Chaos Theory and Fractals

An exciting development has for some years been seen in the use of chaos theory in signal processing. A chaotic signal is not periodic it has random time evolution and a broadband spectrum and is produced by a deterministic nonlinear dynamical system with an irregular 

This sequence appears when you, for example, repeatedly press the cosine button on a pocket calculator and it converges to 0.739 if x is an arbitrary number in radians. Other iterative processes do not converge to a specific number, but produce what seems like a random set of numbers. These numbers may be such that they always are close to a certain set, which may be a. fractal. This set is called a fractal attractor or strange attractor. 

Fractal patterns have no characteristic scale, a property that is formalized by the concept of self-similarity. The complexity of a self-similar curve will be the same regardless of the scale to which the curve is magnified. A so-called fractal dimension D may quantify this complexity, which is a noninteger number between 1 and 2. The more complex the curve, the closer D will be to 2. Other ways of defining the fractal dimension exist, such as the Haussdorff-Besicovitch dimension. 

Self-affine curves resemble self-similar curves, but have weaker scale-invariant properties. Whereas self-similarity expresses the fact that the shapes would be identical under magnification, self-affinity expresses that the two dimensions of the curve may have to be scaled by different amounts for the two views to become identical (Bassingthwaighte et al., 1994). A self-affine curve may hence also be self-similar if it is not, the curve will have a local fractal dimension D when magnified a certain amount, but this fractal dimension will approach 1 as an increasing part of the curve is included. Brownian motion plotted as particle position as a function of time gives a typical example of a self-affine curve. 

The theory of fractal dimension may be used in bioimpedance signal analysis, for example, for studying time series. Such analysis is often done by means of Hurst s rescaled range analysis (R/S analysis), which characterizes the time series by the so-called Hurst exponent H = 2 — D. Hurst found that the rescaled range often can be described by the empirical relation 

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Donahue M.J. HI, An Introduction to Mathematical Chaos Theory and Fractal Geometry. http //www.duke.edu/ mjd/chaos/chaos.html (retrieved October 2004). 

Chaos Theory and Fractals. In the nineteenth century, much science fiction was written about the... 

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