Jane-Valeriane Kimberly BouaChaos Theory Hubert Bray

Math and the Universe - Math 89SDuke University

4/26/2016

Introduction

Chaos theory studies the behavior of dynamic systems that are sensitive to initial

conditions. It is more popularly known as the butterfly effect. Scientifically speaking, chaos is

the apparent lack of order in a system that nevertheless obeys natural laws and rules (Rousee).

Small differences in the initial conditions of the system can yield widely different outcomes in

the future, which makes long term predictions difficult. The large number of outcomes occur

even though these systems are deterministic, meaning their future behavior can be completely

determined by their initial conditions. Although they are deterministic in behavior, these systems

behavior are unpredictable. Systems that exhibit mathematical chaos are deterministic and are

hence orderly, which creates a sort of paradox. If a system is chaotic, how can it hence be

orderly. There is no universal definition for chaos, but a commonly used definition by Robert L.

Devaney states that for there to be chaos, the dynamical system must be sensitive to initial

conditions, must be topographically mixing, and must also have dense periodic orbits (Rousee).

Topological mixing means that a system will change over time in way that space will overlap

over time. This means that although chaotic systems appear to be random and disordered, they

are precisely controlled by physical laws of nature.

Chaotic behavior exists in many natural systems including weather and climate and there

are many applications to sociology, physics, and computer science. The study of chaos theory

and fractal mathematics is important to understanding the physical world. Mankind enjoys

predictability, especially when it comes to abstract systems such as the weather. Understanding

these systems will allow us to not only predict but to also better control weather patterns.

The two major components of chaos theory are that no matter how complex a system

may be, they are an underlying mechanism and order to the system, and that small, simple,

changes in the system can have drastic changes in the system in the future.

Principles of Chaos

The Butterfly effect describes the sensitivity of initial conditions. It theorizes that if a

butterfly flaps in wings in, for example, Brazil, it can result in a hurricane in Florida for example.

The idea exemplifies the idea that small effects can cause great changes in the future (Mendelson

and Blumental). The idea suggests that the flap of a butterfly’s wings will create tiny changes in

the atmosphere, that may ultimately change the course of a tornado or hurricane, move a weather

system from one place to another, or even accelerate the occurrence of a weather system. The

butterfly does not directly cause the hurricane or tornado, but rather creates a chain of events that

will later cause a hurricane or tornado. The butterfly effect is also best exemplified in simple

systems such as throwing a pair of dice (Crystal Links). The randomness that is exhibited when a

pair of dice are thrown are very sensitive to the initial conditions of the systems, making if very

difficult to repeat the same throw twice. The butterfly effect has had a large influence in mass

media. The butterfly effect often times involves time travel, in which an individual travel back in

time in order to fix some part of the past, and their actions drastically alters the events of future.

Unpredictability is also another major component of chaos, there is no way to predict the

ultimate fate of a complex system, even the slight changes in measuring the state of a system will

cause drastic changes in the results, which forces the predictability of such system virtually

impossible (Crystallinks).

Edward Lorenz

Edward Lorenz was a meteorologist, and father of the chaos theory. Dr. Lorenz is

credited with having simulated Chaos theory after using a computer program to model the

weather. Lorenz discovered chaos theory when he was trying to predict the weather using a

computer program. He set initial conditions and let the program run. He input 0.506127 for the

initial run of the program. He then started a longer, second run, of the program in which he input

the same initial conditions, but instead of entering the full 0.506127, he rounded to 0.506

(Chang). He expected the results of the second run to match the results of the first run, but rather,

the results completely diverged from the results of the first run. He initially thought that the

program was malfunctioning but later realized that there was no way to predict the weather

perfectly. He realized that a perfect forecast would have to accompany a complete understanding

of temperature, wind, humidity and other conditions a certain point in time. He also realized that

it would be practically impossible to create a computer system that could predict the behavior of

weather perfectly (Chang). For example, the computer maybe able to predict when Tornado

season is, but how would it predict that a tornado would land in a particular town, on a particular

date and time, weeks in advance of it actually happening? If there was even a small discrepancy

in any condition, it would lead to totally different weather. Dr. Lorenz published his

groundbreaking findings in an atmospheric journal article called “Deterministic Nonperiodic

Flow”.

Atmospheric Chaos

One of the most widely studied areas of chaotic system must be atmospheric, weather

related chaos. There are few natural phenomena that change so radically and unpredictable as the

daily weather. Climate was expected to behave as averages, with small fluctuations over time

that would eventually cancel each other out, but it was found that these large systems were

extremely unstable. It was discovered, mainly by Dr. Lorenz, that these extremely large systems

Edward Lorenz, a meteorologist, made a lot of headway in understanding atmospheric

conditions. His paper is considered the birth of chaos theory (Chang). He suggests three simple

differential equations to describe the behavior of a gaseous system. He simplified the nonlinear

Navior-Stokes Equation which were used to describe fluid motion.

Although these equations are extremely simplified,

they describe chaotic behavior and are sensitive to the

initial conditions of a system. The values that are

commonly used in these equations re a-10, r=28,

b=8/3. These values create the butterfly effect that is shown below. The same formulas are used

to mimic the behavior of the atmosphere, but make it almost impossible to predict. The equations

break down the particles of gaseous systems, are also required to break down the “particles” of

the atmosphere, which is extremely difficult to breakdown. This makes it extremely difficult and

almost impossible forecast for more than a few days at a time (Xeng, Pielke, Eykholt).

Lorenz System and Attractor

Complex systems seem too chaotic to recognize a pattern with the naked eye, but

computers take large arrays of parameters to create points on a graph. Attractors are diagrams

that are the most state trajectories that wind up approaching some common limit (Bradley). The

Lorenz attractor is perhaps one of the best known models of chaos theory. The Lorenz system is

a set of ordinary differential equations used to model atmospheric conventions studied by

Figure 1. The Lorenz Equations

Edward Lorenz. Although it is difficult to analyze, but the results of the differential equations are

modeled by a simple geometric equation. The Lorenz equations also come up in simplified

models for electric circuits, chemical reactions and forward osmosis. They have also been

subject to countless research studies. The Lorenz attractor is an example of a strange attractor

(Bradley). Strange attractors are unique in that it is impossible to determine where on the

attractor the system will be, making them chaotic systems. Two points on the attractor that are

near each other at one time can be arbitrarily far apart at later times. Strange attractors are unique

in that they will never close in on themselves and the motion of the system never repeats. Strange

attractors such as the Lorenz attractor are the epitome of chaotic behavior.

Fractals

Fractals are never-ending patterns. Fractals represent the images of dynamic systems and

depict chaos (Fractal Foundation). One perfect example of a fractal is the Mandelbrot Set. The

Mandelbrot Set is a set of complex numbers c for which the function does not diverge.

Mandelbrot Set images are created by sampling the complex numbers and identifying whether or

Figure 2. Image depicting the geometric results of the Lorenz Attractor. The image is of a figure 8 and resembles the wings of a butterfly

Figure 3. Image depicting the Lorenz attractor at 4 different points in time

not the results approach infinity. The real and imaginary parts of each number is treated with

pixels and is colored according to how rapidly the sequence diverges.

These systems are great examples of chaotic systems. Fractals are very complex and are not

often found in nature, but the underlying patterns of these systems are very obvious (Fractal

Foundation).

Quantum Chaos

Chaotic systems are enticing for theoretical mathematicians and physicists because of

their random like behavior yet they are deterministic. The concept has been introduced to attempt

to understand quantum chaos and understand its mechanism and physical meaning. Quantum

physicists have coined the phenomenon pseudo chaos (Rednick). Quantum chaos is a branch of

physics that studies how chaotic dynamic systems using quantum theory. Based on the

correspondence principle, which states the behavior of systems can be described using quantum

mechanics and is therefore classical physics, chaotic theories must have underlying quantum

mechanics (Rednick). If quantum mechanics does not have sensitivity to initial conditions, which

means there must be some truth in this sensitivity to the correspondence principal, which mean

Figures 4 and 5. These two figures represent fractals or never ending patterns. The one on the left is an example of a Mandelbrot Set.

there is a quantum basis. There are a number of methods that have been employed in the hopes

of trying to understand quantum chaos including solving the perturbation theory.

Applications

The chaos theory has a number of applications from science and mathematics to

sociology. In sociology, an effect known as the ripple effect stems from ideas put forth by chaos

theory. The ripple effect describes the phenomenon where situations can affect the direction of

other situations not directly related to the initial interaction, as when a rock is dropped in a pond

creating ripples in water. For example, a charity event where information and resources is passed

from community to community, increasing the impact (Tracy). A similar phenomenon, known as

the domino effect, is a situation where one event sets off a chain of other similar events. The

stock market has also used chaos theory to explain situations where small fluctuations in the

market effect other stocks. Chaos theory has also been used to describe natural phenomena

including population fluctuations, disease, mental illness, and political unrest (Tracy).

Critics

Some critics argue that weather systems may not be as sensitive to initial conditions as

previously thought. Most critics suggest that the chaotic theory is a radical departure from the

fundamentally deterministic world view, and the notion of “trajectory” obsolete (Gaspard,

Bricmont). J. Bricmont makes a lot of claims about the reliability of chaotic theory. He states,

“The notion of chaos leads us to rethink the notion of “law of nature”. For chaotic systems

trajectories are eliminated from the probabilistic description… The statistical description is

irreducible. The existence of chaotic dynamical systems supposedly marks a radical departure

from a fundamentally deterministic world view, makes the notion of trajectory obsolete, and

offers a new understanding of irreversibility.” (Gaspard, Bricmont). If there is no way to

determine the path that the variables of a system will take, then it is virtually impossible to

reverse it.

Conclusion

Mankind continues to try and understand abstract systems including chaotic ones. It is

often difficult to find the line between what is chaotic, what is random, or if there even is a

difference. What makes chaos theory so difficult to understand is not the deterministic behavior,

but rather the unpredictability of dynamic systems. Chaotic systems are difficult to understand,

yet they appear simple in nature because they are deterministic. Atmospheric chaos is one aspect

of chaotic theory is most interesting to decipher. Edward Lorenz, who was a great proponent of

atmospheric chaos, is considered the father of the theory and one of the first to analyze the

chaotic systems of nature. James Gleick, who is author of a novel called Chaos: Making a New

Science, refers to chaos theory as “a revolution not of technology, like the laser revolution or the

computer revolution, but revolution of ideas. This revolution began with a set of ideas having to

do with with disorder in nature: from turbulence in fluids, to the erratic flows of epidemics, to

the arrhythmic writhing of a human heart in the moments before death. It has continued with an

even broader set of ideas that might be better classified under the rubric of complexity.” Gleick

effectively summarizes what chaotic theory is. Chaotic Theory has roots in not only

mathematical and quantum theories but has greater implications in the universe and in society.

From biological systems, to sociological functions such as the ripple and the domino effect.

Works Cited

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The New York Times. The New York Times, 2008. Web. 22 Apr. 2016.

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2016.

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n.d. Web. 22 Apr. 2016.

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Web. 22 Apr. 2016.

https://www.mcgoodwin.net/julia/mcmloren.gif (Image: Lorenz Attractor) http://pmrb.net/uos/?q=4_3_3 (Image: Lorenz Attractor) http://www.crystalinks.com/chaos.html