also discussed, particularly as related to translational-vibrational energy redistribution.
Fractal superlattices consist of a series of thin layers of two semiconductor materials alternately deposited on each other with widths corresponding to the rules of construction of a fractal set. The scattering of electrons in superlattices is obtained using the transfer matrix method for generalized Cantor fractal potentials that are characterized by a lacunarity parameter. The numerical results show the self-similarity of the reflection coefficient and the appearance of lacunarity-independent energies with perfectly transparent tunneling due to the bound states of the particle in each of the individual potential wells. © 2006 American Association of Physics Teachers.
The present paper explores the physical structure of two fundamental concepts, spacetime and fractals, both of which suffer from a lack of a satisfactory definition. We carefully consider the essential properties of these concepts and attempt to answer the questions: Have space and time an independent existence (with respect to matter and fields) and can we speak of a fractal space or a fractal time? How can we translate into a physical model the geometrically (formal) picture of a fractal?Since spacetime represents the arena in which the rest of physics unfolds it is of interest to determine at how many different levels fractal spacetime can make its influence felt. We distinguish four levels of fractalization. A real intrinsic fractalization of space may arise on the first (quantum gravity) and the fourth (cosmic) level where the texture of space and strong nonlinear physics of space are involved.We show that even if the mathematical fractal curves are nondifferentiable (or finite piece-wise smooth curves in the case of real fractals), we can still study their properties applying the Finsler theory of sprays which approximates physical fractals and may have the same general properties as ideal fractals (self-similarity, self-affinity, etc). This point of view is further supported by generalising the concepts of allometry and hierarchy of possible levels of description for physical fractals. In this context a fractal is a physical system which develops by interaction with the local environment. We also mention that fractal space can be studied within the frame of a conformal relativity.As regards the dimensionality of spacetime, we start from the premiss that we cannot mix the four dimensions of spacetime with (possibly infinite) extra dimensions of quantum geometrodynamics. In the evolution of the universe these two types of dimensions originated at different stages of development and different scales of distances. The situation is related to the fact that we cannot mix quantum or relativistic physics with classical physics. It appears that it is superfluous to look for a demonstration of the assertion that our spacetime is four dimensional. In fact, `spacetime is 4D' is not a theorem but a fundamental axiom or principle confirmed by empirical facts like any other axiom we apply in physics.Following El Naschie's conjecture that gravity is a phenomenon caused by time flowing at varying speeds (`multiple-time scale of time'), we elaborate the concept of time-gravitation induction: `a (nonintegrable) variation of time flow generates gravitation and a variation of gravitation generates variable time flow', a (dual) phenomenon similar to electro-magnetic induction duality. This idea is generalised and it is shown that local (nonintegrable, anholonomic and anisotropic) scale transformations lead to new local (fractal) fields which may explain the generation of physical fields by the fractal structure of spacetime. The existence of empty waves (i.e., a sort of de Broglie waves separated from the associated particle) is also a consequence of an intrinsically fractalized spacetime.Special attention is payed to the interpretation of a double-slit experiment and Feynman's path integral. It is shown that geometric excitons can act as diffusion micro-polarizers and may lead, consequently, to the fractalization (and thus to a nondifferentiability) of a trajectory. A fundamental conclusion of the present paper is that nature (space, time, matter and fields) does not fractalize, it is intrinsically fractal. In fact this is the leitmotiv and raisond'être of our work.
Many techniques have been developed to measure the difficulty of forecasting data from an observed time series. This paper introduces a measure which we call the "forecast entropy" designed to measure the predictability of a time series. We use attractors reconstructed from the time series and the distributions in the regular and tangent spaces of the data which comprise the attractor. We then consider these distributions on different scales. We present a formula for calculating the forecast entropy. To provide a standard of predictability, we define an idealized random system whose forecast entropy will be maximal; we then use this measure to rescale the forecast entropy to lie in the range [0,1]. The time series obtained from several chaotic systems as well as from a pseudorandom system are studied using this measure. We present evidence that the forecast entropy can be used as a tool for determining optimal delays and embedding dimensions used for reconstructing better attractors. We also show that the forecast entropy of a random system has completely different characteristics from that of a deterministic one.
Underwater acoustic transients can develop from a wide variety of
sources. Accordingly, detection and classification of such transients by
automated means can be exceedingly difficult. This paper describes a new
approach to this problem based on adaptive pattern recognition employing
neural networks and an alternative metric, the Hausdorff metric. The
system uses self-organization to both generalize and provide rapid
throughput while utilizing supervised learning for decision making,
being based on a concept that temporally partitions acoustic transient
signals, and as a result, studies their trajectories through power
spectral density space. This method has exhibited encouraging results
for a large set of simulated underwater transients contained in both
quiet and noisy ocean environments, and requires from five to ten MFLOPS
for the implementation described
Several different dimensionlike quantities, which have been suggested as being relevant to the study of chaotic attractors, are examined. In particular, we discuss whether these quantities are invariant under changes of variables that are differentiable except at a finite number of points. It is found that some are and some are not. It is suggested that the word dimension be reversed only for those quantities have this invariance property.
Several definitions of generalized fractal dimensions are reviewed, generalized, and interconnected. They concern (i) different ways of averaging when treating fractal measures (instead of sets); (ii) “partial dimensions” measuring the fractility in different directions, and adding up to the generalized dimensions discussed before.