J. Kalda (Estonia)
Multifractal spectra are derived for 1- and 2-dimensional cross-sections of passively convected scalar fields; 2- and 3-dimensional single-scale velocity fields in the absence of KAM surfaces are considered. Both the Kraichnan model and real flows with non-zero correlation time are studied. The calculation of f(alpha)-curves is based on the probability density function of the stretching factors of fluid elements. It is shown that strict multifractality holds only for small values of alpha. New multifractal scalar field - ``harmfulness'' - is suggested to describe the propagation of environmentally dangerous substances.
V. V. Uchaikin (Russia)
Multiple scattering of classical particles on a system of point scatterers distributed in space in a fractal fashion is considered in the small-angle approximation. The asymptotic regime of this process is described by a fractional differential equation generalizing the ordinary diffusion in the angle space equation. The solution of the problem is presented both in terms of stable distributions and in terms of the Fox-functions. Main features of the obtained solutions are discussed.
Michael I. Tribelsky (Japan)
A qualitatively new type of spatiotemporal chaos at onset is discussed. The chaos may occur in a degenerate system with a family of spatially uniform states reduced to each other by a certain symmetry transformation. If such a system undergoes a symmetry-breaking instability against spatially periodic perturbations with a finite wavenumber, its behavior beyond the instability threshold is quite different from that for the system where the degeneracy is lifted (e.g., with an external field). The degeneracy causes dramatic changes in the spectrum of perturbations of the system, so that a new subband of slowly varying modes centered around zero wavenumber comes into being. Coupling of long-wavelength modes from the new subband and short-wavelength ones associated with the symmetry-breaking instability may result in chaos at onset. Contrary to the known types of chaos at onset the chaos is characterized by smooth interplay of different scales, while defect generation does not play an important role. Turbulent modes are soft, so the chaos may be regarded as a dynamical macroscopic analog of second order phase transition. A number of examples ("free-slip" convection, electroconvection in a homeotropically-aligned nematic layer, a generalized Burgers equation, etc.) is considered. In all the cases quantitative features of the chaos remain the same regardless dramatic difference in the governing equations. For the generalized Burgers equation the detailed study of the chaos is carried out based upon computer simulations. The spectrum of Lyapunov exponents is obtained for several different system sizes for fixed values of the control parameter. The Lyapunov dimension and the Kolmogorov-Sinai entropy are calculated and both shown to exhibit extensive and microextensive scaling, i.e., grow in proportion to the system volume V for large enough V. The distribution functional is shown to satisfy Gaussian statistics at small wavenumbers and small frequency.
Keywords: pattern formation, short-wavelength instability, degeneracy, Goldstone mode, chaos, Lyapunov dimension, ergodicity.
A. A. Tsonis, P. J. Roebber and J. B. Elsner (USA)
Scale invariance (scaling) in a time series of an observable quantity is a symmetry law which, when it exists, can provide unique insights about the process in question. It describes variability and transitions at all scales and is often a result of nonlinear dynamics. It is well known that the spectra of atmospheric and climatic variables possess considerable power at low frequencies. Since spectra often associate with scaling processes, it is reasonable to suppose that a search for scaling laws in climatic data might be fruitful. Consequently, the search for scaling in such data over the past decade has produced some exciting ways to describe climate variability. In the past and lately, there has been a growing interest in the existence of uniform in space temporal scaling laws for observable properties of the climate system, since such a property would provide a common rule describing temporal variability everywhere on the globe. Here we show that in spatially extended systems, uniform in space scaling demands that global averages be time invariant. A corollary to this is that where global averages do exhibit temporal variability, as in our climate system, spatial variation in scaling properties is required.
D. L. Rail, B. I. Henry and S. D. Watt ( Australia)
A simple evolutionary model is introduced for neural development along the lines of the Bak-Sneppen model for biological evolution of an ecology. The model represents a set of neurons and their connections together with associated synaptic weights. Evolution of the system is studied for different model fitness functions of the synaptic weights. The model systems exhibit Darwinian evolution of the synaptic weight space towards maturation.
V. Pean, M. Ouayoun, C.H. Chouard and B. Meyer (France)
Box counting method allows to measure the eventual fractal dimension (D) of a single dimension temporal signal. However its accuracy varies as a function of the frequency sampling (Fs) and the duration of the tested signal (Sd). Consequently, as it is impossible to highly increase Fs, this method is not suitable for short physical signals D measurement. Thus, we designed a semi-continuous box counting method (SCBC) allowing a better approach of the small scales of the signal, especially useful in case of short single dimension temporal signal. Let N = number of samples of the tested signal. SCBC provides with the first M points of the graph log - log owing to the dyadic division of boxes at large scales up to a certain box size SM, such as SM = 2M/Fs. Then, at smaller scales, for each successive point the box size decreases by 1/Fs, that provides the - log with a large number of points. Thus, when N/S(M+x)/Fs is not a whole number, the analyzed signal is peripherally and symmetrically reduced in abscissa and ordinate, so that a whole number of boxes is obtained. But these truncated samples are then reintroduced for designing following boxes. Using SCBC we measured D of mathematical signals which D is known, and compared these results to those obtained using the classic dyadic box counting method.
D.C. Lin and R.L. Hughson (Canada)
Many natural processes can be characterized by their scale-invariance property. In this study, we present the results of potential multiple scalings in the long-term heart rate data from young healthy adults subjected to normal daily activity. Our approach is based on the direct check of the probabilistic structure of the increment process. Results from fractional Brownian motion are compared and the generating mechanism for multiple scaling is discussed in the context of scale-invariance formalism.
Myung-kul Yum, Jung-hye Hwang and Moon-il Park (Korea)
The purpose of this study was to show abnormal fractal correlation and irregularity of heart rate behavior were altered in intrauterine growth restricted fetuses (IUGR group) and fetuses whose mothers had maternal pregnancy induced hypertension (PIH group). We analyzed fetal heart rate data of 5000 points in normal (n=98), IUGR (n=45), and PIH (n=46) fetuses, with their gestational ages > 38 weeks and without any perinatal complication. We calculate approximate entropy for the quantifying irregularity, and short-range (<= 80 beats, alpha1) and long-range (>80 beats, alpha2) fractal scaling exponent for quantifying the fractal correlation properties. We also performed spectral analysis. In the IUGR group, statistical, the spectral measures, alpha2, and alpha2/alpha1 were significantly higher and the approximate entropy was significantly lower than in the normal group. In the PIH group, alpha1 was significantly lower, and alpha2 and alpha2/alpha1 were significantly higher. The fetuses associated with either IUGR or PIH, although they are not severely compromised, showed abnormal fractal and/or irregular heart rate behavior.
P. Streitenberger (Germany)
The process of coarsening by curvature driven interface motion of fractal interfaces in two-dimensional space is studied by analytical and numerical methods. A statistical model is presented, which allows an analytical treatment of the main features of coarsening of a fractal interface. For non-conserved motion, the interface is described by a statistical distribution function of size scales, which obeys a continuity equation in size space. The solution of the continuity equation yields, for a self-similar initial distribution function, the time development of the interface in terms of a time dependent size distribution function, which exhibits a growing lower characteristic length scale and leads to a power-law decay of the total interface length. The effect of coarsening on the scale of observation is discussed.
A. Barbe and F. von Haeseler (Belgium)
We demonstrate how to generate fractals by using a combination of a cellular automaton and a so-called k-invariant substitution. Such a substitution can be considered as a generalization of substitutions related to k-Fermat cellular automata, a notion which allows summarizing the work of several authors on fractals generated by cellular automata in a unifying way. It will be shown that, if both the local evolution rule of a cellular automaton and the k-invariant substitution exhibit certain symmetries, the fractal limit set of a properly rescaled graphical representation of the orbit of the automaton induced by an iterated substitution, inherits these symmetries.
M. Haase (Germany)
Generally, the irregular behaviour of a function is described by its Hoelder exponents quantifying the strength of the singularities. In most cases, where the function does not contain oscillating singularities, the Wavelet Transform Modulus Maxima method (WTMM) allows a reliable estimation of the singularity spectrum [1,2,3,4], i.e. the Hausdorff dimension of the set of all points with the same Hoelder exponent. However, in the presence of oscillating singularities the standard WTMM method gives irrelevant information on the Hoelder regularity of the function. In general, two exponents are necessary to describe the singular behaviour of a function, namely the Hoelder exponent and an oscillation exponent describing the local power law divergence of the instantaneous frequency [5,6]. Therefore, the singularity spectrum of general functions depends on both exponents. In order to extract Hoelder exponents and to quantify at the same time the oscillating behaviour we propose to use a family of complex progressive  wavelets with an increasing number of vanishing moments. Based on these wavelets a partial differential equation for the wavelet transform is derived. An algorithm is developed, which allows the direct tracing of maxima lines. For this purpose we derive a set of ordinary differential equations for the extrema lines which can be integrated directly. Based on the skeleton of maxima lines a partition function is defined. Exploiting the scaling behaviour of this partition function and applying the Legendre transform to the exponent, the two-dimensional singularity spectrum can be calculated. In practice, due to finite length and noise in data sets, the accuracy of the Legendre transform might decrease. To avoid this problem, we propose a generalization of the canonical method described in  to two-dimensional spectra, which allows a direct calculation of the singularity spectrum.
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E. Hocevar (Austria)
Global IFS seem to be suited best for compressed encoding of natural objects which are in most cases self affine even if not always exactly. Since affine transformations - the IFS-Codes - resp. the union of all their orbits generate an object (an IFS-Attractor), the detection of a non minimal set of these orbits solves the inverse IFS-Problem by calculating a superset of IFS-Codes, which has to be minimized. Here a method is presented how these orbits (in particular those on the object boundary) can be calculated. Therefore a generalized convex hull - the Sigma-Hull - is defined. This fractal hull is bounded by log spirals, that curves formed by the orbits. It is shown that log spirals can be represented by a continuos function of powers of affine maps and that by using this "spiral equivalent" the generating transformations of the orbits by which an IFS-object is bound, can be calculated in the x/y-plane. Further more this representation can be used for the classification of the detected orbits, necessary to calculate the IFS-Codes of a minimal IFS from their generating transformations, subsequently.
A. Saucier (Canada) and J. Muller (Norway)
We introduce a method to generate statistical textural transforms that improves the treatment of non-stationarity and leads to a sharper detection of the boundaries between distinct textures (texture segmentation). This method is based on a sliding window processing with fixed size. The basic idea proposed by the authors is to readjust the measuring window around each pixel so as to maximize homogeneity. We use this method with the dimensions Dn(q) that are derived from the Generalized Multifractal Analysis formalism, to show that the Dn(q)s can detect and quantify departures from multifractality, while providing the analogue of the classical generalized dimension if the measure is multifractal.
M. E. Pietrzyk (Germany)
In this paper possibilities of a stabilization of large amplitude fluctuations in an intracavity-doubled Nd:YAG laser are studied. First the stabilization of the by an increase of the number of modes is analyzed. It is shown that for small nonlinearity this dependence is linear and agrees with theoretical predictions. However, for large values of nonlinearity the minimal number of modes is larger than the number which follows from the theoretical predictions. This is because of a cancellation of longitudinal modes, which occurs during the evolution. For very large nonlinearity this cancellation is so big that only few modes remain. Therefore, a large number of simultaneously oscillating modes cannot be achieved and the stabilization cannot be obtained in this way. However, another method based on an increase of the strength of nonlinearity, is proposed. This stabilization occurs for such a large value of nonlinearity, for which during evolution all modes, besides a single one, are canceled. In this case a steady-state solution, stable against small perturbations, arises.
Benhua Fei (P. R. of China)
An experiment of wood drying at different temperatures was conducted to show the fractal nature of the pore space within wood. Cubic blocks made from ginkgo (Ginkgo biloba) and Chinese chestnut (Castanea mollissima ) wood were used. Samples were dried in oven at the temperature of 20, 40,60 and 100 deg. C, respectively. All the drying procedures lasted four hours. The mass was weighted and the dimensions were measured immediately for each sample when every procedure of drying ended. The fractal dimensions of the samples were obtained from the measurement. Results showed that the fractal dimensions increased with the drying temperature from 20 to 60 deg. C and the fractal dimensions keep a constant in the main. Results from different species and for different temperatures, and suggested the fractal dimension was a new parameter to characterize the porosity of wood.
J. M. Li, L. Lu and M. O. Lai ( Singapore)
Applying variation-correlation method to images of fractography obtained from the scanning electron microscope (SEM), it has been found that there exists a fractal characteristic length within which the fractured surfaces are fractal. Investigation shows that the fractal characteristic length can represent the statistical maximum size of texture of the SEM image. Multi-magnification fractal analysis has shown that fractography cannot be described by a single fractal dimension but rather a series of fractal dimensions. Fractal study on the fractured surfaces of gray iron with different grain size has shown that there exists a positive relationship between tensile strength and fractal dimension. However, no essential relationship between impact toughness and fractal dimension for HP26Cr35Ni alloy could be obtained. Therefore, fractal dimension is sensitive to changes caused by geometric factors such as grain size and is suitable to quantitatively describe the irregularity of fractography, and is not universal for a given fracture mode.
Y. Huang (P. R. of China)
Chinese acupuncture-moxibustion, a therapeutic method by needling in or burning moxa (a kind of herb) on the acupuncture points (acupoints) of the body surface, is used to treat diseases. There are over one thousand acupoints on the body surface. The whole acupoint system can be divided into many different hierarchical holographic units. The function of acupoint system possesses fractal structure. This paper studied the distribution and the innervation of acupoints in acupuncture holographic units. We have found that the acupoints distribution graph in acupuncture holographic units affected by fetus development. The acupuncture holographic units on the body surface can be divided into two kinds. One is that their acupoints distribution graphs are similar to the shape of the human nerve-embryo. The acupoints receive segmental innervation or superasegmental (nerve network) innervation. The other is that their acupoints distribution graphs are similar to the shape of the mature fetus. The acupoints receive superasegmental (nerve network) innervation. We have found that the concept of "self-similar system" can be used to describe the former, the concept of "self-affine system" can be used to describe the latter.
P. Woloszyn ( France)
The main topic exposed in this paper concerns the reduction of the diffusion volume to convex isotropic Euclidean structures as polygons and convex polyhedrons, in order to estimate the diffusive acoustic behavior of a fractal structure. By quantifying the multiscale distribution of the shape roughness in the diffusion space, we will attempt to define this diffusion process as a geometric structure formation of the propagation interface.
I. I. Salakhutdinova (Russia)
This paper is devoted to the study of fractal properties of the distribution of large-scale solar magnetic fields in the spatial aspect. The study is based on synoptic maps from the Kitt Peak, Maunt Wilson and Wilcox observatories for the entire observing period from 1986 to 1998, from 1959 to 1976, and from 1976 to 1998, respectively. Maps show the longitudinal-latitudinal distribution of magnetic field flow (or strengths). Maps with gaps were excluded from consideration. As a result, the study made in this paper has shown that the spatial distribution of large-scale magnetic fields on the Sun reveals a fractal character. This manifests itself in the power scaling of statistical characteristics of magnetic fields such as the variance. Also, two typical spatial scales are distinguished, which appear to characterize two systems of solar large-scale magnetic fields organized into giant and supergiant cells. The fractal dimensions of the space of the strength function of magnetic fields show a high degree of their irregularity and alternation on separate synoptic maps. Time variations of the exponents corresponding to the fractal dimension show a cyclic character in many respects, which testifies to a change of statistical and associated physical properties of solar magnetoplasma with cycle phases.
U. Lepik (Estonia)
Nonlinear vibrations of elastic-plastic beams are considered. The end-sections of the beam are pinned so that membrane forces appear. The equations of motion are integrated by the Galerkin method. Chaotic behavior of the solution is discussed; for this purpose displacement time histories, phase portraits and power spectrum diagrams are put together.
I. A. Guerrini, A. J. Spadotto, E. Santos (Brazil)
Diffusion has been considered as a fractal phenomenon once is identified as a Brownian motion type of particle displacement. Recent results have attributed to the soil-water movement a fractal behaviour, either as a regular Brownian motion (rBm) near the soil saturation condition or as a fractional Brownian motion (fBm) in a region of soil far from that condition. Here it is shown that in order to keep the universality of the simple power law that regulates the variation in time for soil-water movement , which is been assigned as a solution to this diffusion process, it is necessary to maintain a minimum of water displacement. This can be understood as a critical point beyond which the power law, and the fractal character as well, does not hold anymore. Long-range infiltration times as well as large negative water pressure entries in the soil lead to soil-water movement that can not be considered as a fractal phenomenon and so is not a real diffusion process anymore. The universality of this movement is then limited to the region where the water flow can be associated with a power law solution what exclude lower water content soil conditions. These results could bring some light into the understanding of the soil-water functions behaviour in dry soil conditions and for long-range infiltration times what has long been a problem for soil physicists. (This work has been partially supported by Brazilian Councils, CNPq and FAPESP).
R. I. Zainetdinov (Russia)
A general model linking dynamics of informational entropy and the self-organization process in open, steady state, nonequilibrium system is proposed. Formulas for dynamics of the informational entropy flow and its rate are developed with respect to random process of influences exerted upon a system. It was revealed that the open system responds to a strong change of conditions by steep growth of the informational entropy flow up to a maximum value at the critical point associated with the self-organization process. An example of self-organization during elasto-plastic deformation of metal is considered.
R. I. Zainetdinov (Russia)
The purpose of the paper is to introduce a new application of the multifractals in the reliability engineering, safety analysis, historical chronology and others fields where we deal with series of events of various natures. The Bernoulli trial process, which is a model for any experiment with two possible outcomes, has been considered. Methods of the fractal theory have been applied to reveal a hidden temporal structure in data sets generated by Bernoulli trials. The multifractal theory is a good basis for analyzing the statistical data on reliability and safety, and provides probabilistic evidence for the existence of a multiplicative cascade hidden in the temporal complexity of a sequence of failures. We have applied the wavelet transform to a series of statistical data obtained from tests and inspections of technical state under real service conditions. For the approbation of the technique, a computer simulation study has been done.
P. S. Addison, B. Qu and G. Pender (UK)
The transport of surface pollutants in the coastal zone is modelled using a modified particle tracking diffusion model. The new model uses fractional Brownian motion (fBm) functions to produce superdiffusive spreading of the synthesised pollutant clouds. The model is tested on a numerical model of a coastal bay recirculation zone.
P. S. Addison, L. T. Dougan, A. S. Ndumu and W. M. C. Mckenzie (UK)
An attempt is made to find the fractal cutoff of crack profiles on the tension face of concrete beams subjected to uni-axial bending. Previous work by the authors has shown that such cracking can be interpreted as a non-Fickian diffusive phenomenon resulting from a self-affine random fractal process: specifically fractional Brownian motion (fBm). In addition, a spatial description of the cracking geometry can be found from experimental data using both a (Hurst) scaling exponent and a diffusion-type coefficient. Herein the authors find that the fractal description of the crack profiles extends down to less than 0.75mm. The use of a scanning electron microscope to probe the crack profile (and surface) at smaller scales is discussed and the synthesis of crack surfaces using fBm is described briefly.
A. S. Ndumu and P. S. Addison (UK)
Monte Carlo simulations of subsurface flow and contaminant transport of a non-reactive solute plume by steady-state flow with a uniform velocity were performed in a two-dimensional synthetic heterogeneous porous media whose hydraulic conductivity is non-stationary and described by multi-scale fractional Brownian motion. Analysis of the flow and transport results indicates that the longitudinal velocity variance is nearly constant in the longitudinal direction while in the transverse direction it assumes a parabolic shape. The velocity variance is maximum at the impervious boundaries and decreases in transverse direction with distance from the boundaries reaching the minimum value at the domain centre. We observe that the displacement covariance is anomalous or non-Fickian at all times in the dispersion process irrespective of the Hurst exponent and its grows temporally faster than linearly.
A. K. Evans, S. J. Nimmo and M. D. London (UK)
The bispectrum is the natural third-order generalisation of the power spectrum. It provides information about correlations between different Fourier components of a signal or image, and about the statistics of Fourier phase. A number of numerical and experimental studies of the bispectra of chaotic systems have been published. In this paper we present the first analytical calculations of the bispectra of chaotic dynamical systems. First, for a generalisation of the classical sawtooth or Renyi map, we calculate the bispectrum using symbolic dynamics. Also, for intermittent systems, we calculate the bispectrum using the relationship between these systems and renewal processes. We review the results of these calculations, drawing some conclusions about the characteristic features of the bispectra of chaotic systems, and compare them with the features of some financial time series.
N. G. Makarenko, L. M. Karimova, A. G. Terekhov (Kazakhstan) and M. M. Novak (UK)
The hypothesis about multifractal nature of radioactive contamination due to nuclear explosions on Semipalatinsk test site (STS) in Kazakhstan is verified. The fields of the terrestrial contamination have extremely high variability, caused by a number of natural processes. All the results we obtained support the existence of multifractal nature in the terrestrial radionuclides contamination in the test site area and two bordering regions. Multifractal formalism and computation topology methods made it possible to distinguish between radionuclide isotopes K, Th, U, Cs, specially for measurements made on test site. Taking into account the multifractal properties of man-made Cs, the existence of dangerous for human health conditions can be established on all contaminated territories. These "hot spots" cannot be detected by the traditional in Kazakhstan technics of measurements from air. Moreover, the finding of self-similarity could be the base of new methods of diagnostics of large territories.
D. N. Belyashov, I. V. Emelyanova, A. V. Tichshenko, N. G. Makarenko, L. M. Karimova (Kazakhstan) and M. M. Novak (UK)
In the framework of the Comprehensive Test Ban Treaty, development of reliable methods to discriminate between underground nuclear explosions and earthquakes at regional distances (less than 2500km) continues to be very important, especially in connection with the last (in May, 1998) nuclear explosions conducted at Indian and Pakistan test sites. Since the lithosphere is a fractal, we suppose the signals, which propagate through the media, inherit its 'self-similar' (scaling) features. We assumed that these features of explosions and earthquakes or their topological reconstructions (embeddings) have to be different. Scaling reflects correlations of higher order then it is possible to estimate by linear discriminating methods and can be used as base of non-linear discrimination. We propose to build a universal geometrical model of a seismic signal using the canon algorithm of F. Takens and to estimate scaling of the model. The scaling features were used as patterns of seismic signals for entering them into an artificial neural network. Records of nuclear explosions and earthquakes from different regions were included into the training set. The net was trained to classify types of seismic events. Results have shown 89% correct classification of the unknown signals. As additional tools for distinguishing between nuclear explosions and earthquakes we propose to use Hurst's method and the cross correlation method. Results of using these methods are demonstrated on examples of some explosions and earthquakes.
N.K. Indira (India)
The nature of non-linearity can be understood by analysing
observations varying over time given as a time series. Sea level
variations are known to occur at various time scales. They may be caused
by the host of geo-dynamic and climatic factors. Sea level variations
reflect the dynamics of the nonlinear atmosphere-ocean-lithosphere system.
The tide gauge data, which is nothing but the time series data of annual
variations of sea level, has been obtained and analysed. The analysis is
carried out for the coastal stations in India, Singapore, Thailand and
China. The aim is to compare the variations in different oceans like
Indian Ocean, Pacific Ocean etc. Since sea level changes reflect the
dynamics of the nonlinear system, fractal analysis of the data at
different stations through IFS (Iterative Function System) determines the
fractal dimensions. It is inferred that the range of the fractal dimension
of sea level variations remain unchanged for the stations in Singapore,
Thailand and China when compared to those obtained earlier for the Indian
coastal stations (Indira et al (1996)).
Indira, N.K.; Singh, R.N. and Yajnik, K.S. 1996, 'Fractal analysis of sea level variations in coastal regions of India', Current Science, Vol. 70, No. 8, pp. 719-723.