CHAOS

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CHAOS

• Todd Hutner

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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.

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The Butterfly Effect

• Edward Lorenz was an American Meteorologist.
• In the Early 1960s, 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.

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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.

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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.

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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_Course/Lesso
  n1/Demos.html

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Fractal geometry

• Benoit Mandelbrot was a Scientist at IBM in the
  1960s.
• 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.

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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.

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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.

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Fractal Geometry

• This reminded Mandelbrot of the Cantor Set.

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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.

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Fractal Geometry
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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 xn1rxn(1-xn). In this case, r is rate of
  growth.

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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.

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Bifurcation
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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.

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Punctuated Equilibrium

• Darwins 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.

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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.

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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.

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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.

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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.

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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.

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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

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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.