GEM2505M

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Taming Chaos

• GEM2505M

Frederick H. Willeboordse frederik_at_chaos.nus.edu.s
g
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The Butterfly Effect

• Lecture 2

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Todays Lecture

• The Oscar Award
• Homoclinic Points
• What a Bug!
• Sensitive Dependence
• The Butterfly Effect

Who is King Oscar?
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The Oscar Award
Once upon a time in a galaxy far far away well,
not exactly actually it was more around 1889 in
Sweden.
To commemorate the 60th birthday of King Oscar
the II, a grand challenge was posed to the
scientific community the solution of which would
be rewarded with a big prize (and perhaps more
importantly great fame).
As scientists like to do when unsupervised, the
challenge was worded uninhibited by common sense.
It was something like
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The Oscar Award
The Challenge
Given a system of arbitrary mass points that
attract each other according to Newton's laws,
under the assumption that no two points ever
collide, try to find a representation of the
co-ordinates of each point as a series in a
variable that is some known function of time and
for all of whose values the series converges
uniformly. This problem, whose solution would
considerably extend our understanding of the
solar system, . . . .
Could you repeat that?
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The Oscar Award
Well thats that. However, what it basically
means is figure out the paths of more than two
celestial objects (without them bumping into each
other).
All right, pretty cool. We know that the planets
have regular motions. So all we need to do is
some clever calculations.
Are you sure? Quite! Solar Eclipses, the setting
of the sun, etc. all such things can be predicted
rather well.
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The Oscar Award
Excellent! One of the 19th centurys brightest
mathematicians, Henry Poincaré
He won the competition and collected the prize of
2,500 kroner!
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The Oscar Award
Ohoh! His answer was wrong! Fortunately, he
discovered the error himself and hence
frantically worked to correct his mistakes.
Finally in 1890 he published a 270 page revision.
This time he was correct and what he found was
not quite what one had expected. The first signs
of CHAOS!
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The Oscar Award
In fact he found that even for a rather idealized
and simplified system of three bodies the Oscar
challenge cannot be solved.
Negligible mass. How does it move under Newtons
laws?
Circle around each other regularly.
Hence, the sun, moon and earth system (which is
more complicated) cannot be solved!!!
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Homoclinic Points
In order to understand why well need two
concepts.
1) Stable and Unstable Manifolds
If we have a point in a plane at a certain time
n, and we want to know where it is at time n1.
How can we describe this?
With the help of a matrix.
This matrix is, so-to-speak, the rule by which
the point moves.
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Homoclinic Points
Matrices

• vector x describes the point (x is called the
  state usually)
• matrix A is the rule (called a map usually)
• the time counter is n

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Homoclinic Points

• If p is larger than 1, then it will stretch the
  x-direction.
• Conversely if p is smaller than 1, it will
  shrink the x-direction.
• Similarly for q and the y-direction.

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Homoclinic Points
Example
If we have
?
and set
What happens to the various points in the plane?
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Homoclinic Points
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Homoclinic Points
x direction is unstable y direction is stable
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Homoclinic Points
A fixed point is a point that does not change
when applying A. I.e. x is a fixed point when
Ax x.
Consider
When do we have
In this case when x0 and y0.
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Homoclinic Points
Manifolds

• The stable manifold of the fixed point r is the
  set of points s such that they are attracted to r
  asymptotically (when n ??).
• The unstable manifold of the fixed point is the
  set of points u such that they are repelled from
  r asymptotically.

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Homoclinic Points
2) Homoclinic Points
Homoclinic points
Easily destroyed configuration.
The interesting thing is that one can prove that
if theres one homoclinic point, then there are
infinitely many.
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Homoclinic Points
Homoclinic points do not know where they belong
to and since there are infinitely many, it
becomes impossible to say what applying A
repeatedly will lead to.
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70 years later
While of paramount importance, Poincarés work
was mainly forgotten outside of some rather
specialized areas.
Roughly seventy years later computers started to
become available as research tools to somewhat
more mainstream scientists.
One of them was
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What a Bug!

• 1960, E Lorenz was doing weather prediction
  research at MIT.
• He managed to get funding to acquire a Royal
  McBee LGP-30 computer with 16 KB of memory that
  could do 60 multiplications per second.

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What a Bug!

• Lorenz set the new computer to solve a system of
  12 differential equations that model a miniature
  atmosphere.

• To speed up the output, Lorenz altered the
  program to print only three significant digits of
  the solution trajectories, although the
  calculations themselves were carried out with a
  somewhat higher precision

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What a Bug!

• After seeing a particularly interesting run, he
  decided to repeat the calculation.
• He typed in the starting values from the printed
  output and started the program.
• Lorenz went for a coffee break, and when he
  returned, he found that the results we completely
  different.

?????
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What a Bug!

• At first he thought that some vacuum tubes in the
  computer were not working.
• Upon careful check, he realised that the
  discrepancies between the original and re-started
  calculations occurred gradually First in the
  least significant decimal place and then
  eventually in the next, and so on.

E.g. start first with
0.165
then with
0.1653
then with
0.16538
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What a Bug!
What Lorenz has discovered is that tiny
differences in the starting conditions can have
big effects.
This has become known as sensitive dependence on
initial conditions.
Lets have a bit a closer look at what this means.
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Sensitive Dependence
A small change has a big effect
Sensitive
Dependence on
A small change in what? (i.e. what does the big
change depend on?)
It depends on the values with which you start the
calculations
Initial Conditions
And on what do these initial conditions have a
big effect?
The system.
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Sensitive Dependence
Growth of an error
Previously we saw that a matrix is applied over
and over again.
Now let us say that we have a very small error
which doubles every time the matrix is applied.
How quickly will this error grow?
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Sensitive Dependence
Growth of an error
Really quickly!!!!!
Try it out!
0.00000000000000000000000001 10-26
How many times do you need to double to get to
around 1.0?
0.00000000000000000000000002
0.00000000000000000000000004
0.00000000000000000000000008
0.00000000000000000000000016
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Sensitive Dependence
Growth of an error
Is that a lot? NO! I can double 87 times in less
than a minute on a pocket calculator.
About 87!
How can we know?
Log10 1026
Log2 of 1026
86.37
Log10 2
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The Butterfly Effect

• When the initial conditions change a bit, does
  the flap of a butterfly's wings in Brazil set off
  a Tornado in Texas?

Edward Lorenz Dec 1972, Talk given in Washington
DC
?
Do you think this is true?
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The Butterfly Effect
The answer is
YES!
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The Butterfly Effect
But!
There is a common misconception as with regards
to the words set off (or cause in other
formulations of the same idea).
You cannot call uncle Eddie in Brazil and ask him
to let his pet-butterflies flap their wings so
that they cause a rain storm in Ang Mo Kio to
soak your boy/girl-friend whom you are angry at.
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The Butterfly Effect

• What is means is that you have to imagine two
  identical worlds.
• In one of the worlds you place a butterfly and
  let it flap its wings.
• In the other world you dont place the butterfly
• Now you wait a while (a few months or more
  perhaps) and will see that the global weather
  patterns on your two worlds are completely
  different.

Sensitive dependence on initial conditions!
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Key Points of the Day

• Homoclinic Points
• Sensitive Dependence on Initial Conditions

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Think about it!

• Could there be situations when the butterfly
  effect doesnt apply?

Butter, Fly, Holidays, Resort Island .