1 - Georgia Institute of Technology



Is it all about Chaos?

Final Project Report

MATH 6514: Industrial Mathematics I

Fall 2002

Georgia Tech

Authored by:

Manas Bajaj

Voice: 404-385-1674

manas.bajaj@eislab.gatech.edu

Zhang, Qingguo

Voice: 404-463-0886

gtg088j@mail.gatech.edu

Sripathi Mohan

Voice: 404-463-0886

gtg755i@mail.gatech.edu

Thao Tran

Voice: 404-463-0886

gtg234j@mail.gatech.edu

Evaluated by:

Dr. John McCuan

Professor

Department of Mathematics

Skiles 265, Georgia Tech

Atlanta, GA 30332

USA

Voice: (404) 894 4752

mccuan@math.gatech.edu

Table of Contents

1. Introduction………………………………………………………..3

2. Some examples of “Chaos”………………………………………5.

3. Trying our hands at “Chaos”…………………………………...13

4. Conclusion………………………………………………………..25

5. The Road ahead………………………….…...……………….…26

6. References……………………………………………………….. 27

1. Introduction

What we imagine is order is merely the prevailing form of chaos.

Kerry Thornley, Principia Discordia, 5th edition

Reflecting on the quote above and investigating the true nature of how things behave around us, it might not be bad idea to understand what Chaos is. Is it something that we see each day and disregard? Does it have any explanation? Is it a science? Is it something merely supernatural that provokes one’s intellect to unravel its mystery? Is it a theory, completely antipodal to Deterministic Sciences?

By the way, if you take a careful look at the paragraph above, you would see a bunch of question marks (?). That’s Chaos! ( With its presence, there shall always be those queries as to how will a system behave.

a.) What is Chaos?

Let’s ask a simple question first? What does a chaotic system do? A chaotic system is not stable (its behavior changes with time) and it’s not periodic (it doesn’t trace its path back and forth and hence it’s not periodic). So is it irregular motion? NO! A Chaotic system is an unpredictable system, whose behavior is highly dependent on its initial conditions and there is order in which it evolves to its unpredictable disordered behavior. You might think that it’s somewhere on the boundary between order and disorder but there is more to it.

Chaos is "orderly disorder created by simple processes."

The genesis of this theory started with a study into instabilities in dynamical systems by physicist Henri Poincaré in 1900. He was interested in the mathematical equations that would describe the motion of planets around the sun.

The philosophy of determinism speaks about the fact that every action in this universe is causal and can be predicted. The universe, in its behavior, unfolds with time in a predefined manner, according to the laws of physics. Every action in future or retrospect can accurately be predicted form the current happenings. Newton’s Laws a glaring example of the deterministic theory. The three laws of motion talk about the behavior of systems in a manner described by mathematical equations. The laws of motion, combined with the body of the knowledge concerning a specific science, were believed to provide an accurate insight into the future behavior of a system. Newton believed that this could be applied to even study the motion of planets and described their paths in the future.

For predicting the exact behavior of any system, whether it the convection currents in the atmosphere, the motion of a heavenly body or even a falling object, one needs to know the initial conditions of the system. The Newtonian theory says that the behavior can completely be determined and is exactly the same for the same initial conditions. The question is can we have exactly the same initial conditions?

Real world systems behave based on measurements taken in reality. The initial conditions of a system can never be infinitely accurate. The scientific faculty has been lead to the belief that an almost accurate initial condition will lead to an almost accurate prediction of the behavior of the system, which is not the true case and this is what Chaos defies. We can never measure the initial conditions of a system accurately and some systems are so sensitive to these initial conditions that over a period of time, they completely diverge from what their behavior would been had they started with an initial condition seemingly indistinguishable from the current one. A good example of this is the tossing of a coin. No matter how hard one may try to let the system have the same initial conditions, it shall always be different and hence the result would vary from “head” or “tail” from seemingly same initial conditions.

From the above, we can infer that Uncertainty in dynamical systems doesn’t arise due to the equations of behavior of the system since they are completely deterministic but due to the shift in the initial conditions of the system.

The investigations of Henri Poincaré challenged the tacit assumption that an almost accurate measurement of the initial conditions was enough to predict their behavior correctly. When used for the case of measuring planetary movements, this can have a large impact on the predictions made, based on some initial conditions, since there will always be a greater degree of inaccuracy in measuring the initial conditions of such systems.

2. Some examples of “Chaos”

Real life is abuzz with examples from this realm and we would definitely like to touch upon a few, including the ones that were simple enough for us to try out within the time frame of the project.

(Example: 1)

The most striking revelations about the existence of chaotic systems started with Edward Lorenz’s simulations in 1960 wherein he was trying to model a problem on weather prediction and he had a system of twelve equations that his computer setup was trying to solve. Based upon the results of one day, he wanted to check a particular pattern again and to save time, he decided to start from somewhere in the middle of the iteration rather than the beginning by looking at the value from a computer printout of the iteration. When he saw the new results, he was astonished since the predictions completely diverged from what they had been the day before. He noticed that the memory of the computer stored the numbers (solution of the equations) to six digits of decimal and he had, in order to save paper, printed only the first three digits. He had entered 0.506 instead of 0.506127 and hence, over a period of iteration, the current simulation had completely diverged from the results that he had the day before. A figure is included herewith, taken from Ian Stewart’s book on The Mathematics of Chaos.

[pic]

Figure 1: Results from Lorenz’s atmospheric model. This has been taken from Ian Stewart’s book, “Does God play Dice? The Mathematics of Chaos”

The difference between the starting points of the two curves is so small that it can be compared to flapping of the wings of a butterfly and rightly so, this effect came to be known as The Butterfly Effect and as Ian Stewart puts it in his book,

The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)

(Example: 2)

A second set of example that sprung from Lorenz’s work thereafter was the Waterwheel experiment. He reduced down the set of twelve equations from the weather model and made a system of equation simpler to observe this behavior. Later on, it was found that he his new set of equation predicts the dynamics of a Waterwheel.

Essentially, it consists of a set of cups and a stream of water flowing from some place at the top. Depending upon the speed of flow of water and the resulting motion and thereby the resulting angular velocity and the amount of water that collects in the cups, this system turns to show chaos and is highly dependent on the initial

.[pic]

Figure 2: Lorenz’s Waterwheel experiment, the dynamical equations for which were unknowingly derived by Lorenz by stripping down hi model for atmospheric weather prediction

(Example 3)

Another example is regarding the prediction of biological populations. The equation would be simple if population just rises indefinitely, but the effect of predators and a limited food supply make this equation incorrect. The simplest equation that takes this into account is the following:

Next’s population = r * this year's population * (1 - this year's population)

In this equation, the population is a number between 0 and 1, where 1 represents the maximum possible population and 0 represents extinction. R is the growth rate. The question was, how does this parameter affect the equation? The obvious answer is that a high growth rate means that the population will settle down at a high population, while a low growth rate means that the population will settle down to a low number. This trend is true for some growth rates, but not for every one.

Robert May, a biologist, decided to see what would happen to the equation as the growth rate value changes. At low values of the growth rate, the population would settle down to a single number. For instance, if the growth rate value is 2.7, the population will settle down to .6292. As the growth rate increased, the final population would increase as well. Then, something weird happened. As soon as the growth rate passed 3, the line broke in two. Instead of settling down to a single population, it would jump between two different populations. It would be one value for one year, go to another value the next year, then repeat the cycle forever. Raising the growth rate a little more caused it to jump between four different values. As the parameter rose further, the line bifurcated (doubled) again. The bifurcations came faster and faster until suddenly, chaos appeared. Past a certain growth rate, it becomes impossible to predict the behavior of the equation. However, upon closer inspection, it is possible to see white strips. Looking closer at these strips reveals little windows of order, where the equation goes through the bifurcations again before returning to chaos. This self-similarity, the fact that the graph has an exact copy of itself hidden deep inside, came to be an important aspect of chaos.

The diagram below is the one being discussed above.

[pic]

Figure 3: The Bifurcation Diagram for the population equation (James Gleich, “Chaos - Making a new Science”, Pg 71)

(Example 4)

This example is something for which we were able to set up an experiment and investigate the mathematical model too. The next chapter discussed about this example.

The motion of a DOUBLE PENDULUM is a good example of a chaotic system.

As shown below, the motion of such a system can be highly dependent upon the initial conditions ((1 and (2). From the plot below, we can see that the behavior for such a system can be highly non periodic.

[pic]

[pic]

Figure 4: The Double Pendulum Experiment and the response of (1 and (2 as function of time

(Example 5)

Another example of a chaotic system is the Magnetic Pendulum. The bob of the pendulum has a magnet attached to it and is under the effect of the underlying magnets, of reverse polarity, and is free to move. The motion of the pendulum is highly sensitive on the initial position from where this system is left to move. The pendulum will, after some time interval, hang exactly above one of the underlying magnets and which magnet the pendulum will hang on top of is very much sensitive to the initial conditions.

We did perform this experiment too and were able to observe this behavior. We plan to demonstrate this behavior in our Final presentation. The next chapter, where we discuss our results, will talk about the same.

[pic]

Figure 5: Setup for the Magnetic Pendulum Experiment

(Example 6)

One of the most interesting examples that we sighted on Chaos was regarding the behavior of the solution of a system of two equations.

y = x2 + c (1)

y = x (2)

We may start from some initial condition (initial value of x) and then get the corresponding value of “y” from the first equation. Now, this becomes our new “x”, as per the second equation and we keep on iterating hence.

Basically, we have developed the following mapping:

f: x --> x2 + c.

The path followed by “y” is called the orbit of the system and the initial value of “x” with which we started is called the seed of the orbit. Hence {x, f(x), f2(x),….fn(x)} is the trajectory of the orbit.

Investigating the system for c=0,

will yield the following results:

[pic]

All orbits approach either zero or infinity except for those with seed x = +/-1. The points zero and infinity are called attracting fixed points or sinks because they attract the orbits of the points around them while +/-1 are called repelling fixed points or sources for the opposite reason.

Investigating the system for values of c around c=1/4,

Y

[pic][pic][pic] X

c >1/4 c = 1/4 -3/4  ................
................

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