Fractals

We present an algorithm, an extension of Lavaurs' algorithm for z²+c, for determining the combinatorial arrangement of the hyperbolic components of the generalized Mandelbrot set for zⁿ+c. Several external rays are plotted to support the algorithm.

The complicated dynamics of logistic maps with four fundamental operations in the polar coordinates are analyzed in this paper. Iterations of single logistic map in polar coordinates exhibit similar characteristics as in the Cartesian coordinates. Iterations of double logistic maps, however, are more interesting in polar coordinates than those in Cartesian coordinates. Under different iterative rules, the dynamics of double logistic maps show some funny artistic patterns. Some of them look like the formation of an embryo, and some even look like Taiji diagram, which is the representative symbol of Taoism in ancient China. It is interesting to see that Taiji, which is believed in Taoism to be the mysterious origination of the world, is an interim phenomenon within the evolution of the systems governed by the law of the logistic equations.

Scaling behavior of a self-affine fractal interface observed in a tree line is examined. A series of relatively horizontal tree lines are photographed, and each 9 cm × 13 cm color print is scanned and digitized to obtain the interface height h(x) as a function of position x. From these, the interface with w(L) is determined as a function of system size L. It is found that the interface width scales with system size L as w(L)~L^α with α=0.66±0.03. The result is assessed in comparison with other relevant experimental and theoretical results.

We calculate the box-counting dimension of a self-affine version of the Sierpiński triangle. This is done by investigating the singular values of the affine transformations. We also investigate multifractal features of self-affine measures supported by certain generalized Sierpiński triangles.

A general distribution function for the heights of anisotropic engineering surfaces is obtained by extending earlier work on surface profiles. The derivation starts from a functional description of surface heights that involves fractal quantities and is comprehensive enough to include almost all of the mathematical models for surface topography that have appeared in the literature. It is found that the distribution is in the form of a Gaussian function multiplied by a convergent power series, and the terms in the series depend in a fundamental way on the fractal parameters of the surface. This distribution is used to predict the dependence of bearing-area on fractal parameters, and is compared with other approaches to anisotropic surfaces in the literature. Two truncated approximate versions of the distribution function are introduced in order to test the theoretical model against experimentally obtained distributions of engineering surfaces; the results show good agreement between theory and experiment.

We study electron tunneling between two infinte potential square wells connected via an opaque barrier and find that time evolution of the probability of the presence of a Gaussian wave packet, localized initially in one of the wells, shares the fractal behavior of tunneling in a quartic potential, discovered by Dekker (H. Dekker, Phys. Rev.A35, 1825 (1987). However, the fractal dimensions are found to be closer to those of a conventional Weierstrass function than those appropriate to the quasi-Weierstrass behavior of Dekker. It is argued that the usual exponential decay predicted by conventional relaxation processes can be recovered only as an effect of thermal fluctuations.

By analyzing international company data we find that company size distributions are not universal but clearly depend on country. In each country, the size distributions for different categories of business are quite similar. In order to understand the country dependence we introduce a numerical model of company size which is based on two effects: stochastic competitive growth by capital exchange, and deterministic protection of small companies by equi-partition of taxed wealth. A power law distribution is realized when the protection effect is negligible. The model is also consistent with the empirical laws for company's growth rates.

Importance of the influence of neighboring canopy gaps upon new gap creation has been clarified by the ecological study of a neotropical forest on Barro Colorado Island (BCI), Panama. A stochastic lattice model for the forest dynamics with interacting canopy gap expansion was introduced by Kubo et al. We give a theorem showing a condition that this model can be regarded as a stochastic Ising model, and that its stationary state is exactly given by a Gibbs state. Using this theorem, we obtain a Gibbs state which remarkably well approximates the real gap-size distribution in BCI.

Photographs of the human bronchial trees were obtained from resin casts. After data editing, the fractal dimension of the contour of the bronchial tree was determined by the 2D box-counting-method (2D-BCM). Mathematical theorems about projections and intersections allow us to determine the fractal dimension of the spatial human bronchial tree.