Chaos and Fractals

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Chaos and Fractals
Frederick H. Willeboordse http//staff.science.nus
.edu.sg/frederik
SP2171 Lecture Series Symposium III, February 6,
2001
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Today's Program
Part 1 Chaos and Fractals are ubiquitous - a
few examples Part 2 Understanding Chaos and
Fractals - Some Theory - Some Hands-on
Applications Part 3 Food for Thought
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Part 1 A few examples ..

• The world is full of Chaos and Fractals!
• The weather can be ...chaotic
• The ocean can be ...chaotic
• Our lives can be ...chaotic
• Our brains can be ...chaotic
• Our coffee can be ...chaotic

Well
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Chaos and Fractals in Physics
The motion of the planets is chaotic. In fact,
even the sun, earth moon system cannot be solved
analytically!
In fact, the roots of Chaos theory go back to
Poincare who discovered strange propertied when
trying to solve the sun, earth moon system at the
end of the 19th century
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Chaos and Fractals in Physics
Tin Crystals
Molten tin solidifies in a pattern of tree-shaped
crystals called dendrites as it cools under
controlled circumstances. From Tipler,
Physics for Scientists and Engineers, 4th Edition
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Chaos and Fractals in Physics
Snowflake
The hexagonal symmetry of a snowflake arises from
a hexagonal symmetry in its lattice of hydrogen
and oxygen atoms. From Tipler, Physics for
Scientists and Engineers, 4th Edition
A nice example of how a simple underlying
symmetry can lead to a complex structure
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Chaos and Fractals in Physics
The red spot on Jupiter.
Can such a spot survive in a chaotic environment?
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Chaos in and Fractals Physics
An experiment by Swinney et al
One of the great successes of experimental chaos
studies.A spot is reproduced.
Note these are false colors.
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Chaos and Fractals in Chemistry
Beluzov-Zhabotinski reaction
Waves representing the concentration of a certain
chemical(s). These can assume many patterns and
can also be chaotic
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Chaos and Fractals in Geology
Satellite Image of a River Delta
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Chaos and Fractals in Biology
Delicious!
Broccoli Romanesco is a cross between Broccoli
and Cauliflower.
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Chaos and Fractals in Biology
Broccoli Romanesco
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Chaos and Fractals in Biology
Would we be alive without Chaos?
The venous and arterial system of a kidney
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Chaos and Fractals in Paleontology
Would we be here without Chaos?
Evolutionary trees as cones of increasing
diversity. From Wonderful Life by Stephen Jay
Gould who disagrees with this picture (that
doesnt matter as with regards to illustrating
our point).
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Chaos and Fractals in Paleontology
Replicate and Modify
Built from similar modified segments?
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Chaos and Fractals in Paleontology
Would we be alive without Chaos?
Is there a relation to stretch and fold?
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Simple? Complex?
Complex
The phenomena mentioned on the previous slide are
very if not extremely complex. How can we ever
understand them?
Try to write anequation for this.
Simple
Chaos and Fractals can be generated with what
appear to be almost trivial mathematical formulas
You could have done this in JCRight!??
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Part 2 Understanding Chaos and Fractals
In order to understand whats going on, let us
have a very brief look at what Chaos and Fractals
are.
Chaos
Fractal
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Chaos
Chaos
What is Chaos?
Chaos is often a more catchy name for
non-linear dynamics.
Non-linear (roughly) the graphof the function
is not a straight line.
Dynamics (roughly) the time evolutionof a
system.
Are chaotic systems always chaotic?
No! Generally speaking, many researchers will
call a system chaotic if it can be chaotic for
certain parameters.
Parameter (roughly) a constant in an equation.
E.g. the slope ofa line. This parameter can be
adjusted.
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Chaos
Chaos
What is Chaos?
Quiz Can I make a croissant with more than
15000layers in 3 minutes?
Try it!
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Chaos
Chaos
What is Chaos?
The key to understanding Chaos is the concept of
stretch and fold. Or Danish Pastry/Chinese
Noodles
Quiz-answer Can I make a croissant with more
than 15000layers in 3 minutes? Yes stretch
and fold! Or perhaps I should say kneed and roll
?.
Sensitive dependence on initial conditions
Two close by points always separate yet stay in
the same volume. Inside a layer, two points will
separate, but, due the folding, when cutting
through layers, they will also stay close.
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The Butterfly Effect
Chaos
Sensitive dependence on initial conditions is
what gave the world the butterfly effect.
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The Butterfly Effect
Chaos
Sensitive dependence on initial conditions is
what gave the world the butterfly effect.
The butterfly effect describes the notion that
the flapping of the wings of a butterfly can
cause a typhoon at the other side of the world.
How? We saw with the stretch and fold Chinese
Noodle/Danish Pastry example, where the distance
between two points doubles each time, that a
small distance/difference can grow extremely
quickly. Due to the sensitive dependence on
initial conditions in non-linear systems (of
which the weather is one), the small disturbance
caused by the butterfly (where we consider the
disturbance to be the difference with the
no-butterfly situation) in a similar way can
grow to become a storm.
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Logistic Map
Chaos
The logistic map can be defined as
Looks simple enough to me! What could be
difficult about this?
Let's see what happens when we increase the
parameter alpha from 0 to 2.
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Iteration
Chaos
Iteration is just like our Danish Pastry/Chinese
Noodles.
In math it means that you start with a certain
value (given by you) calculate the result and
then use this result as the starting value of a
next calculation.
given
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Logistic Map
Chaos
The so-called bifurcation diagram
Plot 200 successive values of x for every value
of a
As the nonlinearity increases we sometimes
encounter chaos
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Logistic Map
Chaos
What's so special about this? Let's have a closer
look.
Let's enlarge this area
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Logistic Map
Chaos
Hey! This looks almost the same!
Let's try this again...
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Logistic Map
Chaos
Let's enlarge a much smaller area!
Now let's enlarge this area
Hard to see, isn't it?
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Logistic Map
Chaos
The same again!
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Logistic Map
Chaos
Indeed, the logmap repeats itself over and over
again at ever smaller scales
What's more, this behaviour was found to be
universal!
Yes, there's a fractal hidden in here.
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Chaos and Randomness
Chaos
Chaos is NOT randomness though it can look pretty
random.
Let us have a look at two time series
And analyze these with some standard methods
Data Dr. C. Ting
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Chaos and Randomness
Chaos
Power spectra
No qualitativedifferences!
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Chaos and Randomness
Chaos
Histograms
No qualitativedifferences!
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Chaos and Randomness
Chaos
Autocorrelation Functions
No qualitativedifferences!
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Chaos and Randomness
Chaos
Well these two look pretty much the same.
A They are both Chaotic B Red is Chaotic and
Blue is Random C Blue is Random and Red is
Chaotic D They are both Random
Let us consider 4 options
Random???
Random???
Chaotic??
What do you think?
Random???
Chaotic??
Chaotic??
Chaotic??
Chaotic??
Random???
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Chaos and Randomness
Chaos
Return map (plot xn1 versus xn )
Red is Chaotic and Blue is Random! As we can see
from the return map.
Henon Map
xn1 1.4 - x2n 0.3 yn yn1 xn
Deterministic
Non-Deterministic
White Noise
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Fractals
Fractals
What are Fractals?
(roughly) a fractal is a self-similar geometrical
object with a fractal dimension.
self-similar when you look at a part, it just
looks like the whole.
Fractal dimension the dimension of the object
is not an integer like1 or 2, but something like
0.63. (well get back to what this means alittle
later).
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Fractals
Fractals
The Cantor Set
Take a line and remove the middle third, repeat
this ad infinitum for the resulting lines.
This is the construction of theset!The set
itself is the result ofthis construction.
Remove middle third
Then remove middle third of what remains
And so on ad infinitum
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Fractals
Fractals
Fractal Dimension
Let us first look at a regular line and a regular
square and see what happens when we copy the
these and then paste them at 1/3 of their
original size.
We see that our original objectcontains 3 of the
reduced pieces.
We see that our original objectcontains 9 of the
reduced pieces.
of pieces
Reduction factor
Apparently, we have
Dimension
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Fractals
Fractals
Fractal Dimension
Now let us look at the Cantor set
Reduced Copy
Original
This time we see that our original
objectcontains only 2 of the reduced pieces!
A fractal dimension.. Strictly, this just one of
several fractal dimensions, namely the
self-similarity dimension.
If we fill this into ourformula we obtain
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Fractals
Fractals
The Mandelbrot Set
This set is defined as the collection of points c
in the complex plane that does not escape to
infinity for the equation
Note The actual Mandelbrot set are just the
black points in the middle!All the colored
points escape (but after different numbers of
iterations).
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Fractals
Fractals
The Mandelbrot Set
Does this look like the logistic map? It should!!!
Take z to be real, divide both sides by c,
,
.
then substitute
to obtain
Defining
we find the logistic map from before
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Fractals
Fractals
The Mandelbrot Set
The Madelbrot set is strictly speaking not
self-similar in the same way as the Cantor set.
It is quasi-self-similar (the copies of the whole
are not exactly the same).
Here are some nice pictures from http//www.geoci
ties.com/CapeCanaveral/2854/
What Id like to illustrate here is not so much
that fractals can be used to generate beautiful
pictures, but that a simple non-linear equation
can be incredibly complex.
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Fractals
Fractals
The Mandelbrot Set
Next, zoom into this Area.
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Fractals
The Mandelbrot Set
Next, zoom into this Area.
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Fractals
The Mandelbrot Set
Next, zoom into this Area.
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Fractals
Fractals
The Mandelbrot Set
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Chaos and Fractals
How do they relate?
Fractals often occur in chaotic systems but the
the two are not the same! Neither of they
necessarily imply each other.
Roughly
A fractal is a geometric object
Chaos is a dynamical attribute
Let us have a look at the logistic map again.
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Chaos and Fractals
In the vertical direction we have the points on
the orbit for a certain value of a.
How do they relate? -gt Not directly!
This orbit is chaotic, but if we look at the
distribution, it is definitively not fractal. It
approximately looks like this
Self Similar, adinfinitum. This can be used to
generate a fractal.
probability
-1 value of x 1
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Coupled Maps -Why on 'earth'
Chaos
A short detour into my research.
Universality
The logistic map has shown us the power of
universality. It is hoped that this universality
is also relevant for Coupled Maps.
Coupled Maps are the simplest spatially extended
chaotic system with a continuous state (x-value)
Simplicity
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Coupled Maps -What they are
Chaos
The coupled map discussed here is simply an array
of logistic maps. The formula appears more
complicated than it is.
f is the logistic map
Time n
Time n1
Or in other words
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Coupled Maps -Phenomenology
Chaos
Even though coupled maps are conceptually very
simple, they display a stunning variety of
phenomena.

• Patterns with Kinks
• Frozen Random Patterns
• Pattern Selection
• Travelling Waves
• Spatio-temporal Chaos

Coupled Map have so-called Universality classes.
It is hoped that these either represent essential
real world phenomena or that they can lead us to
a deeper understanding of real world phenomena.
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Pattern with Kinks
Chaos
No Chaos lattice sites are attracted to the
periodic orbits of the single logistic map.
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Frozen Random Pattern
Chaos
Parts of the lattice are chaotic and parts of the
lattice are periodic. The dynamics is dominated
by the band structure of the logistic map.
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Pattern Selection
Chaos
Even though the nonlinearity has increased and
the logistic map is chaotic for a1.7, the
lattice is entirely periodic.
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Travelling Waves
Chaos
The coupled map lattice is symmetric, yet here we
see a travelling wave. This dynamical behaviour
is highly non- trivial!
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Spatio-Temporal Chaos
Chaos
Of course we have spatio-temporal chaos too. No
order to be found here ... or ??? . No, despite
the way it looks, this is far from random!
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My quasi-logo
Chaos
The bifurcation diagram, the source of complexity
Now we can guess what it means
The strength of the non-linearity
The logistic map, the building block of the
coupled map lattice
A coupled map lattice with travelling domain
walls, chaos and orderly waves
The strength of the coupling
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Simple? Complex?
We can now come back to the question posed
previously
How do these two seemingly contradictory aspects
relate?
The study of Chaos shows that simplicity and
complexity can be related by considering
universal properties of simple iterative
processes.
Universality
It was discovered that certain essential
properties of chaotic systems are universal. This
allows us to study a simple system and draw
conclusion for a complex system
Iteration
Repeat a (simple but non-linear) recipe over and
over again
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Understanding Chaos and Fractals
Some applications/hands-on demonstrations
Video Feedback
Chinese Noodles
Double Pendulum
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Part 3Food for Thought
More Coffee!
(Scientists at work ?)
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Chaos in Biology
Evolution?
Replicate and Modify What does that mean?
Is the human body a fractal?
Could it be that fractals hold the key to how so
much information can be stored in DNA?
Life?
Is the fact that Chaos can look like randomness
essential for life?Could it be that chaos is an
optimization algorithm for life in an unstable
environment?
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Chaos in Physics
Stunningly common at the macroscopic level
linear
non-linear
But! Quantum Mechanics is a linear science!
It was discovered that certain essential
properties of chaotic systems are universal. This
allows us to study a simple system and draw
conclusion for a complex system
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Chaos in Meteorology
What are the implications of the butterfly effect
for the prediction of weather?
Perhaps its up in the clouds
Sunny?
Rainy?
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Chaos and Philosophy
Determinism
Classical (and in a sense also Quantum Physics)
seems to imply that the world is deterministic.
If we just had the super-equation, we could
predict the future exactly. In essence, there is
not freedom of the mind.
Good-bye?
Can non-linear science contribute to the
discussion on determinism versus free will?
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Conclusion

• Almost everything in our world is chaotic, yet
  order is also everywhere.
• Understanding this dichotomy is a fabulous
  challenge.
• The study of Chaos can help us on our way.

Chaos is fun!