Multifractal Phenomena Research Articles

Multifractal detrended fluctuation analysis of pressure fluctuation signals in an impinging entrained-flow gasifier

Abstract On a laboratory-scale testing platform of impinging entrained-flow gasifier with two-burner, the pressure signals are measured at two axial positions with stainless steel water-cooled probes. The probes are traversed radially from the wall to the reactor centerline (spaced 15 cm apart) for six radial positions (0, 3, 6, 9, 12, and 15 cm) to acquire pressure signals. Multifractal detrended fluctuation analysis (MF-DFA) is a method which is accurate and easy to implement, therefore it is used on the pressure fluctuation to examine the multifractal characteristics. It is found that the pressure fluctuation exhibits multifractal characteristics, the analysis of the pressure fluctuation signals under “unstable operating state”, “stable operating state” and also “cold state” suggests that the combustion system is characterized by a dynamical change from heterogeneity toward homogeneity, revealed by a loss of multifractality. Moreover, the analysis confirms the existence of multifractal characteristics in the investigated pressure fluctuation series. Origin of multifractal phenomena of the pressure signal measured in the entrained-flow gasifier is interpreted in terms of the multiplicative cascade process.

Multifractal analysis of pressure fluctuation signals in an impinging entrained-flow gasifier

A study of pressure fluctuation signals in an impinging entrained-flow gasifier was undertaken using multifractal analysis. By applying the method of direct determination of the f( α) singularity spectrum, the values of different radial position pressure fluctuation signals’ singularity spectra were calculated. It was found that the pressure fluctuation exhibited the multifractal characteristics, the comparison of the pressure fluctuation signals from “unstable operating condition” to “stable operating condition” suggests that the combustion system is characterized by a dynamical change from heterogeneity toward homogeneity, revealed by a loss of multifractality. Origin of multifractal phenomena of the pressure signal measured in the entrained-flow gasifier was interpreted in terms of the multiplicative cascade process.

Lq spectra and Rényi dimensions of in-homogeneous self-similar measures

Let for i = 1, …, N be contracting similarities. Also, let (p1, …, pN, p) be a probability vector and let ν be a probability measure on with compact support. We show that there exists a unique probability measure μ on such that The measure μ is called an in-homogeneous self-similar measure.In this paper we study the Lq spectra and the Rényi dimensions of in-homogeneous self-similar measures. We prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the in-homogeneous case. In particular, we show that in-homogeneous self-similar measures may have phase transitions, i.e. points at which the Lq spectra are non-differentiable. This is in sharp contrast to the behaviour of the Lq spectra of (ordinary) self-similar measures satisfying the open set condition.We also present a number of applications of our results. Namely, we obtain non-trivial upper bounds for the multifractal spectrum of an in-homogeneous self-similar measure, and we provide applications to the study of box-dimensions of in-homogeneous self-similar sets.

Scaling characteristics in aftershock sequence of earthquake

The possible scale-invariant behavior and the clustering characteristics in aftershock sequence of Chi-Chi (Taiwan) main earthquake (ASCCME) that occurred in 1999/9/20/17/47 were investigated by means of some statistical tools: histogram, spectral analysis, and fractal theory. The examined data were constructed from the aftershocks that occurred at the locations defined at longitude 120.1–121.3 and latitude 23.3–24.5 during the 1999/9/20/17/47–1999/9/24/08/13 period. It was found that the aftershock sequence exhibited the characteristic right-skewed frequency distribution and could be well described with the lognormal distribution. Long-term memory and the possibility of scale invariance were first roughly identified through the analysis of autocorrelation and power spectrum, respectively. Scale invariance was clearly revealed with the aid of box-counting method and the box dimension was shown to be a decreasing function of the threshold magnitude level, i.e., the weak and intense regions scaled differently. To verify the existence of multifractal characteristics, the aftershock sequence was transferred into a useful compact form through the multifractal formalism, namely, the τ( q)– q and f( α)– α plots. The analysis confirmed the existence of multifractal characteristics in the examined aftershock sequence. The origin of both the pronounced right-skewness and multifractal phenomena in aftershock sequence might be interpreted in terms of the multiplicative cascade process of the stress in the Earth's crust. A simple two-scale Cantor set with unequal scales and weights was then used to fit the calculated τ( q)– q plot. This model fitted remarkably well the entire spectrum of scaling exponents of the examined ASCCME.

Scaling characteristics in the Taiwan stock market

Some statistical tools, including histogram, spectral analysis and fractal theory, were used on the daily Taiwan stock price index (TSPI) from 1987 to 2002 to examine the possible scale-invariant behavior and the clustering characteristics in Taiwan stock market. It was found that the TSPI data exhibited the characteristic of right-skewed frequency distribution. The long-term memory and the possibility of scale invariance were roughly identified through the analysis of autocorrelation and power spectrum, respectively. The monofractal analysis was then performed by the box-counting method. Scale invariance was clearly found in the time series and the box dimension was shown to be a decreasing function of the threshold index level, implying multifractal characteristics, i.e., the low and high regions scale differently. To test this hypothesis, the time series were transferred into a useful compact form through the multifractal formalism, namely, the τ( q)– q and f( α)– α plots. The analysis confirmed the existence of multifractal characteristics in the investigated time series. The origin of multifractal phenomena in Taiwan stock market might be interpreted in terms of the multiplicative cascade process of stock market information.

Multifractal Characteristics in Air Pollutant Concentration Time Series

One-year series of hourly average air pollutant concentration(APC) observations, including O3, CO, SO2 and NO2,were analyzed by means of statistical tools: histogram, variancemethod, and multifractal analysis, to examine the possible scale-invariant behavior and the clustering characteristics. It was found that all APC measurements exhibit the characteristicright-skewed unimodal frequency distribution and long term memory. A monodimensional fractal analysis was performed by thebox counting method. Scale invariance was found in these time series and the box dimension was shown to be a decreasing function of the threshold APC level, implying multifractal characteristics, i.e., the weak and intense regions scaledifferently. To test this hypothesis, the APC time series weretransferred into a useful compact form through the multifractalformalism, namely, the τ(q)-q and f(α)-α plots. The analysis confirmed the existence of multifractal characteristics in the investigated APC time series. It wasconcluded that the origin of both the pronounced right-skewnessand multifractal phenomena in APC time series may be interpretedin terms of the Ott (1995) proposed successive random dilution (SRD) theory and the dynamics of APC distribution process can bedescribed as a random multiplicative process.

Multifractal critical phenomena in traffic and economic processes

Multifractal critical phenomena with infinite-temperature critical point and with complex coexistence of the infinite and finite temperature critical points are considered and it is shown that strange attractors generated by cascades of period-doubling bifurcations (Feigenbaum scenario) as well as fields of velocity differences in fluid turbulence belong to the former subclass of the multifractal critical phenomena, while the real traffic processes and real currency exchange processes belong to the last (complex) subclass of the multifractal critical phenomena. Data obtained by different authors are used for this purpose.

MULTIFRACTAL DIMENSION SPECTRA IN POLYMER PHYSICS

The notion of non-integer dimension although developed by mathematicians in late 19th – early 20th centuries has found its wide application in physics only in the last few decades. This application was mainly initiated by the pioneering work of B. Mandelbrot who attracted attention to the fact that “fractal” geometry provides an appropriate framework for the description of a whole range of complex structures that are formed from smaller subunits [1]. Whereas such structures are characterized by appropriate fractal dimensions, their growth and spatial correlations [2] are described by a (non-trivial) spectrum of multifractal (MF) dimensions [3,4]. In this paper, we study the properties of diffusion phenomena in the presence of an absorbing polymer. This provides another example of a multifractal phenomenon in condensed matter physics. To this end we use the model proposed by Cates and Witten [5] and derive the MF spectrum in the frames of a field theoretical formalism using the renormalization group [6,7] method. The MF spectrum is related to the spectrum of exponents governing scaling properties of copolymer stars [8]. We calculate this spectrum to the third order of perturbation

Two inverse problems induced by multifractal phenomena

In frame of the phenomena revealing multifractality, two inverse problems are formulated and their solution is outlined; namley, (i) to what extent the indices characterizing scaling properties of dynamical fluctuations (specified by factorial moments) are determined by means of power law indicesτ(q) which characterize scaling properties of the associated frequency moments, the later moments representing asymptotics (for sufficiently large multiplicities) of the former ones, and (ii) to what extent the scaling indicesτ(q) are determined by their asymptotic properties (at large absolute value of the orderq ascribed to the associated frequency moments and taken at a finite and fixed energy).

Multifractal structure in high energy collisions and finite multiplicity corrections

There are established rigorous relations between scaling indices which reveal on one hand the presence of intermittency and on the other hand the presence of multifractal phenomena represented by the frequencyG-moments. In the procedure applied in present paper, also the corresponding intercepts as well as an effective average multiplicity are involved. The last mentioned quantity is introduced by extending the relation which characterizes appearance of the multifractality. It is shown that the relation between scaling indices and corresponding slopes is satisfied with sufficient accuracy by the data available so far on deep-inelastic muon-nucleon scattering at 280 GeV.

The rank-size distribution of settlements as a dynamic multifractal phenomenon

The rank-size distribution in a system of settlements is considered as a multifractal phenomenon. The occurence and temporal stability of the rank-size distribution is obtained as the result of a dynamic self-organization process in the nested system of settlements. The inherent time scale of this self-organization process and the obtained hierarchical structure are related to low intersettlement mobility of urban population, a tendency of the population to agglomerate, as well as distance and preference effects. A well-known migration model is used with three different approaches for the attractivity of a settlement for comparative purposes.

Multifractal phenomena in physics and chemistry

The neologism 'multifractal phenomena' describes the concept that different regions of an object have different fractal properties. Multifractal scaling provides a quantitative description of a broad range of heterogeneous phenomena.