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CHAOS
Todd Hutner
Background
► Classical
physics is a deterministic way of looking
at things.
► If you know the laws governing a body, we can
predict where it will be at any given time
thereafter.
► Classical physics also disregards nonlinearity. If the
system behaves nonlinearly, find a linear
approximation for it.
► That was the prevailing way of thinking about
physics in 1960.
The Butterfly Effect
► Edward
Lorenz was an American Meteorologist.
► In the Early 1960’s, computer technology made
solving complex systems of Differential Equations
easier.
► Lorenz, in an attempt to better predict the
Weather, had his computer solve a system of
three nonlinear differential equations.
► Thus, given the data of the current weather,
Lorenz could calculate the weather conditions far
into the future.
The butterfly Effect
► Lorenz
then wanted to extend the date of
an earlier run.
► So, he started in the middle of the previous
run.
► The assumption was that the computer
would duplicate the earlier run.
► This, however is not what happened.
The Butterfly Effect
► Instead,
the forecast diverged. Slowly at first, but
eventually, the two weather patterns were
extremely different.
► It turns out, the original value was .506127, yet
the value Lorenz fed into the computer for this
subsequent run was rounded to .506.
► This led to a new idea regarding complex systems:
sensitive dependence on initial conditions. This is
why we can not predict the weather more then a
few days in advance.
The Butterfly Effect
► Lorenz
would eventually map his findings in
phase space, giving rise to the Lorenz
Attractor.
► http://www.cmp.caltech.edu/~mcc/Chaos_C
ourse/Lesson1/Demos.html
Fractal geometry
Benoit Mandelbrot was a Scientist at IBM in the 1960’s.
► The 1st problem was to find any meaning from the
fluctuations of Cotton Prices.
► Classical Economics predicted the ratio of small fluctuations
to high ones to be very high. The data did not agree.
► Mandelbrot realized that while each price change was
random, the curve for daily price change was identical to
that for monthly, and yearly.
► This relationship held fast for over 60 years, including two
world wars.
► This effect is called scaling. Patterns are the same on a
large scale and a small scale, and all scales inbetween.
►
Fractal Geometry
► The
second problem Mandelbrot
encountered was so called “noise” in
telephone lines used to transmit information
between computers.
► The noise was random, but no mater how
strong the signal, the engineers could never
drown out the noise.
► Every so often, this noise would drown out
a piece of information, creating an error.
Fractal Geometry
According to the Engineers, there would be extended
periods with out errors, followed by periods with.
► Mandelbrot realized you could break the periods with
errors into smaller groups, some with errors, and
some without.
► Mandelbrot then discovered that within any burst of
errors, there would be smaller, error free periods.
► This relationship followed a scaling pattern, just as the
cotton prices. The proportion of error free periods to
error ridden periods was constant over any time scale.
►
Fractal Geometry
► This
reminded Mandelbrot of the Cantor Set.
Fractal Geometry
► Mandelbrot
then asked, what other shapes
are similar.
► He came up with fractal geometry, a way to
place an infinite line in a finite area.
► Fractal geometry gets its basis from the
scaling principles he discovered in the two
problems previously discussed.
Fractal Geometry
Bifurcation
► Robert
May was interested in, among other
things, the question of what happens to a
populations as the rate of growth passes a
critical point?
► The equation he was used to study this
question was: xn+1=rxn(1-xn). In this case, r
is rate of growth.
Bifurcation
When r is low, the population tends toward extinction.
► As r increase, the population tends towards a single,
steady state.
► As r continues to rise, the state splits, so that there
are two steady states that the population oscillates
between.
► As r continues to rise, the states split again, and
again, into 4, 8, 16, etc.
► At some critical value of r, the population becomes
chaotic, never tending toward a single steady state.
► The state of the population just before turning chaotic
is called “self organized criticality.”
►
Bifurcation
Comets
► Most
comets have an elliptical orbit, placing them
just beyond the orbit of Jupiter.
► These orbits remain, for the most part, unchanged
over time.
► Yet, a slight, new gravitational tug from Jupiter
can send comets careening of into space, or
plunge them into the inner solar system, flying by
the orbit of Earth, and in toward the sun.
► This is another example of sensitive dependence
to initial conditions.
Punctuated Equilibrium
Darwin’s theory of Evolution describes natural
selection as a gradual process, each individual
difference building up over long periods of time.
► Yet, the fossil records shows long periods of
stagnation followed by short periods of rapid change.
► Natural Selection alone can not account for this.
► We have seen how small changes initially can produce
huge changes later on.
► So, a single organism adapting could cause larger
scale adaptive changes over a short period of time.
►
The Extinction of the Dinosaurs
► Most
scientist agree that a meteor striking the
Earth was probably responsible for the extinction
of the Dinosaurs.
► The commonly held belief is that it had to be a
very large meteor to cause such a large extinction
event.
► Yet, Chaotitians tend to disagree. They believe
that the dinosaurs were in a self organized
criticality.
► Once again, a very small change can drive this
system into the chaotic realm of the bifurcation
diagram.
Earth Processes
► The
Earth, over time, has seen large scale
fluctuations in its average global temperature, and
its magnetic field.
► Most scientist are at a loss for an explanation why
there are ice ages, as well as shifting magnetic
fields.
► Chaotitians, on the other hand, believe that the
Earth lies in the bifurcation diagram. Thus, the
Earth can oscillate between the two (or more)
values with out any reason.
Ventricle Fibrillations
► Fibrillation
is where the heart no longer beats in a
correct manner. This electrical pulse stimulates
some parts of the heart to beat, while others do
not.
► This can, if not corrected quickly, lead to death.
► Yet, after further investigation, it appears that no
part of the heart is malfunctioning. Each cell works
correctly on its own.
► But, the heart, as a whole, does not work.
Ventricle Fibrillations
► Chaotitians
believe that there are two
steady states to the heart.
► One is the correct beating pattern, and the
other is fibrillation.
► Thus, it is believed that chaos will help us
determine new, and improved ways to help
patients going through ventricle fibrillations.
Conclusions
► Since
the emergence of chaos, physics does
not view nonlinearity as a problem.
► Also, physics does not believe it can predict
the outcome of even the simplest of
systems, to an infallible degree.
► Thus, nonlinear dynamic systems is one of
the fastest growing fields in physics today.
Further Reading
► Chaos:
Making a New Science by James
Gleick
► The Essence of Chaos (The Jessie and John
Danz Lecture Series) by Edward Lorenz
► Fractals and Chaos: The Mandelbrot Set and
Beyond by Benoit Mandelbrot
► Stability and Complexity in Model
Ecosystems by Robert May
Sources
► Gleick,
James. Chaos: Making a New
Science. Penguin Books, New York, 1987.
► Gribbin, John. Get a Grip on Physics. Ivy
Press, East Sussex, 1999.