Fractals

We analyze the adhesion probability distribution of diffusion-limited aggregation (DLA) assuming that a random walker directly diffuses to the adhesion position of DLA. The distribution is divided into two regions. In the region that is closer to a seed particle, the adhesion probability is described as the product between the Poisson distribution and the adhesion process. In the region that is farther from a seed particle, the distribution is understood to be the product of the Gaussian and the adhesion process. Both results are physically reasonable.

We compute the spectra of the recurrence dimension for dynamically defined subsets of Rauzy fractals, in the case when these sets are totally disconnected. These subsets of the Rauzy fractals are defined by subadic systems, therefore there is a well-defined dynamical system defined on them. The spectra of the recurrence dimension for adic and subadic systems was computed by the author in Sirvent,¹ and in the present article we extend those results to the dynamical systems mentioned before, which are geometrical realizations of these symbolic systems.

This dimension is characterized by the Poincaré recurrence of the system, and the corresponding spectrum is invariant under bi-Lipschitz transformations.

We also address the question, when the dynamically defined subsets of Rauzy Fractals are totally disconnected.

A self-similar process has power spectrum with power law depending on its self-similarity parameter H and we use this property for its generation by the method of surrogate data. The surrogates are a set of random data with the same distribution as that of the increments of the process. These are iteratively rearranged according to the rank order of a time series obtained with the same power law spectrum. The method is fast, and reliable as shown by the characteristics reproduced. It is also possible to extend this method to prediction of a self-similar process, the success of which will depend on the number of available data.

Iterative function systems are often used for investigating fractal structures. The method is also referred as Chaos Game Representation (CGR), and is applied for representing characteristic structures of DNA sequences visually. In this paper, we proposed an original way of plotting CGR to easily confirm the property of the temporal evaluation of a time series. We also showed existence of spurious characteristic structures of time series, if we carelessly applied the CGR to real time series. We revealed that the source of spurious identification came from non-uniformity of the frequency histograms of the time series, which is often the case of analyzing real time series. We also showed how to avoid such spurious identification by applying the method of surrogate data and introducing conditional probabilities of the time series.

We consider a certain type of random substitution and show that the sets generated by it have almost surely the same box-counting and Hausdorff dimension, and that box-counting and Hausdorff dimension coincide.

Fragmentation processes can produce isolated patches from an originally continuous habitat. The shapes of the resulting patches are usually highly irregular. Due to this irregularity, their ability to endure external disturbances is different, which is an expression of their ecological quality. Traditionally, patch area and interior-to-edge ratios are used to characterize the quality of habitats but the latter one is not applicable under varying external disturbances. In this paper, we show that the internal Minkowski-Bouligand dimension (and the corresponding form-factor) can be used as a simple interior-to-edge function applicable on a variable edge width.

A static statistical approach to the Bak, Tang and Wiesenfeld (BTW) sandpile model is proposed. With this approach, the exact avalanche distribution of the one-dimensional BTW sandpile is given concisely. Furthermore, we investigate the two-dimensional BTW sandpile and obtain some interesting results. First, the total particle number of the two-dimensional BTW sandpile obeys some kind of stable distribution. With the increase of the sandpile scale, the stable distribution transits from Gamma to Normal distribution. Second, when the total number of particles is fixed, the avalanche distribution is not power law. The system, however, shows a kind of "negative temperature" phenomenon when the particle number increases. Third, power law distribution of the avalanche could be viewed as the result of the superposition of a series of weighted distributions which do not yield power law.

Lightning data, measured from April to October 2002 in an area of northern Italy, have been analyzed by using the Allan Factor statistics. The sequence of lightning time-occurrences is characterized by two scaling regions, which reveal intra-cluster (inside an individual thunderstorm) and inter-cluster (among successive thunderstorms) time-clustering properties of lightning data. We investigated intra- and inter-cluster correlations for three classes of lightning: (i) the whole series, (ii) the sequence of only negative flashes and (iii) that of only positive flashes. Furthermore, we performed the analysis of the scaling behavior of the three classes of lightning varying the current intensity threshold.

This paper describes fractal behaviors in a soccer game according to the player's position. It is quite important for us to characterize the fractal motion behaviors of the objects during the game. We obtained two-dimensional coordinates of the objects using standard video processing techniques from a computer soccer game. We calculated values of regularization dimensions of the time series to characterize their fractal behaviors. To see positional dependence, we averaged individual player's values over the same position in the same team. When a team is one-sidedly experiencing a severe attack, its defenders have higher fractal dimensions than those of the opponent's corresponding players. We propose a new measure of relative dominance in attack against the opponent team.