Fractals

In this article, we continue to explore some specific results in bicomplex dynamics. In particular, we give a bicomplex version of the so-called Fatou-Julia theorem. In fact, we give a complete topological characterization in ℝ⁴ of the bicomplex filled-Julia set for a quadratic polynomial in bicomplex numbers of the form w² + c.

Cumulative frequency-size distributions associated with many natural phenomena follow a power law. Self-organized criticality (SOC) models have been used to model characteristics associated with these natural systems. As originally proposed, SOC models generate event frequency-size distributions that follow a power law with a single scaling exponent. Natural systems often exhibit power law frequency-size distributions with a range of scaling exponents. We modify the forest fire SOC model to produce a range of scaling exponents. In our model, uniform energy (material) input produces events initiated on a self-similar distribution of critical grid cells. An event occurs when material is added to a critical cell, causing that material and all material in occupied non-diagonal adjacent cells to leave the grid. The scaling exponent of the resulting cumulative frequency-size distribution depends on the fractal dimension of the critical cells. Since events occur on a self-similar distribution of critical cells, we call this model Self-Similar Criticality (SSC). The SSC model may provide a link between fractal geometry in nature and observed power law frequency-size distributions for many natural systems.

We have questioned whether a complex behavior, such as fish swimming, can be better described quantitatively as a sequence of discrete events or states than with classical kinematic measures which can be compromised by inherent variability. Here, the different states, expressed as combinations of symbols, were defined on the basis of the animal's location (A: periphery, and B: inner part of the aquarium) and speed (Fast and Slow). We observed that the distributions of time intervals spent in the successive states were not gaussian. Rather, they were fit by power laws associated with an underlying Lévy-like process which has more long intervals, primarily due to prolonged periods of relative inactivity. Furthermore, our data suggest that the swimming behavior can be attributed to interactions between two intrinsic systems. One is represented by the matrix of transition of probabilities between states and controls their sequential organization while the second, which is defined by interval distributions, determines the time spent in each state. This kinetic model detects subtle effects of low doses of neuroactive compounds, and identifies their specific locus of action. We propose that this paradigm can be applied to characterize normal behavior and its modifications by genetic or pharmacological manipulations.

Three aspects of time series are uncertainty (dispersion at a given time scale), scaling (time-scale dependence) and intermittency (inclination to change dynamics). Simple measures of dispersion are the mean absolute deviation and the standard deviation; scaling exponents describe how dispersions change with the time scale. Intermittency has been defined as a difference between two scaling exponents. After taking a moving average, these measures give descriptive information, even for short heart rate records. For this data, dispersion and intermittency perform better than scaling exponents.

Let X be a stationary multifractal measure in the unit cube S₀ of R^d and X_r = X(S₀/r), r ≥ 1, be the measure in a smaller cube. We develop methods to calculate the marginal distributions of X_r and the coefficients C_j,i of the Haar wavelet representation of X. The methods are exact for discrete cascades and approximate for continuous-scaling multifractal measures. In the small scale limit r → ∞, the distribution of log[X_r] or log[X_r|X_r > 0] converges under renormalization to one of three different distribution types: normal, stable non-normal, or non-stable of a particular type. The normal distribution has the largest domain of attraction. The distribution of the absolute wavelet coefficients |C_j,i| at level j has similar asymptotic properties as j → ∞. We also develop a way to directly obtain the integer moments of X_r and C_j,i without preliminary calculation of the marginal distributions. Using the marginal distribution of the measure density ε_r = r^dX_r, we assess the accuracy for finite r of the asymptotic scaling relation P[ε_r > r^γ] ~ r^-c(γ), where c(γ) is the co-dimension function of X.

In this paper, we introduce a dendritic crystal, formed in matt glaze of LiAlSiO₄-SiO₂, having "Sierpinski gasket" fractal morphology. The crystal structure of this "Sierpinski gasket" dendrite is β-quartz. β-quartz can grow two kinds of fractal patterns: snow-shaped dendrite and "Sierpinski gasket" dendrite, depending on different supercooling conditions. These two kinds of fractals can develop together in one dendritic crystal. The evolution of the boundary morphologies between these two kinds of fractal dendrites can be described by another fractal — Koch curve. The "Sierpinski gasket" dendrite is a rather new fractal growth pattern which can introduce new opportunities to fractal growth research of nonlinear sciences.

We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in ℝ⁴ and its projection in ℝ³ is taken. The extra degree of freedom coming from ℝ⁴ provides hidden variable, which is an important factor for flexibility and diversity in the interpolated surface. In the present paper, we construct an IFS that generates both self-similar and non-self-similar FIS simultaneously and show that the hidden variable fractal surface may be self-similar under certain conditions.

In this paper, we sought to determine whether fractal parameters alone are good enough in classifying astronomical images. A fuzzy membership function which follows the model of a parabola was chosen for the purpose and the success rate was found to be 73.45%. Also, we have investigated how the grade of membership functions affect the performance of a neural network classifier. For this we included the parameter, spectral flatness measure in addition to fractal dimension and the grades of both the parameters were given as input features to the neural net. It could be observed that when grades were given as inputs to the classifier, performance of the classifier has increased to 80.53%.

Advanced statistical approaches have been applied to determine the temporal scaling-invariance laws in annual mean yields registered in Mexico during the period 1897–1999. When the maize, bean, wheat and rice series are treated as self-affine profiles through power spectrum analysis, the temporal fluctuation f^-β is detected in each case for time scales of 2 to 103 years. β values (p < 0.0001) of -1.84 ± 0.15, -1.71 ± 0.218, -1.62 ± 0.194 and -1.7 ± 0.139 for maize, bean, wheat and rice profiles, respectively, suggest that short-term variation is important and are influencing the dynamic evolution of the yearly mean yields. Thus, in addition to the linear trends driven by endogenous factors of agricultural systems, grain yield variation may be due to external forcings such as climatic oscillations and/or solar signals. Through the spectral analysis using the maximum entropy method, the influence of external forcings on the behavior of the annual mean yields in all four crops was evidenced. The detected main external forcings are: a 6- to 8-year political oscillation, the magnetic and sunspot cycles of the Sun, and El Niño Southern Oscillation (ENSO). Taking into account the number of significant periodicities, the order of annual mean yield series' sensitivity to external forcings influencing their short-term variation is as follows: maize < wheat < rice < bean.