Chaos Theory Explained

Chaos Theory Explained Essay, Research Paper

The Stability of our universe in the face of Chaos

-By Josh Allouche & Chance Vanguard

?Traditionally, scientists have looked for the simplest view of the world around us. Now, mathematics and computer powers have produced a theory that helps

researchers to understand the complexities of nature. The theory of chaos touches all disciplines.?

-Ian Percival, The Essence of Chaos

Part I: The Basics of Chaos.

Watch a leaf flow down stream; watch its behavior within the water? Perhaps it will sit upon the surface, gently twirling along with the current, dancing around

eddies, slightly spinning, then all of a sudden, it slaps into a rock or gets sucked beneath the water by a small whirlpool. After doing this enough times one will realize

it is nearly impossible to accurately predict a leaf?s travel downstream, as the slightest change in its position can result in a severe deviation from it?s original path. A

small change in one variable can have a disproportional, even catastrophic, impact on other variables; this is the signature of chaos. By no means, though, is that the

extent.

Scientists used to, before the chaos theory, believe in the theory of reductionism, many still do. Reductionism imagines nature as equally capable of being assembled

and disassembled. Reductionists think that when everything is broken down a universal theory will become evident that will explain all things. Reductionism implied

the rather simple view of chaos evident in Laplace?s dream of a universal formula: Chaos was merely complexity so great that in practice scientists couldn?t track it,

but in principle they might one day be able to. When that day came there would be no chaos, everything in existence would be perfectly predictable, no surprises,

the world would be safely mutable. The universe would be completely controlled by Newton?s laws. 1

Chaos touches all things in existence, and all sciences, mathematics, physics, biology, anthropology, entomology, astronomy, even the Ivory Tower science of

Newtonian physics?.

In the last years of the 19th century French mathematician, physicist and philosopher Henri Poincare? stumbled headlong into chaos with a realization that the

reductionism method may be illusory in nature. He was studying his chosen field at the time; a field he called ?the mathematics of closed systems? the epitome of

Newtonian physics.

A Closed system is one made up of just a few interacting bodies sealed off from outside contamination. According to classical physics, such systems are perfectly

orderly and predictable. A simple pendulum in a vacuum, free of friction and air resistance will conserve its energy. The pendulum will swing back and forth for all

eternity. It will not be subject to the dissipation of entropy, which eats its way into systems by causing them to give up their energy to the surrounding environment.

Classical scientists were convinced that any randomness and chaos disturbing a system such as a pendulum in a vacuum or the revolving planets could only come

from outside chance contingencies. Barring those, pendulum and planets must continue forever, unvarying in their courses.2

It was this comfortable picture of nature that Poincare? blew apart when he attempted to determine

The stability of our solar system?

For a system containing only two bodies, such as the sun and earth or earth and moon, Newton?s equations can be solved exactly: The orbit of the moon around the

earth can be precisely determined. For any idealized two-body system the orbits are stable. Thus if we neglect the dragging effects of the tides on the moon?s

motion, we can assume that the moon will continue to wind around the earth until the end of time. But we also have to ignore the effect of the sun and other planets

on this idealized two-body system.3

Poincare?s problem was that when an additional body was added to the situation, like the influence of the sun, Newton?s equations became unsolvable. What must

be done in this situation is use a series of approximations to close in on an answer. In order to solve such an equation, physicists were forced to use a theory called

?Perturbation?. Which basically works in a third body by a series of successive approximations. Each approximation is smaller than the one before it, and by adding

up a potentially infinite amount of these numbers, theoretical physicists hoped to arrive a working equation.

Poincare? knew that the approximation theory appeared to work well for the first couple of approximations, but what about further down the line, what effect would

the infinity of smaller approximations have? The multi-bodied equation Poincare? was attempting was essentially a Non-linear equation. As opposed to a

differential or linear equation.

For science, a phenomenon is orderly if its movements can be explained in the kind of cause-and-effect scheme represented by a differential equation. Newton first

introduced the differential idea throughout his famous laws of motion, which related rates of change to various forces. Quickly scientists came to rely on linear

differential equations. Phenomena as diverse as the flight of a cannonball, the growth of a plant, the burning of coal, and the performance of a machine can be

described by such equations. In which small changes produce small effects and large effects are obtained by summing up many small changes.4

A non-linear equation is quite different. In a non-linear equation a small change in one variable can have a disproportional, even catastrophic impact on other

variables.5 Behaviors can drastically change at any time. In linear equations the solution of one equation allows the solver to generalize to other solutions; in

non-linear equations solutions tend to be consistently individual and unrelated to the same equation with different variables.

In Poincare?s multi-bodied equation, he added a term that added nonlinear complexity to the system (feedback) that corresponded to the small effect produced by

the movement of the third body in the system. As he experimented, he was relieved to discover that in most of the situations, the possible orbits varied only slightly

from the initial 2-body orbit, and were still stable? but what occurred during further experimentation was a shock. Poincare? discovered that even in some of the

smallest approximations some orbits behaved in an erratic unstable manner. His calculations showed that even a minute gravitational pull from a third body might

cause a planet to wobble and fly out of orbit all together.6

PART II: Chaos in the solar system, The end of Earth.

Poincare?s discovery was not fully understood until 1953 by Russian physicist A. N. Kolmogorov.

Initially scientists believed that in theory they could break up a complicated system into its components before experimentation because any changes in patterns

would be small and not effect an established construct such as an orbit. Kolmogorov was not prepared to accept that the whole universe is a fraction of a decimal

point away from self-destruction. Unfortunately his research didn?t help.7

Kolmgorov concluded, from his own calculations, that the solar system won?t break up under its own motion provided that the influence of an additional

gravitational source was no bigger than a fly approximately 7000 miles away, and the cycles per planetary ?year? did not occur in a simple ratio like 1:2 1:3 or 2:3

and so on.7

But, what happens when the planet?s years form a simple ratio? Well, that would mean that with each orbit, the disturbance is amplified due to a steady input of

gravitational energy. It creates a resonance feedback effect much like a normal microphone amplifier.

Say you lie an amplifiers input mic directly in front of its output speaker. Any sound that enters the microphone will be played back through the speaker louder, that

playback will be picked up by the mic and amplified once again, eventually the volume will reach its critical point and the speaker will blow out.

Well, if this were so, is there proof? Does this really happen in space? Could this occur in our solar system? The answer is yes. Between mars and Jupiter there is

an asteroid belt, merrily flying around our solar system, minding its own business. Every once in a while, the asteroids will shift within their belt, as asteroids due, to

correspond with all the inherent gravities of our solar system. Well, as chaos goes, at random intervals Jupiter and one of these asteroids will form a relative orbit

with simple ?year? ratio, once centrifugal force reaches its critical limit, the asteroid will fly out of orbit and create a gap in the belt. By studying these gaps, scientists

have given validity to Kolmgorov?s simple ratio orbit hypothesis. In fact, Jack Wisdom of MIT determined from voyager?s flyby results that Hyperion, a moon of

Saturn, is in such a chaotic relationship at this time. The resonance theory may also explain the gaps in the rings of Saturn. 8

As it turns out, the solar system isn?t the simple cosmic clock pictured in Newton?s day, but a system of constantly changing, infinitely complex variables capable of

self-destruction if a little friction is applied. Is the solar system stable? Right now, there?s no positive answer.

Part III Identifying chaos

Chaos is everywhere. In fact the only place where there is no chaos is in scientific Ideals. But where does one find chaos in life? Well, in a sense, life is chaos;

chains of events that occur in life are the subjects of an infinite amount of variables in a multibodied equation so big it would take exactly your lifetime to write.

Chaos theory has popped up in Hollywood, as the intriguing science of Jeff Goldblum?s character on the blockbuster ?Jurassic Park?. In one scene he poses the

question of which way a drop of water will run when place upon the back of his hand.

This is a perfect example of chaos theory, being a non-linear equation, the variables are nearly infinite, and a small change in any one can dramatically affect the

result. Variables for this include the pressure of blood flow through his hand, imperfections in the surface of his skin, his body temperature, the waters initial

temperature, the height from which it was dropped, movement of his hand, ambient wind, ambient temperature. The only control for the experiment is the

gravitational constant of the earth, but that?s moving around space, so you have to account for the positioning of the planets and the earth?s moon.

If you could understand the nature of chaos, you could clean up in Vegas, but you?d have to account for all the variables present in a dice roll, besides the basics

such as angle of throw, intensity of throw and what numbers were face up when thrown, you have to account for table friction, tectonic plate movement, and ambient

air friction (taking into consideration the amount of cigarette smoke present). Sure some of these variables seem a bit unlikely of affecting the outcome, so lets

analyze one. Say tectonic plate movement?

In coincidence with Kolmgorov?s findings, most tectonic plate movements under Las Vegas will be like Kolmgorov?s fly at 7000 miles, and not affect the overall out

come, but there?s always Pincer?s chance, that a change in even the most unlikely or unimportant variable could have a catastrophic effect.

Say that at the precise moment that the dice first hit the table, there was a north western magma current directional shift, the plate resting upon this unstable mass

would shift, imperceptible to us, but not to a precariously balanced die deciding which direction to fall.

Some other instances of chaos in nature would be the motion of an earthquake, tornado, wind currents in a hurricane, the movements of the ocean, the growth of a

tree, the spasms of an epileptic, the spasmodic activity of a heart in cardiac arrest? But what is chaos? What does it look like?

Well, in attempts to graph the capabilities of chaotic equations with computers have produced something amazing, a beautiful form of art called fractals. Fractals are

considered the face of chaos, as each is a representation of a different non-linear equation.

If you were to graph the distance traveled by a free-falling ball at short time intervals, you would get a curve, because the ball is accelerating. While it is not easy to

compute exactly where the ball will be three seconds from now, your curve will tell you with a simple computation.

But now, we hit a block. Something so complex, we cannot find a curve to match it. Graph the weather over the past ten years and what do you get A seemingly

random set of fluctuations that apparently cannot be represented by an equation. This is chaos. There appears to be no pattern, and the only way to say for sure

where the graph will be in the future is to you have to wait until tomorrow

At first glance, fractals seem the same way. They are extremely complex, and they appear to have a random shape. But many fractals are generated through simple

mathematical equations. We may be able to use fractals as additional types of equations to which we can map our data. Fractals and Chaos are relatively new

branches of math, since they cannot be explored without powerful computers invented only recently. Without a doubt they have already improved our precision in

describing or classifying “random” or organic things. 9

The most well known of all fractals is the Mandelbrot Set, the Mandelbrot set iterates the equation (z=z^2+c) with z starting at 0 and c varying. Then the Julia Set,

which is essentially the same, but z is imaginary and c is real, creating a paradoxical iteration. 10

Part IV: Farewell

Chaos scientist?s hope that by understanding chaos they can accurately predict weather patterns, neutralize tornado-effects, and predict earthquakes. Essentially

predicting the future to an approximation of possible futures. Physicists hope to use chaos to understand feedback and resonance. Engineers would use knowledge

of chaos to build flawless machinery. Architects would use understanding of the chaos theory when determining locations to build.

The chaos theory holds infinite value to the average citizen as well as the lab scientist. Predicting chaotic outcomes would lead to better pool games, golf scores,

bowling ability. Of course, that would mean predicting the unpredictable, if it wasn?t impossible it would take all the fun out of the games.

In the infinite swirling entity that is chaos there is possibility. In fact, it has been said that chaos itself is nothing more than infinite possibility. Chaos brings to us the

wondrous world of chance, of likely and unlikely, of risks and challenges. If it weren?t for chaos, where would be the fun in life? Where would be the excitement?

We owe chaos at least some understanding, considering all chaos has done for us? Like, for example, scientists postulate that chaos is the nameless force

responsible for the creation of the universe, the formation of the planets, and the origin of life. Now, how is it that a destructive, completely unpredictable force such

as chaos have done all this?

Well, chaos allows all things to happen, making the unlikely or impossible infinitely possible, but infinitesimally improbable. So the probability that energy would

suddenly exist was met within the swirling chaos, and then further randomness occurred over the next infinite amount of time that created matter, caused the big bang,

randomly placed the planets, set them into motion, and spawned life, all purely by chance.

From the infinite nothingness of nonexistence our universe sprang into existence, filling with matter, energy and void. However unlikely, at least one planet spawned

single cellular life. Over time, elements were introduced to this life form that caused evolution, and over billions of years, these developing events continued, and the

life fell into sentience. The chances of something like this naturally occurring in the proper order is of course very rare, else we would most likely be aware of other

sentient species we share the universe with.

Within the grand kingdom of infinity, everything that we were, are and are going to become, amount to nothing more than a grain of sand on the beach compared to

the timeframe of infinity, and during the infinity in which we exist, lies infinite possibility. The same infinite possibility that has created and maintained life for billions of

years can just as quickly destroy it, and all of its other creations, at any time. But the beauty about chaos is that just as there are an infinite amount of possibilities that

we will be destroyed, there are also an infinite amount of possibilities that say that were here until the end of infinity.

Sources- General

Chaos: Making a New Science, James Gleick, Touchstone, 1995

The Matter Myth, J. Gribbin, Princeton, 1987

The Physical Universe, Konrad B. Krauskoph, Arthur Beiser, 1997, WCB/McGraw hill

Chaos in Dynamical Systems, Edward Ott, Cambridge University Press, 1993.

www.lib.rmit.edu.au/fractals/exploring.html -Understanding Chaos and Fractals

www-chaos.umd.edu – the Maryland chaos page, magazine publications and articles + diagrams and explanations

The Meaning of Quantum Theory – Jim Baggot, Oxford, 1992

Chaos in Wonderland – Clifford A. Pickover, St. Martins, 1994

Turbulent Mirror – John Briggs, Harper&Row, 1993

Exploring Chaos – Nina Hall, W.W. Norton & Company, 1993

The Essence of Chaos – Lorenze, Washington University Press, 1993

Footnote Legend-

1. The Essence of Chaos

2. Turbulent Mirror

3. Turbulent Mirror

4. Chaos in Wonderland

5. Exploring Chaos

6. Turbulent Mirror

7. Exploring Chaos

8. The Matter Myth

9. www.lib.rmit.edu.au/fractals/exploring.html

10. Chaos in Wonderland

Bibliography