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_Chaos Theory Explained _
By: Chance Vanguard
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, 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 _
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
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