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Frederick H. [email protected]

Taming Chaos

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The Butterfly Effect

Lecture 2

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Today’s Lecture

The Oscar Award

Homoclinic Points

What a Bug!

Sensitive Dependence


Who is King Oscar?

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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).

The Oscar Award

Once upon a time in a galaxy far far away … well, not exactly … actually it was more around 1889 in Sweden.

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

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, . . . .

The Challenge

Could you repeat that?

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The Oscar Award

Well … that’s 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 century’s brightest mathematicians, Henry Poincaré:

He won the competition and collected the prize of 2,500 kroner!

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The Oscar Award

Oh–oh! 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.

Circle around each other regularly.

Negligible mass. How does it move under Newton’s laws?

Hence, the sun, moon and earth system (which is more complicated) cannot be solved!!!

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Homoclinic PointsIn order to understand why we’ll 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 n+1. 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

dycx

byax

y

x

dc

ba:Ax

nn Axx 1

• vector xx describes the point (x is called the state usually)

• matrix AA is the rule (called a map usually)• the time counter is n

Matrices

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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.

q

p

0

0A

n

nn

q

p

0

0A

yq

xp

y

x

q

px

n

n

n

nn

nn0

001 xAAx

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Homoclinic Points

5.00

02A

nn Axx 1If we have:

and set:

What happens to the various points in the plane?

Example

?

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Homoclinic Points

012

1 ..... xAxAAxx nnnn

00

0

5.00

02 n

n

nnA

y

x

y

xn

n

n

nn

5.0

2

5.00

02xA

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Homoclinic Points

y

x

y

x

5.0

2

5.00

02Ax

x direction is unstabley direction is stable

y

x

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Homoclinic PointsA fixed point is a point that does not change when applying A. I.e. x* is a fixed point when Ax* = x*.

xxA n

Consider:

When do we have:

y

x

y

xn

n

n

nn

5.0

2

5.00

02xA

In this case when x=0 and y=0.

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Homoclinic Points

The stable manifold of the fixed point rr is the set of points ss such that they are attracted to rr asymptotically (when n ).

The unstable manifold of the fixed point is the set of points u u such that they are repelled from rr asymptotically.

y

x

Manifolds

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Homoclinic Points

Homoclinic points

2) Homoclinic Points

The interesting thing is that one can prove that if there’s one homoclinic point, then there are infinitely many.

Easily destroyed configuration.

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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, Poincaré’s 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: 0.165then with: 0.1653then 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 on

Initial Conditions

A small change has a big effect

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

And on what do these initial conditions have a big effect?

The system.

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Growth of an error

Previously we saw that a matrix is applied over and over again.

012

1 ..... xAxAAxx nnnn

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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Growth of an error

Really quickly!!!!!

0.00000000000000000000000001 = 10-26

Try it out!

0.00000000000000000000000002

0.00000000000000000000000004

0.00000000000000000000000008

0.00000000000000000000000016

How many times do you need to double to get to around 1.0?

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Growth of an error

About 87!

How can we know?

Log2 of 10+26 =

Is that a lot? NO! I can double 87 times in less than a minute on a pocket calculator.

Log10 1026

Log10 2= 86.37

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

Edward LorenzDec 1972, Talk given inWashington DC

Do you think this is true? ?

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The answer is:

YES!

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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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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 don’t place the butterflyNow 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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Homoclinic Points

Sensitive Dependence on Initial Conditions

Key Points of the Day

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Could there be situations when the butterfly effect doesn’t apply?

Think about it!

Butter,Fly,Holidays,Resort Island ….

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