Chaos Theory Essay, Research Paper
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, and, no matter how long he recorded the positions and contents of the buckets, there was never an instance where the waterwheel was in the same position twice. The waterwheel would continue on in chaotic behavior without ever repeating any of its previous conditions. A graph of the waterwheel would resemble the Lorenz Attractor.
Chaos and randomness are no longer ideas of a hypothetical world; they are quite realistic. A basis of chaos is established in the Butterfly Effect, the Lorenz Attractor, and the Lorenz Waterwheel; therefore, there must be an immense world of chaos beyond the basic fundamentals. This new form mentioned is highly complex, repetitive and full of intrigue.
The extending and folding of a chaotic systems give strange attracts, such as Lorenz Attractor, the distinguishing characteristic of a non-integral dimension. This non-integral dimension is most commonly referred to as a fractal dimension. Fractals appear to be more popular in the world of mathematics for their aesthetic nature that they are for their mathematics. Everyone who has seen a fractal has admired the beauty of a colorful, fascinating image, but what is the formula that makes this image? The classical Euclidean geometry that one learns in school is quite different than the fractal geometry mainly because fractal geometry concerns non-linear, non-integral systems while Euclidean geometry mainly is concerned with linear and integral systems. Euclidean geometry is a description of lines, ellipses, circles, etc. However, fractal geometry is a description of algorithms. There are two basic properties that constitute a fractal. First, is self-similarity, which is to say that most magnified images of fractals are essentially indistinguishable from the unmagnified version. A fractal shape will look almost, or evenly exactly, the same no matter what size it is viewed at. This repetitive pattern gives fractals their aesthetic nature. Second, as mentioned earlier, fractals have non-integer dimensions. This means that they are entirely different from the graphs of lines that we have learned about in fundamental Euclidean geometry classes. By taking the mid-points of each side of an equilateral triangle and connecting them together, one gets an interesting fractal known as the Sierpenski Triangle. The iterations are repeated an infinite number of times and eventually a very simple fractal arises.
The Sierpenski Triangle:
In addition to the Sierpenski Triangle, the Koch Snowflake is also a well-known simple fractal image. The end construction of the Koch Snowflake resembles the coastline of a shore.
The Koch Snowflake:
These two fundamental fractals provide a basis for much more complex, and elaborate fractals. Two of the leading researchers in field of fractals were Gaston Maurice Julia and Benoit Mandelbrot.
Gaston Maurice Julia was injured in World War I and was forced to wear a leather strap across his face for the rest of his life to protect and cover is injury. He spent a large majority of his life in hospitals; therefore a lot of his mathematical research took place in a hospital. At the age of 25, Julia published a 199 page masterpiece entitled Memoire sur l iteration des fonctions. The paper dealt with the iteration of a rational function. With the publication of this paper came his claim to fame. Julia spent his life studying the iteration of polynomials and rational functions. If f(x) is a function, various behaviors arise when f is iterated or repeated. If one were to start with a particular value of x, say x=infinity, then the following would result:
a, f(x), f(x)), f(f(f(x))), etc.
Repeatedly applying f to infinity yields large values. Hence, the set of numbers is partitioned into two parts, and the Julia set associated to f is the boundary between the two sets. The filled Julia set includes those numbers x=infinity for which the iterates of f applied to a remain bounded. The following fractals belong to Julia s set.
Julia became famous around the 1920 s, however upon his death, he was essentially forgotten. It was not until 1970 that the work of Gaston Maurice Julia was revived and popularized by Polish born Benoit Mandelbrot.
Benoit Mandelbrot was born in Poland in 1924. When he was 12 his family emigrated to France and his uncle, Szolem Mandelbrot, took responsibility for his education. It is said that Mandelbrot was not very successful in his schooling; in fact, he may have never learned his multiplication tables. When Benoit was 21, his uncle showed Julia s important 1918 paper concerning fractals. Benoit was not overly impressed with Julia s work, and it was not until 1977 that Benoit became interested in Julia s discoveries. Eventually, with the aid of computer graphics, Mandelbrot was able to show how Julia s work was a source of some of the most beautiful fractals know today, The Mandelbrot set is made up a connected points in the complex plane. The simple equation that is the basis of the Mandelbrot set is included below:
Changing number + Fixed number = Result
In order to calculate points for a Mandelbrot fractal, start with one of the numbers on the complex plane and put its value in the Fixed Number slot of the equation. In the Changing Number slot, start with zero. Next, calculate the equation. Take the number obtained as the result and plug it into the Changing Number slot. Now, repeat this operation an infinite number of times. When iterative equations are applied to points in a certain region of the complex plane, a fractal from the Mandelbrot set result. A few fractals from the Mandelbrot set are included below:
George Cantor, a nineteenth century mathematician, became fascinated by the infinite number of points on a line segment. Cantor began to wonder what would happen when an infinite number of line segments were removed from an initial line interval. Cantor devised an example portrayed classical fractals made by an iteratively taking away something. His operation created dust of points; hence, the name Cantor Dust. In order to understand Cantor Dust, start with a line; remove the middle third; then the remove the middle third of the remaining segments; and so on. The operation is shown below:
The Cantor set is simply the dust of points that remain. The number of these points are infinite, but their total length is zero. Mandelbrot saw the Cantor set as a model for the occurrence of errors in an electronic transmission line. Engineers saw periods of errorless transmission, mixed with period when errors would come in gusts. When these gusts of errors were analyzed, it was determined that they contained error-free periods within them. As the transmissions were analyzed to smaller and smaller degrees, it was determined that such dusts, as in the Cantor Dust, were indispensable in modeling intermittency.
There are many other uses of Chaos Theory that apply to every day life. For example, fractals make up a large part of the biological world. Clouds, arteries, veins, nerves, parotid gland ducts, and the bronchial tree, all show some type of fractal organization. In addition, fractals can be found in regional distribution of pulmonary blood flow, pulmonary alveolar structure, surfaces of proteins, mammographic parenchymal pattern as a risk for breast cancer, and in the distribution of anthropod body lengths.
Some other more common uses of Chaos Theory are the Chaos washing machine, Stock Market Chaos, and Solar System Chaos. In 1933, Goldstar Co. created a washing machine that utilized the Chaos theory. This washing machine supposedly produced cleaner and less tangled clothes. Stock Market analysts have found evidence of chaos in the stock market. Chaos Theory is also, very familiar to astronomers. Most have long known that the solar system does not run with precision of a Swiss watch.
The fractals and iterations are fun to look at; the Cantor Dust and Koch Snowflakes are fun to think about, but what breakthroughs can be made in terms discovery? Is chaos theory anything more than a new way of thinking? The future chaos theory is unpredictable, but if a breakthrough is made is will be huge.
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