Chaos Theory

An Introduction to Mathematical Chaos Theory and Fractal Geometry

» Posted by on Dec 3, 2014 in References | 0 comments

Chaos Theory

Editor’s note: Manus J. Donahue, III composed this essay in December, 1997 as a student at Duke University majoring in physics, mathematics and philosophy. It has been cited in The New York Times and has been awarded technology site of the day. The article has been reprinted here with the author’s permission.

This page has also been selected by the researchers at Lightspan StudyWeb® to receive their award as one of the best educational resources on the Web.



The stock markets are said to be nonlinear, dynamic systems. Chaos theory is the mathematics of studying such nonlinear, dynamic systems.

The world of mathematics has been confined to the linear world for centuries. That is to say, mathematicians and physicists have overlooked dynamical systems as random and unpredictable. The only systems that could be understood in the past were those that were believed to be linear, that is to say, systems that follow predictable patterns and arrangements. Linear equations, linear functions, linear algebra, linear programming, and linear accelerators are all areas that have been understood and mastered by the human race. However, the problem arises that we humans do not live in an even remotely linear world; in fact, our world should indeed be categorized as nonlinear; hence, proportion and linearity is scarce. How may one go about pursuing and understanding a nonlinear system in a world that is confined to the easy, logical linearity of everything? This is the question that scientists and mathematicians became burdened with in the 19th Century; hence, a new science and mathematics was derived: chaos theory.

The acceptable definition of chaos theory states, chaos theory is the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems. A dynamical system may be defined to be a simplified model for the time-varying behavior of an actual system, and aperiodic behavior is simply the behavior that occurs when no variable describing the state of the system undergoes a regular repetition of values. Aperiodic behavior never repeats and it continues to manifest the effects of any small perturbation; hence, any prediction of a future state in a given system that is aperiodic is impossible. Assessing the idea of aperiodic behavior to a relevant example, one may look at human history. History is indeed aperiodic since broad patterns in the rise and fall of civilizations may be sketched; however, no events ever repeat exactly. What is so incredible about chaos theory is that unstable aperiodic behavior can be found in mathematically simply systems. These very simple mathematical systems display behavior so complex and unpredictable that it is acceptable to merit their descriptions as random.

An interesting question arises from many skeptics concerning why chaos has just recently been noticed. If chaotic systems are so mandatory to our every day life, how come mathematicians have not studied chaos theory earlier? The answer can be given in one word: computers. The calculations involved in studying chaos are repetitive, boring and number in the millions. No human is stupid enough to endure the boredom; however, a computer is always up to the challenge. Computers have always been known for their excellence at mindless repetition; hence, the computer is our telescope when studying chaos. For, without a doubt, one cannot really explore chaos without a computer.

Before advancing into the more precocious and advanced areas of chaos, it is necessary to touch on the basic principle that adequately describes chaos theory, the Butterfly Effect. The Butterfly Effect was vaguely understood centuries ago and is still satisfactorily portrayed in folklore:

For 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!

Small variations in initial conditions result in huge, dynamic transformations in concluding events. That is to say that there was no nail, and, therefore, the kingdom was lost. The graphs of what seem to be identical, dynamic systems appear to diverge as time goes on until all resemblance disappears.

Perhaps the most identifiable symbol linked with the Butterfly Effect is the famed Lorenz Attractor. Edward Lorenz, a curious meteorologist, was looking for a way to model the action of the chaotic behavior of a gaseous system. Hence, he took a few equations from the physics field of fluid dynamics, simplified them, and got the following three-dimensional system:




Delta represents the “Prandtl number,” the ratio of the fluid viscosity of a substance to its thermal conductivity; however, one does not have to know the exact value of this constant; hence, Lorenz simply used 10. The variable “r” represents the difference in temperature between the top and bottom of the gaseous system. The variable “b” is the width to height ratio of the box which is being used to hold the gas in the gaseous system. Lorenz used 8/3 for this variable. The resultant x of the equation represents the rate of rotation of the cylinder, “y” represents the difference in temperature at opposite sides of the cylinder, and the variable “z” represents the deviation of the system from a linear, vertical graphed line representing temperature.

If one were to plot the three differential equations on a three-dimensional plane, using the help of a computer of course, no geometric structure or even complex curve would appear; instead, a weaving object known as the Lorenz Attractor appears. Because the system never exactly repeats itself, the trajectory never intersects itself. Instead it loops around forever. I have included a computer animated Lorenz Attractor which is quite similar to the production of Lorenz himself. The following Lorenz Attractor was generated by running data through a 4th-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 parameters were a =16, r =45, and b = 4 for the following equations (similar to the original Lorenz equations):




The initial position of the projectory was (8,8,14). When the points were generated and graphed, the Lorenz Attractor was produced in 3-D. The attractor will continue weaving back and forth between the two wings, its motion seemingly random, its very action mirroring the chaos which drives the process. Lorenz had obviously made an immense breakthrough in not only chaos theory, but life. Lorenz had proved that complex, dynamical systems show order, but they never repeat. Since our world is classified as a dynamical, complex system, our lives, our weather, and our experiences will never repeat; however, they should form patterns.


Lorenz, not quite convinced with his results, did a follow-up experiment in order to support his previous conclusions. Lorenz established an experiment that was quite simple; it is known today 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 Ferris-wheel seats, so that the buckets would always point 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 and 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.

Now it may be accepted from Lorenz and his comrades that our world is indeed linked with an eerie form of chaos. Chaos and randomness are no longer ideas of a hypothetical world; they are quite realistic here in the status quo. A basis for 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 rudimentary fundamentals. This new form mentioned is highly complex, repetitive, and replete with intrigue.

“If we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and the universe exist.”
— Stephen Hawking

The extending and folding of chaotic systems give strange attractors, such as the Lorenz Attractor, the distinguishing characteristic of a nonintegral dimension. This nonintegral dimension is most commonly referred to as a fractal dimension. Fractals appear to be more popular in the status quo for their aesthetic nature than 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 up this glitzy image? The classical Euclidean geometry that one learns in school is quite different than the fractal geometry mainly because fractal geometry concerns nonlinear, nonintegral systems while Euclidean geometry is mainly oriented around linear, integral systems. Hence, 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 even 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 and conic sections that we have learned about in fundamental Euclidean geometry classes. By taking the midpoints 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 or times and eventually a very simple fractal arises:


tri1 tri2 tri3 tri4


In addition to the famous Sierpenski Triangle, the Koch Snowflake is also a well noted, simple fractal image. To construct a Koch Snowflake, begin with a triangle with sides of length 1. At the middle of each side, add a new triangle one-third the size; and repeat this process for an infinite amount of iterations. The length of the boundary is 3 X 4/3 X 4/3 X 4/3…-infinity. However, the area remains less than the area of a circle drawn around the original triangle. What this means is that an infinitely long line surrounds a finite area. The end construction of a Koch Snowflake resembles the coastline of a shore.


The two fundamental fractals that I have included provided a basis for much more complex, elaborate fractals. Two of the leading researchers in the field of fractals were Gaston Maurice Julia and Benoit Mandelbrot. Their discoveries and breakthroughs will be discussed next.

On February 3rd, 1893, Gaston Maurice Julia was born in Sidi Bel Abbes, Algeria. Julia was injured while fighting in World War I and was forced to wear a leather strap across his face for the rest of his life in order to protect and cover his injury. He spent a large majority of his life in hospitals; therefore, a lot of his mathematical research took place in the 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 for x, say x=a, then the following would result:

a, f(a), f(f(a)), f(f(f(a))), etc.

Repeatedly applying “f” to “a” yields arbitrarily 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=a for which the iterates of “f” applied to “a” remain bounded. The following fractals belong to the Julia set.

fr1 fr2

Julia became famous around the 1920’s; however, upon his demise, 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 him 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.

“I coined ‘fractal’ from the Latin adjective fractus. The corresponding Latin verb frangere means “to break”: to create irregular fragments.”
— Benoit Mandelbrot

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