# Chaos Theory: On a Disorderly Note

This is a glimpse into the orderly world of disorder―a study of the chaos theory.

Anirban Ray Choudhury

*The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen...*―

**Ian Stewart**

The Chaotic Universe

To understand chaos, let us first begin with linear and non-linear systems. Linear systems, simply defined, are a set of repetitive events, where the principal event is the sum of all the secondary events following a linear relationship. Periodicity is the most important factor in establishing a linear system. Take, for example, the motion of a bicycle. The forward motion of the vehicle is determined by the circular movement of its wheels, which is periodic in nature (i.e., any given point on the wheel would rest on the surface of the road at periodic intervals).

The Initial Steps

The first signs of thinking in 'chaotic' terms were observed way back in 1900, when Henri Poincaré came up with the idea that, in case of three bodies in mutual gravitational attraction, there can be orbits which are not periodic, though not perpetually expanding or contracting. In 1961, a meteorologist by the name of

**Edward Lorenz**came to the realization that, seemingly minuscule events may have a large bearing on subsequent events, i.e., there need not be a linear relationship between two events affecting one another. While experimenting with a twelve equation model of the weather, Lorenz observed that the same set of data yielded surprisingly dissimilar results, depending upon the number of digits in use after a decimal point. After further observations, Lorenz concluded that it was impossible to predict the weather with accuracy even though the seasons followed an order.The Mathematician & His Fractals

**Benoit Mandelbrot**, a mathematician working with IBM, was studying the fluctuations in cotton prices, when he observed that, whatever be the mode of analyzing the data on the prices, the results invariably refused to fit into a normal distribution, even though they fit perfectly into trend models. Thus, although each price change was random and unpredictable, a scaled-up graph of the price changes showed that there were surprising similarities between the daily and monthly price variation trends, regardless of the fact that the period over which the data had been accumulated had seen two world wars and a depression. His observations led to the conclusion that there was a scaled-down self duplication as the reference frame grew smaller, i.e., there is order hidden within chaos and vice versa. It was the study of this non-periodic self similarity that gave rise to the idea of fractal dimensions.

Fractal Dimensions

A fractal is simply any image that has the attribute of self similarity. Though nearly impossible to conceive, a fractal dimension is easy to understand. Take for example Koch's Curve, which is nothing but equilateral triangles, being added on to each side of another equilateral triangle, the process being repeated an infinite number of times. The result is a star-like formation, with an infinite number of star like arms, which in turn nest an infinite number of star formations, and so on and so forth. Owing to this crinkly, star-like formation, a Koch curve takes up a lot more space than a one dimensional line. At the same time, since it does not have an area (area being a two-dimensional concept), it is not as effective in filling up space as a rectangle or a square. Therefore, the dimension of a Koch curve fractal lies somewhere between one and two.

The Chaotic Contributions

Okay, so we now have a reasonable picture of the properties of a chaotic system; it is bounded, sensitive to initial conditions, transitive, and aperiodic. But how does it aid the advancement of science, or for that matter, mankind?