GEM2505M

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Taming Chaos

• GEM2505M

Frederick H. Willeboordse frederik_at_chaos.nus.edu.s
g
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The Essence of Chaos

• Lecture 11

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Todays Lecture
The Story
Now that we have obtained some understanding of
the phenomena encountered in chaotic and complex
systems, let us bring the essential points
together. What are the key features of such
systems?

• Sensitive Dependence
• Stretch and Fold
• Homoclinic Points
• Chaos and Randomness
• Universality

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Sensitive Dependence
Sensitive dependence on initial conditions means
that initially tiny differences grow rapidly to
the order of the system size.
As a consequence, in real life, systems that
display sensitive dependence on initial
conditions cannot be predicted long term due to
the inevitable presence of noise.
But there is a problem here. Real systems are not
infinitely big!
How can errors keep on growing in a finite system?
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Stretch and Fold
The answer to that question lies in stretch and
fold.
With layers
Stretch
Fold
Back to the original shape
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Stretch and Fold
If it funny once, its funny twice!
Stretch
Fold
Back to the original shape
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Stretch and Fold
The distance between points on opposite end of
the bar.
The distance grows exponentially!
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Stretch and Fold
The distance between nearby points. Sensitive
Dependence!
The distance grows exponentially!
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Stretch and Fold
Here, after stretching and folding, the top and
bottom layer are merged together (as is the case
in a 1-D map).
On a line
Stretch
Start
After Folding and Merging
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Stretch and Fold
Mathematically
The Tent Map
if
if
Bifurcation Diagram
Cobweb
Lyapunov Exponents
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Stretch and Fold
In real life
An excellent example of stretch and fold is the
making dough!
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Stretch and Fold
Give me some flower, water and a tiny bit of oil.
After some mixing and kneading, Ill have a
hopefully nice piece of dough.
Next Ill use this dough to make a croissant.
In 3 minutes, a croissant with how many layers
can I make?
?

1. Between 10 and 100
2. Between 100 and 1,000
3. Between 1,000 and 10,000
4. Between 10,000 and 100,000

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Stretch and Fold
In the logistic map
The same as stretch and fold with the stretch
being nonlinear.
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Homoclinic Points
Now that we have seen stretch and fold at work,
we can get a bit a better understanding of why
the homoclinic points lead to chaotic orbits.
Let us see what happens to a small area near the
stable manifold
After a few steps it will be near the fixed point.
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Homoclinic Points
After arriving at the fixed point the rectangle
will be stretched and pushed away along the
unstable manifold.
original square
Eventually, it will be near the starting point
again and overlap the original area.
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Homoclinic Points
Hence we see stretching and folding at work.
Where does the luck go in this case ?
original square
Note in these simplified drawings other
deformations due to the homoclinic points etc.
have been ignored.
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Chaos and Randomness
What is the relationship between chaos and
randomness? Are they the same?
Let us have a look at two time series
Data Dr. C. Ting
And analyze these with some standard methods
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Chaos and Randomness
Power Spectra
No qualitativedifferences!
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Chaos
Chaos and Randomness
Histograms
No qualitativedifferences!
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Chaos and Randomness
Random???
Chaotic??
Chaotic??
Chaotic??
Well these two look pretty much the same.
Random???
Random???
Random???
Chaotic??
Chaotic??
?
What do we have here?

1. Both are chaotic
2. Red is chaotic and blue is random
3. Red is random and blue is chaotic
4. Both are random

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Chaos and Randomness
Return map(plot xn1 versus xn )
Red is Chaotic and Blue is Random!
Henon Map
Deterministic
xn1 1.4 - x2n 0.3 yn yn1 xn
White Noise
Non-Deterministic
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Universality
A key motivation for the study of chaos is the
notion of universality. In this context it means
that a certain feature or a certain constant is
applicable to a whole range of systems which are
said to be a class of systems.
It is important to note that universality in this
sense does not mean everywhere in all conceivable
cases.
The most well known universal constant in chaos
theory is the Feigenbaum constant. It applies to
all single hump functions.
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Universality
Some Examples
Experimental verifications of the Feigenbaum
constant.
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Universality
Self-organized criticality
Some systems for which a sand pile is the
standard model are composed of many parts.
this is why its called self-organized
criticality
A sand pile turns out to naturally evolve to a
critical state in which a small event can trigger
a chain reaction of tumbling sand grains. This
chain reaction can stop rapidly but can also
become a so-called catastrophe where a large
number of sand grains forms an avalanche.
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Universality
Self-organized criticality
The minor events are much more common than the
major events, but their underlying mechanism is
the same.
An essential aspect of self-organized critical
systems is that their global features do not
depend on the details of the components dynamics.
Let us look a bit more closely at a sand pile
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Universality
Self-organized criticality
Drop one grain of sand slowly onto a circular
surface and see what happens

1. Grains stay close to where they land
2. Slowly form a pile with a gentle slope
3. Slope stops getting steeper

Critical state is reached. Avalanches of all
sizes occur.
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Universality
Self-organized criticality
Mass fluctuations
If one plots the number of avalanches versus
their size, one obtains a so-called power law.
Outcome of the sand pile experiment
how steep is a pyramid?
Power law
When a quantitys parameter dependence is a
straight line in a log-log plot. L(d) c da
(with c and a constants)
Avalanches are a big risk in alpine countries
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Universality
Zipfs law
An interesting power law, known as Zipfs law is
the relative ranking of cities in the world
around 1920 versus their population.
10M
Frequency vs Rank
Population
1M
In a more general sense, nowadays, a power law
describing the frequency of something versus its
rank is often called a Zipfs law.
100K
1
10
100
Frequency
M Million, K Thousand
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Universality
Zipfs law
For example, Zipf also discovered a power law for
the occurrences of words in the English language.
The most common word is the with a frequency of
about 9. The tenth most common word I has a
frequency of 1. This independent of the text as
long as the text is long enough. E.g. it holds as
well for Ulysses as for news papers.
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Key Points of the Day

• Stretch and Fold
• Universality

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Think about it!

• Is there a Zipfs law for Innovation?

Stretch, Fold, Exercise, Fitness, Chaos is
healthy!
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References
http//www.cmp.caltech.edu/mcc/Chaos_Course/Lesso
n4/Demo1.html
http//www.expm.t.u-tokyo.ac.jp/kanamaru/Chaos/e/