Go to Math Dept Main Page | Go to Mission College Main Page

This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.

To explore other such papers go to the Math G Projects Page.
 
 
 
 
 

Chaos, so what····. So what is it? If I were nine years old again, I would say that it is an evil force of people fighting against the good forces of control. Now, if my question was based on the popular 1960's t.v. series, Get Smart, I would have been right, but my question deals with something a bit more complicated than a t.v. show. The chaos I will be discussing involves what is technically known as non-linear dynamics, and in contrast to the previously mentioned t.v. show, instead of chaos being the opposing system of control, it is simply a subset of the overall non-linear function of a system.

My goal for this paper is to define chaos in the most elementary way possible and still get the key points across. There is standard vocabulary when describing chaos that, I will define, so total understanding will not be so laborious (teacher excluded).

To best describe the name chaos, I would first say that "chaos" is clearly a scientific/mathematical oxymoron, just as "jumbo shrimp" is to food and not without reason. Even though the concept of what we now call chaos is fairly new to the population in general, the roots of this phenomenon go way back to at least the 19^th century. Now that it has been definitively identified, people find it everywhere; in weather studies, fluid flow in rivers and large pipelines, traffic patterns, and biological behavior just to name a few. It happens that the availability of computer power makes possible many approaches to gaining some understanding of chaos that would not have been available fifty or sixty years ago. That is one reason why there is so much work going on in the area now.

To better understand the chaos theory, I thought it might be helpful to define some keywords in relation to its description. The underlying key to these words is that they are the conditions inherent to the creation of chaos.

NONLINEAR

Lets begin with the definition for "nonlinear" since it is the actual essence of our subject matter (although not all-nonlinear patterns produce chaos.) To understand a concept it is sometimes necessary to define the opposite one, which is linearity. Linear equations in math have been used for centuries; the concept, set of instructions and results of an equation do not change in theory. In other words, we can rely on our knowledge of linear equations to solve applicable questions. A linear equation is one in which changes in initial variables produce equal changes in variables later in the process. Solutions display themselves by way of straight lines on a graph. Linear equations are solutions for cause and effect situations: a small cause yields a small effect, and a large cause yields a large effect. With all that said, I will describe nonlinearity. Nonlinear equations do not produce straight, somewhat predictable lines, but just as the name implies, these equations produce unpredictable line or shape patterns and move in any direction possible. Nonlinear situations are more prevalent in our world than linear situations. To explain this concept in a mathematical way, I will substitute this quote, (5) "Mathematically, the signature of a nonlinear system is the breakdown of the superposition principle which states that the sum of two solutions of the equation(s) describing the system is again a solution." What makes a nonlinear system difficult to determine is the fact that any small disturbance in its already established, initial pattern can result in totally different behaviors of the system at a time progressively later.

DYNAMIC SYSTEMS

A dynamic system is one of which is constantly changing. Chaos can develop from regularity and regularity can develop from chaos. This statement is an essential aspect of a dynamical system. In relation to the nonlinear concept, a dynamic system can be compared to the movement of a pendulum, breaking wave or a rock rolling down the hill. In my work, telecommunications, a dynamic system can be related to the traffic of calls utilizing different phone numbers and all sharing the same T1 facility (24 call paths), randomly filtering through the system on any one of those paths at any given time.

DETERMINISTIC

A deterministic system is, (3)"a system in which later states evolve from earlier ones according to a fixed law." When applying this rule to an equation or explaining the process of a nonlinear movement, deterministic sequences are those that "decide" on a particular choice or path dictated by precise laws. In other words, a chaotic system may look random, but in reality, the deterministic laws it follows are quite decisive.

ITERATIONS

Iterations describe a procedure present in chaotic systems. Iteration is simply repeating the same thing over and over. In working with fractal images its apparent that "self-referential" iterations occur to create the symmetry of a design.

SENSITIVE DEPENDENCE

"Sensitive dependence on initial conditions." This phrase is indicative of the sensitivity in changes related to chaotic behavior that produce drastic differences. Weather is one area of study that is very linked to this term. When experimenting, if initial conditions change, even slightly, then the rest of the system takes a metamorphic turn, nowhere near similar to the system of the first initial condition.

BIFURCATION

An analogy of bifurcation can be shown by the branches of a tree and by the complicated paths each branch takes to its point. Bifurcations are changes; changes in a systemâs behavior from chaotic to more chaotic or steady to chaotic or chaotic to steady. Small disturbances will amplify and yield new behaviors. (1) "A bifurcation in a system is a vital instant when something as small as a single photon of energy, a slight fluctuation in external temperature, a change in density or the flapping of a butterfly's wings in Hong Kong is swelled by iteration to a size so great that a fork is created and the system takes off in a new direction." There is also a type of bifurcation that is called "saddle-node" bifurcation. This is a label given to modes of behavior that cease to exist rather than become unstable.

PHASE SPACE & TRANSITION

The phase space is defined by the variables of a mathematical equation plotted on a graph of multidimensions. Each point on the graph displays the state of a dynamical system. Phase transition is when the state of a system becomes drastically different, for example, when water boils or metal melts. The molecular structure takes on a state of change. In phase transition, fluids become turbulent. The transition is when order turns to chaos. Fractal geometry displays pictorially the changes in a system.

BEGINNINGS OF A PHENOMENON

How did the discovery of chaos begin? Well, the concept of nonlinear systems did not exist in ancient times, but the concept of chaos did. It was considered "formless matter" of complete confusion, existing before the universe. The Egyptians, Babylonians, Greeks and Romans all had comments about the fundamental ideas of creation, and with it, chaos. Chaos was always a force or entity related to order in a converse and counteracting way. Theological discussions of chaos began with the creation of the universe, where chaos and order are reflections of God.

When Newton came along, mathematical equations were applied to many systems of the world. Newton defined the laws of gravity and the law of motion. According to him, everything in the universe functioned like a "well oiled machine", and was considered very predictable. He supported these ideas with differential equations that explain rates of change. Newton came up with a solution for the deterministic universe through linear mathematics. Newton did not know that future discoveries in math and nonlinear concepts would prove his idea of the universe wrong. It took until the 19^th century for theories, defining chaos, to develop. A French mathematician, Jacques Hadamard, proved a theorem on the sensitive dependence on initial conditions. His examples used ideas from frictionless motion of a point on a surface to geodesic flow on a surface with a negative curvature. A French physicist, Pierre Duhem, took to Hadamardâs discovery and determined that, (6) "·prediction was Îforever unusableâ because of the necessary present uncertain initial conditions in Hadamardâs theorem."

Henri Poincare is the name most recognized with the application of nonlinear systems. In 1846, the planet Neptune was discovered. Laws of Newton still prevailed. In 1889, the King of Norway offered a prize for the solution proving a stable universe. Poincare stepped up to the challenge and submitted his solution and won the prize, but an error in his calculations required him to revisit his theory. (6) "In his 1890 paper, he showed that Newtonâs laws did not provide a solution to the Îthree-body problem,â in other words, how one deals with predictions about the earth, moon and sun." This idea was not well accepted by the powers that reigned at the time, because there was no definitive solution to replace Newtons. Poincare was probing into equations as a result of the "three-body problem." He proved, through calculations that small gravitational pulls from a third body might make another planet lose control in its orbit and behave unpredictably. Poincare uncovered that chaos is the essence of nonlinear systems of which unpredictability reigns.

INTO THE 20^TH CENTURY

Up until the 1970s, computers were not widely used, but existed. Before computers, all research involving chaos was achieved without the aid of mechanical processing from computers. Scientists, mathematicians and physicists from around the world conducted experiments involving characteristics inherent to chaos. Experiments in turbulence, period doubling, bifurcation and biological studies in population added to the understanding of chaotic order. Two mathematicians from France, Pierre Fatou and Gaston Julia conducted research in the area of shapes, known as fractals. The two did not work together, but combined, their research covered much in the area of complex analysis. By applying a calculus formula, Julia discovered that simple mappings of "many" or "complex" numbers led to intricate, complicated geometric shapes.

The birth of computers, no doubt, aided mathematician Benoit Mandelbrot in his research of just about everything! In the 1960s, he began to realize that the culmination of his work in different areas produced a common thread: a geometric structure, evolving from the dynamics of nonlinear design that was soon to be known as a fractal. In the 70s Mandelbrot was able to exploit his work through the use and aid of computer graphics, thus refining both his and the previous work of Julia and Fatou. Mandelbrotâs work became so popular that the name, "Mandelbrot Set" refers to a specific set of points on a complex plane. A mathematical example of this set is described as follows, (13) "Pick a point Z in the complex plane

Calculate: Z= Z + Z, Z= Z + Z, Z= Z + Z, Z= Z + Z

If the sequence Z,Z,Z,Z· remains within a distance of 2 of the origin forever, then the point Z is said to be in the Mandelbrot Set."

Picture of Mandelbrot Set

Also in the 1960s, an American by the name of Edward Lorenz made discoveries that are now thought to be way ahead of his time. He deduced that nonlinear differential equations, in a finite system of deterministic nature, could describe and represent certain patterns in nature, mostly related to weather. Through the use of computers and dimensional representation of the phase space, Lorenz had demonstrated his theory for all to see.

THE BUTTERFLY EFFECT

Even though all previous discoveries laid a foundation for the phenomenon of nonlinear dynamics, only one event actually culminated all the theories into one; basically a new concept emerged called chaos. The butterfly effect was the result of Lorenzâs computer experiment on weather conditions. He made the significant discovery of how minute changes in initial conditions changed the dynamics of a total given system, when stretched out over a long period of time. This was an accidental discovery for the man who was already ahead of his time. Applying his new discovery, he concluded that, conditions in weather that are forming in one direction could take a drastic change and reinvent themselves into something entirely different. This is the idea he coined the butterfly effect.

. http://library.thinkquest.org/26242/full/index.html

The Lorenz attractor-Although it might not seem so right away, this figure is a fractal since magnifying it does not decrease the amount of detail:

FRACTALS

As Chaos began to gain new audience, it was discovered to exist in many areas of our world. As mentioned earlier, Mandelbrot was instrumental in defining fractals. The complicated mathematical formulas for fractals create endless beauty and intrigue for the aesthetic observer. A factal is exactly what the name implies, a fraction or small chunk of a larger, similar parent object. The fractal adheres to no scale, but when viewed microscopically, the small patterns that make up the whole are commonly referred to as "self-similarâ objects. If the scale of the object is iterated down in size, the same object seems to appear, but in a rougher, more grooved form, just like many objects in nature. Geometric shapes appear in nature in the form of snowflakes, clouds, coastlines and mountains. Now computers can simulate these fractals. Computerized pictures are produced on screen by mathematical formulas. A symmetrical formula produces a symmetric picture. Pixels on the computer screen follow the rules of a given formula and construction of a beautifully colored geometric design unveil. So fractals consist of the same ingredients described for chaos. That is, dynamic movement within a determined environment based on sensitivity to initial conditions and propelled by strange attractors. The symmetry created looks random and chaotic but it is not. Of course, there is much more to it than that, but again, my goal is just to give a general description.

CHAOS IS EVERYWHERE

Weather, geometric structures in nature, and celestial systems all exemplify patterns of chaos. But there is more, chaos thrives in areas you might never have been aware of. The stock market is one of those areas. Stock prices always appear to be random, and in the short term they are, but continual observation has revealed long term, deterministic behavior.

In business organizations, chaos lurks as an insidious perpetrator. An example is in the Information Systems department. Historically, IT systems never quite repeat their past behavior, they are always reinventing their structure, flourishing, dying off, and rebuilding. They exist amongst perpetual change to compensate for the unexpected.

Understanding chaos has also led modern medicine to apply its concepts to irregularities found in the heartbeats of patients. In addition, other biological and even theological studies have been made with the use of this theory and I am sure more are to come.

CONCLUSION

Conditions of chaos may be distinct and applications of living chaos can be analyzed, but there is still an abstract characteristic of chaos that is hard to place in reality. To sum it up using my own analogy, I would say that chaos could be viewed as the space within a shoebox. On the outside, the box is defined by its rectangular shape, of known composition and structure, but inside, the space could contain any number of ingredients, twisting and turning in what seems like unpredictability and randomness. These ingredients perform their bifurcations and iterations all within a determined structure, the box. Therefore, chaos lives within the structure.

Without chaos, our world would be boring and robotic. Any initial action would produce a common reaction. All of nature could be predicted, so why would we need to think very hard? If all systems behaved in a methodical order, it would probably throw the balance of the world off kilter. Ironically, it seems the presents of chaos ultimately yield control!

1) Briggs and Peat. Turbulent Mirror, N.Y.: Harper and Row, 1989

2) Stewart, Ian. Does God Play Dice?, Massachusetts: Blackwell, 1989

3) Lorenz, Edward, N. The Essence of Chaos, Seattle: University of WA Press, 1993

4) Field and Golubitsky. Symmetry in Chaos, N.Y., Oxford University Press, 1992

5) Lam, Lui. Non-Linear Physics for Beginners, N.J.: World Scientific Publishing Co, 1998

6) www.wfu.edu/~petrejh4/HISTORYchaos.html

7) www.mwmbwrs.home.net/jason.yiin/content.html

8) www.cms.dmu.ac.uk/~nkm/CHAOS.html

9) www.santafe.edu/~gmk/MFGB/node2.html

10) www.life.csu.edu.au/complex/tutorials/tutorial3.html

11) www.easyweb.co.uk/~zac/chapt17.html

12) www.chaos.umd.edu/misc/poincare.html

13) www.library.thinkquest.org/3120/text.html