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Chaos Theory Essay, Research Paper

Chaos Theory

and Fractal Phenomena

Chaos theory is the qualitative study of unstable aperiodic behavior in deterministic nonlinear systems. To understand the definition of chaos can be understood if broken down: A dynamical system may be defined to be a simplified model of the time-varying behavior of an actual system and an aperiodic behavior is the behavior that occurs when no variable describing the state of the system undergoes a regular repetition of values. An aperiodic behavior will never repeat itself and continues and therefore the prediction of this system is impossible; although, patterns are present. A good example of an aperiodic behavior is history. Yes, history repeats itself but never exactly as it was before. These behaviors can be found in simple mathematical systems but they display very complex and unpredictable behaviors that the description of them can be called random.

It has just been recently that the study of the Chaos theory has arose. The reason for this is technology and computers. The calculations involved in studying this theory is extremely repetitive and can number in the million. This job cannot be done by a human but can easily be done with a computer. They are extremely good at endless repetition and that is exactly what Chaos theory entails. One said that computers are the telescope to studying Chaos.

There is a basic principle that describes chaos theory and that is known as the Butterfly Effect. The butterfly effect means a small variation in initial conditions, resulting in huge, dynamic transformations in concluding events. The term butterfly is obviously used due to the transformation from a caterpillar to a butterfly. A folklore that has been used to better explain this butterfly effect goes like this:

For a want of a nail, the shoe was lost;

For want of a shoe, the horse was lost;

For want of a horse, the rider was lost;

For want of a rider, the battle was lost;

For want of a battle, the kingdom was lost!

This started with a small variation: no nail and ended in a huge transformation the kingdom was lost.

An identifiable symbol linked with the Butterfly Effect is the Lorenz Attractor, by Edward Lorenz. He was a curious meteorologist who was looking for a way to model the action of the chaotic behavior of a gaseous system. The Lorenz attractor is based on three differential equations, three constants, and three initial conditions. The attractor represents the behavior of gas at any given time, and its condition at any given time depends upon its condition at a previous time. If the initial conditions are changed by even a tiny amount, say as tiny as the inverse of Avogadro s number, the number of atoms in a mole, checking the attractor at a later time will yield numbers totally different. This is because small differences will propagate themselves recursively until numbers are entirely dissimilarly to the original system with the original initial conditions. However, the plot of the attractor will look very much the same. Both systems will have totally different values at any given time, and yet the plot of the attractor the overall behavior of the system will remain the same. His three simple equations were taken from the physics field of fluid dymanics. He simplified these equations and came up with the three-dimensional system:

dx/dt = delta * (y-x)

dy/dt = r * s-y-x * z

dz/dt = s * y-b * z

The delta in the above equation represents the Prandtly number, which is the ratio of the fluid viscosity of a substance to its thermal conductivity. You do not have to know the exact value of this constant and therefore Lorenz decided to use 10. The r represents the difference in the temperature between the top and bottom of the gaseous system. Lorenz plugged 8/3 for this variable. The x represents the rate of the rotation of the cylinder and the y is the difference in the temperature at the opposite sides of the cylinder. The z represents the deviation of the system from a linear, vertically graphed line representing temperature. If this was graphed no geometric system would appear, instead, a weaving object known as the Lorenz Attractor would appear. Since the system never exactly repeats itself, the trajectory never intersects itself. Instead it loops around forever. Here is a Lorenz Attractor which is run through a fourth order Runge-Kutta fixed-timestep integrator with a step of .0001, printing every 100th data point. It ran for 100 seconds and only took the last 4096 points. The original parameter were a=16, r=45 and b=4.

These were used in equations very similar to Lorenz s equations:

x = a(y-x)

y = rx-y-xz

z = xy-bz

Lorenz was not quite convinced of his results and he did a follow up experiment in order to support his previous conclusions. Lorenz established an experiment that was quite simple; it is known as the Lorenzian Waterwheel. Lorenz took a waterwheel; it had about eight buckets spaced evenly around its rim with a small hole at the bottom of each. The buckets were mounted on swivels, similar to a Ferris-wheel seat, so that the buckets would always pint upwards. The entire system was placed under a waterspout. A slow, constant stream of water was propelled from the waterspout; hence, the waterwheel began to spin at a fairly constant rate. Lorenz decided to increase the flow of water, and, as predicted in his Lorenz Attractor, an interesting phenomena arose. The increased velocity of the water resulted in a chaotic motion for the waterwheel. The waterwheel would revolve in one direction as before, but then it would suddenly jerk about and revolve in the opposite direction. The filling and emptying of the buckets was no longer synchronized; the system was now chaotic. Lorenz observed his mysterious waterwheel for hours,

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