Chaos Theory: On a Disorderly Note

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

Sounds too much like the insane ramblings of a lunatic, doesn't it? Well, my friend, welcome to the world of chaos, a world where order is a tailor-made creation of disorder.

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).

A non-linear or chaotic system, on the other hand, is a set of non-linear, non repetitive events, resulting in the principal event which is not predictable as the sum of the individual events. In other words, chaos is the randomness originating from sensitivity to initial conditions.

For example, let us consider a long line of bicycles parked next to each other, the last bicycle being parked against a explosive detonator. Now, if there is a piece of brick lying about a foot away from the first bicycle, upon which John happens to stumble, the result would follow a sequence of events as shown below:

John Stumbles on the brick, and falls on the first bicycle. The first bicycle topples over onto the second bicycle, eventually, the last bicycle lands on the detonator, and there is an explosion five hundred meters away.

Now, we can see from the above example, that while there is seemingly no relationship between a brickbat and an explosion a few hundred meters away from it, on rolling back for 'n' number of iterations, we find that a chain of events interlinks the two.

With a little imagination, the example can be stretched to the theory of chaos, which states that any uncertainty in the initial state of a given system would give rise to rapidly growing errors in the effort to predict the future behavior - Gollub and Solomon).

And when we speak of the butterfly calmly flapping its wings, we are actually speaking of tiny errors being inserted into the wind flow at the point of origin, which would gradually avalanche into a much larger error, causing a tornado somewhere that was not supposed to happen at all, had the weather stuck to its initial state of motion.

The first signs of thinking in 'chaotic' terms were observed back in 1900, when Henri Poincaré came up with the idea that, in case of 3 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 no. 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.

Encouraged by the uniqueness of the results, Lorenz then proceeded to analyze the behavior of convection currents, and after making several observations with varied data sets, he developed a three equation model for the water wheel. On proceeding to graph the observations, Lorenz observed that the curve maintained the shape of a double spiral.

This was a surprising discovery indeed, inasmuch that the curve deviated from the principles of the two known order states - the steady state, and the state of periodic behavior (where the system indefinitely repeats itself). While his curve was ordered, it was neither in a steady state nor was it repetitive (and therefore not in periodic motion).

Thus began the Theory of Chaos. Lorenz proceeded to write a paper on his discovery, but failed to cause much stir in the scientific community - the class bias that persisted in those days did not encourage the idea of treating a meteorologist as a mathematician!

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.

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.

Why does a fractal have to exist? What is it that creates a fractal? Well, at the core of all chaotic motion, there are strange attractors - attractors that form the 'nucleus' of the motion curve. When a complex dynamical chaotic system becomes unstable, these attractors draw the stress, and the system splits.

This is known as bifurcation. In Lorenz's tri-equation formula for the curve of a water wheel's motion, the spiral distribution of the motion has a narrow base which fans out towards the exterior, and then again contracts back towards the center.

The attractors, known as Lorenz attractors, split the stress of the motion in two directions. Actually, it is these attractors that cause order to be maintained in chaotic motion; without them, there would be an unbounded state, i.e., the motion curve would be forever expanding.

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?

Before the advent of the chaos theory, the consensus of the scientific community was that, if the uncertainty in initial conditions could be marginalized, the uncertainty in the final conditions would shrink proportionately. Chaos theory has shaken the fundamentals of this belief to the core, meaning that, probably nothing is sacrosanct anymore. This has also opened up a whole new way of looking at systems; a degree of dynamism has been introduced to what was earlier considered static.

There are several ways in which science can benefit from a proper understanding of chaos. For example, in any living being, the genetic code defines the species, the structure, and the identity of an individual.

However, while the function of the gene code is understood, it is not known as to how the basic building blocks, i.e., DNA, distribute the information required to create a complex organism. The chaos theory could perhaps hold the key to this query.

Again, while earlier, all systems were considered as non-chaotic, thereby increasing the possibility of fatal errors in judgment, now, we can distinguish between chaotic and non-chaotic systems. The theory can also explain the turbulence in fluid motion, and non-periodic oscillations in radio circuits.

The fractal nature of blood vessels can also be studied, thanks to chaos. While a lot of work has already been done in this field, there remain many dark corners which are yet to be explored, and theories such as the one of chaos lead the way towards the ultimate aim of mankind - the understanding of everything.