Fractals

The Mandelbrot-like sets appear in the parameter spaces of many one-parameter complex mappings. We find that the type of these sets depends on the multiplicity of the critical points of the mappings. We give the effective map for parameters in these sets, and accordingly calculate the positions, the sizes, and the orientations of these Mandelbrot-like sets for any one-parameter complex mappings.

Multifractal measures have the property of being statistically invariant under some group of space and field transformations. In general, the measures lack invariance for transformations outside this group, but we show that an important exception exists. We consider measures that are invariant under isotropic contraction and examine how their marginal distribution changes under affine space transformations. We find that, for regions that are highly elongated in one direction, the marginal distribution satisfies a multifractal scaling relation that depends on the direction of elongation. We use numerical simulation to investigate the transition range between asymptotic scaling regimes, when the regions do not have extreme elongation in any one direction. The results have practical implications on the inference of multifractal properties from observations in regions with one or more fixed dimensions.

An approach to apply the fractal concept to estimate hydrologic response is proposed in this paper by matching suitable self-similar networks (SSNs) to a specific watershed, and modeling the runoff with a width-function based geomorphologic instantaneous unit hydrograph (WF-GIUH). In order to work out the identification between a specific basin and SSNs that are generated by an interior generator cooperating with an exterior generator, a generalized width function is derived. Subsequently, cumulative width functions on the basis of the derived function, as well as the informational entropies are used as criteria to decide the best patterns of the two cooperating generators for the specific watershed. The WF-GIUH model is then applied to calculate the runoff of this watershed as an outcome of the estimation. To assess the adaptability of the estimation model, San-Hsia watershed of Northern Taiwan is selected as a study area, where the analytical results of the outflow estimation indicate that the fractal algorithm has been implemented successfully for the calculation of hydrologic responses.

Consider a pure recurrent positive renewal process generated by some interarrival waiting time. The waiting time paradox reveals that, asymptotically, the time interval covering one's arrival in the file is statistically longer than the typical waiting time. Special properties are known to hold, were the waiting time to be infinitely divisible, two particular subclasses of interest being the exponential power mixtures' and the Lévy's ones. These models are revisited in some detail. Questions related to these problems are investigated and special examples of interest are underlined.

In this paper, we generalize Barenblatt's cohesive fracture theory for fractal cracks. We discuss the difficulties of generalizing the concept of traction on a fractal surface. Borodich's modification of Griffith's theory for fractal cracks is reviewed. Irwin's driving force is generalized for fractal cracks and a fractal driving force (G_f) is defined. It is shown that to generalize Barenblatt's theory for fractal cracks it is necessary to introduce a new quantity, D-fractal cohesive pseudo-stress. This new quantity is cohesive force per unit of a fractal measure. Fractal modulus of cohesion is seen to be a function of both the material and the fractal dimension of the crack. Equivalence of fractal Barenblatt's and Griffith's theories is discussed. It is seen that the order of stress singularity at the tip of a fractal crack cannot be obtained using modified Barenblatt's theory because this theory is a local theory and assumes the order of stress singularity a priori.

Fractal subsets of ℝⁿ with a highly regular structure are often constructed as a limit of a recursive procedure based on contractive maps.

The Hausdorff dimension of recursively constructed fractals is relatively easy to find when the contractive maps associated with each recursive step satisfy the Open Set Condition (OSC). We present a class of random recursive constructions which resemble snowflake structures and which break the OSC. We calculate the associated Hausdorff dimension and conjecture that an a.s. deterministic exact Hausdorff function does not exist.

A previous paper¹ has introduced a class of random functions in Rⁿ called iterated random pulse (IRP) processes. IRP processes are sums of pulses whose locations form an iterated clustered point process and whose deterministic shapes, sizes and amplitudes satisfy affine scaling relations. In the simplest case, at each finer scale the pulse support is isotropically contracted by a factor r>1 and the amplitude is multiplied by r^-γ, where γ>-n is a given constant and n is the space dimension. We consider extensions in which the pulse amplitudes are random variables with a multiplicative cascade structure. This means that, if a parent pulse at a certain level has amplitude A, its offspring pulses at the next level have amplitudes A_i=Ar^-γ η_i, where the η_i are positive iid variables with mean value 1. Interest is in the generalized field X(h)=lim_j→∞X_j(h) where h(t) is a test function, X_j(h)=∫ h(t)X_j(t)dt, and X_j(t) is the pulse field at resolution level j. We show that X(h) is multifractal at small scales. We also derive its scaling properties and the fractal dimension of the support. In the case of non=isotropic contraction of the pulse support from level j to level j+1, X(h) has a more general form of multifractality known as generalized scale invariance (gsi). This seems to be the first known construction of gsi random fields.

We consider the attractor T of injective contractions f₁, …, f_m on R² which satisfy the Open Set Condition. If T is connected, then T's interior T° is either empty or has no holes, and T's boundary ∂T is connected; if further T° is non-empty and connected, then ∂T is a simple closed curve, thus T is homeomorphic to the unit disk {x∈R²: |x|≤1}.

We introduce a new fractional iterative map with two control parameters of β and γ in the range of the fractional value between one and two. Particularly, we present basically chaotic properties and multifractals of this map. The fractal dimension obtained from multifractal measures is taken to be near one as γ→4 and η→2 for fixed values of β, while it is expected to take the minimum value of the fractal dimension 0.538 as γ→γ_∞ for β=1 and η=1. Our result obtained from multifractal measures are also compared with other well-known results.

Recent studies have established that macroeconomic time series exhibit fractal properties. Empirical tests here demonstrate that interest rates, exchange rates, output and prices all show evidence of a non-integer fractal dimension. Several classes of volatility models widely used in econometrics can give rise to fractality. In the paradigm proposed here, fractality results from multiplicative relationships between residual noise terms in simultaneous equation systems. The emergence of fractality in a large-scale econometric model is analyzed. The model uses well-established structural equations, so that all variables converge toward their equilibrium paths in the long run. The forecasted paths are then embedded in noise, and the model is re-simulated at a higher frequency. The simultaneity of the model equations causes the embedding noise to take on fractal properties. Multi-scaling demonstrates that the model simulations reproduce the fractal properties of the real-world time series reasonably well. Finally, it is possible to forecast at short horizons using an algorithm that exploits two aspects of fractality, scaling symmetries and intermittency. Ratios of rates of change capture proximate symmetries. A logit regression is used to predict the conditional probability of extreme events.