Fractals

Fractal time series with scaling properties expressed through power laws appear in many contexts. These properties are very important from several viewpoints. For instance, they reveal the nature of the correlations present in the fractal signals. It is common that the scaling properties characterized by means of invariant quantities suffer changes along with the dynamical evolution of the studied systems. One of these changes is a crossover in the scaling properties reflecting an important change in the system dynamical behavior. In this article, we present two cases of crossover behavior corresponding to interbeat and electroseismic time series, we observe the crossovers in time series of experimental data and their corresponding simulation with simple models. We suggest a possible explanation of the observed crossovers in terms of the models considered.

Natural river channel networks have been shown in empirical studies to exhibit power-law scaling behavior characteristic of self-similar and self-affine structures. Of particular interest is to describe how the distribution of distance to the outlet changes as a function of network size. In this paper, networks are modeled as random self-similar rooted tree graphs and scaling of distance to the root is studied using methods in stochastic branching theory. In particular, the asymptotic expectation of the width function (number of nodes as a function of distance to the outlet) is derived under conditions on the replacement generators. It is demonstrated further that the branching number describing rate of growth of node distance to the outlet is identical to the length ratio under a Horton-Strahler ordering scheme as order gets large, again under certain restrictions on the generators. These results are discussed in relation to drainage basin allometry and an application to an actual drainage network is presented.

We propose a new method for calculating fractal dimension (DF) of a signal y(t), based on coefficients , mean absolute values of its nth order derivatives (consecutive finite differences for sampled signals). We found that logarithms of , n = 2,3,…,n_max, exhibited linear dependence on n:

We characterize the path length set of asymmetric binary fractal trees in terms of the scaling ratios, r and ℓ. We show that if r + ℓ < 1, then the path length set is a Cantor set, and if r + ℓ ≥ 1, then the path length set is an interval.

In the present work, we focus on the multifractal structure of the microtemperature time series monitored at depth in boreholes, namely in two holes drilled in Kamchatka (Russia). Two monitoring series were performed for approximately two weeks with a 5-second reading interval; thus, each series contains about 230,000 data points. The observed temperatures displayed sharp gradients and large fluctuations over all observed time ranges, reflecting fine structure of the heat transfer process in the shallow subsurface. Recurrence plot (RP) technique was applied for detecting hidden rhythms that generated the time series. The most characteristic feature of the RPs is their web-like structure, indicating quasi-periodic occurrence of the sharp changes. The spectral and the local growth of the second moment techniques were used for distinguishing between the potentially different heat transfer processes. Both spectra show "red noise" behavior, however, exhibit distinct scaling exponents for different frequency domains from near 1.33 for low frequencies to near 3 at the high frequency end of the spectra. Local growth of the second moment technique has revealed the presence of temperature forming process with the characteristic time of approximately 1 minute, that smoothes out generally anti-persistent behavior on shorter time scales, but has a little effect on longer time scales.

The investigation was accomplished by the calculation of universal multifractal indices, which characterize temperature fluctuation upon scales in the range from minutes to weeks. Both time series show very similar multifractal behavior. The Hurst exponent, characterizing the degree of non-stationarity, equals to 0.18–0.20, the measure of intermittence C₁ amounts to 0.10, and the α-index, characterizing the degree of multifractality, amounts to 1.32 and 1.24 for both boreholes. We speculate on the origin of these common features intervening all over the observed range of time scales in a borehole.

We study the distribution of the maximum M of multifractal measures using discrete cascade representations. For such discrete cascades, the exact distribution of M can be found numerically. We evaluate the sensitivity of the distribution of M to simplifying approximations, including independence of the measure among the cascade tiles and replacement of the dressing factor by a random variable with the same distribution type as the cascade generator. We also examine how the distribution of M varies with the dimensionality of the support and the multiplicity of the cascade. Of these factors, dependence of the measure among different cascade tiles has the highest effect on the distribution of M. This effect comes mainly from long-range dependence. We use these findings to propose a simple approximation to the distribution of M and give charts to implement the approximation for beta-lognormal cascades.

Different fractal methods (Factorial moment, Takagi moment) have been used for the analysis of fractal behavior of multiparticle production in π^--AgBr interaction at 350 GeV/c of compound multiplicity distribution in emission angle (cos θ) space and azimuthal angle (ϕ) space. The generalized fractal dimensions D_q are determined according to two methods. Results indicate multifractal behavior in compound multiplicity distribution. From the knowledge of D_q, the multifractal specific heat is calculated. Results are compared with the corresponding results of shower multiplicity.