Lecture 19

FRACTALS - One Perspective on Chaos

Benoit Mandelbrot

Mandelbrot is a mathematician with a strong visual sense. He is interested in spatial patterns of variability, such as the length and shape of coastlines, and temporal patterns of variability such as cotton prices.

Mandelbrot noted two effects which xould not be accounted for by linear systems analysis. He termed these the:

Noah Effect - time series that have discontinuous changes (steps in time rather than smooth changes)

Joseph Effect - persistense of extreme values (an area with many droughts should expect more)

Cotton Prices: Mandelbrot noted that daily and longer term discrete measurements of cotton prices were hard to correlate with other economic indicators. They were characterized by discontinuous steps (Noah Effect) and variability at the daily scale that looked just like variability at the monthly scale (with different amplitudes). The second effect can be called scale invariance or self-similarity (at different scales).

Mandelbrot was also intrigued with one type of noise that occurred in telephone transmission lines. The noise occurred irregularily - long periods of no noise separated by shorter intervals of intermittent noise. When he looked more closely at the noisy intervals, he found that they too had l(relatively) long intervals of no noise separated by shorter intervals of intermittent noise (self-similarity).

Mandelbrot was able to explain both cotton price variations and transmissionnoise by considering a Cantor Set. One forms a Cantor Set by the following rule: take a line segment, divide it into three equal pieces and rmove the center piece. Then do it again for the two remaining pieces. Then do it again..... One ends up with irregular short line segments with long irregular intervals in between. If one adds numbers cumulatively every time one sees a short line segment, the resulting cumulative curve loks like the cotton prices.

Lengths of Coastlines: If one measures the perimeter (length of coastline) around Great Britain from a map, and then do it again with increasingly higher resolution maps, the perimeter will increase. In fact, arguably the perimeter of Great Britain is essentially infinite by the area is quite finite. Why?

Fractal Dimension

Both spatial and temporal patterns of variability can be characterized by plots relating size/occurence to number observed.

When size/occurence versus number observed is plotted on a log-log scale (logrithms of numbers, not the numbers themselves), there is often a straight-line relationship between the number. This is termed a power-law relationship. If the slope and dimension of the line is integer, then we are in the linear world. A slope of two, for example, indicates a proess where Y is proportional to X squared. Cantor Sets and other related data types have fractional slopes and dimension. For example, Y might be proportional to X to the 3.45 power. Power-law relationships with fractional dimentsions are called FRACTALS.

Fractal dimension implies scale invariance or self-similarity. If you look at a process at one scale, you see the same thing if you change the scale.

Fractals imply nonlinearity in the process which created the fractal process. Fractals turn out to be one way to measure chaotic systems. By themselves, fractals do not have any unique cause or source.