GEM2505M

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Taming Chaos

• GEM2505M

Frederick H. Willeboordse frederik_at_chaos.nus.edu.s
g
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The Mandelbrot Set

• Lecture 4

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Todays Lecture

• Fractal Dimension
• Complex Numbers
• Prisoners and Escapees
• Julia Set
• Mandelbrot Set

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Fractal Dimension
Usually when we talk about dimension we think of
lines or surfaces.
Line 1 Dimensional Plane 2 - Dimensional
But what does this actually mean? It is (to a
certain degree) related to self similarity!
Previously, we saw that the meter stick is
perfectly self-similar. Consequently, one can
rescale it easily.
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Fractal Dimension
For example
1m 10dm
1dm 10cm
Magnify by a factor 10.
To turn this around, when magnifying by a factor
10, the new stick contains 10 times the number of
original sticks.
I.e. Magnify 1dm by a factor 10 to obtain 1m
which contains 10 pieces of 1dm.
1m 10 dm
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Fractal Dimension
Thats quite obvious of course. Now what happens
if we do the same trick with a square?
1 sq. m 100 sq. dm
If we magnify the small red square by a factor
10. What we see is that the new big square does
not contain 10 times as many small squares but
102 times as many squares!
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Fractal Dimension
This leads us to an important observation
a number of pieces S scaling factor (i.e.
magnification) D dimension

In other words, the dimensions is
Defined in this way, D is called the
self-similarity dimension.
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Fractal Dimension
Can we apply this to the Cantor Set? Yes!
If we take a chunk and make it 3 times bigger,
how many copies of the original do we have? 2!!!

So for the Cantor Set we have
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Complex Numbers
In order to talk about the Mandelbrot set we need
to know what complex numbers are.
Complex sounds like this is very complex but in
fact its not.
We all know that its quite easy to solve an
equation like this one
Seeing that, its not particularly far-fetched to
wonder what the solution is of
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Complex Numbers
Ah, easy enough!
ergo
Oops! Theres no root for -1 (even if we try very
hard indeed). Were stuck and it would really be
useful to be able to solve such equations.
The solution is simple, if theres no number in
existence which is the root of -1, we can just
introduce one. Clever solutions need not be
complicated!
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Complex Numbers
All right, we can define
Twisting history a bit we can say Since we
imagined this solution lets call it imaginary.
Then the solution to
or
is just
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Complex Numbers
Excellent, well if we can do the trick once, we
can do it twice! How about the solution to
That must be
Stuck again! Now we dont know what that is well
perhaps we can think of some equation whose
square is
Indeed
or
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Complex Numbers
This brings us quite closely to the general
notation of a complex number
with a and b real numbers (like 0.231, 1.949,
2.000)
Consequently, a complex number can be drawn as a
point in a plane where the x-axis is the real
part a, and the y-axis the imaginary part b.
a
b
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Complex Numbers
If we have
Then the modulus is
conjugate
And the conjugate is
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Complex Numbers
Addition

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Complex Numbers
Subtraction

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Complex Numbers
Multiplication

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Complex Numbers
Multiplication as rotation

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Prisoners and Escapees
Now let us consider the map
Prisoners
What happens if we start with r 0.8 and f 10O
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Prisoners and Escapees
Escapees
Next, lets consider what happens if we start
with r 1.5 and f 50O
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Prisoners and Escapees
Guards?
And lastly lets set r 1.0 and f 10O
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Prisoners and Escapees
Graphically
Prisoner
Escapee
Boundary
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Julia Set
Definition

The Julia Set is the boundary between the
escapees and prisoners of a complex iterative map.
In the case of
this means the unit circle
Often the inside of the Julia Set is filled in
and the escaping points are colored according to
how long it takes for the point to become larger
than a certain value.
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Julia Set
Adding a constant
The situation changes drastically when a constant
is added to the iterative map.
Connected Julia Sets
Disconnected Julia Sets
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Julia Set
Some famous Julia sets of the complex quadratic
map
Dendrite Fractal
Rabit Fractal
c at the boundary of the Mandelbrot Set (for this
picture, c i)
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Another Chaos Game
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Another Chaos Game
Result
A Julia Set!
Dendrite Fractal
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Another Chaos Game
Explanation
The Julia Set is the boundary between escapees
and prisoners. Hence all points not exactly on
top of it move away from it. If we go backwards
in the iteration, we will get closer to it.
Forward iterate
Dendrite Fractal
Backward iterates
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Mandelbrot Set
There are two types of Julia Sets, connected
Julia Sets and disconnected Julia Sets.

The Mandelbrot Set is defined as the set of
parameters c that lead to a connected Julia Set.
Alternatively
M c Î C c c2c remains bounded
I.e. parameters c for which the orbit of z0 is
bounded.
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Mandelbrot Set
The difference
Mandelbrot Set Parameters c
Julia Set Initial conditions z0
Find boundary between prisoners and escapees.
Find connected Julia Sets.
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Mandelbrot Set
Escape?
How can we know that an orbit escapes to infinity?
Answer if z gt r(c) max(c,2), an orbit will
escape.
Proof
Take a number such that z gt r(c) is true.
z2 c
Iterating once we obtain
z2
z2 c c z2c c
Applying the inequality we get
Ergo
z2c ³ z2 - c z2 - c ³ z2 - z
(z-1)z (1e)z
So when iterating z it grows and thus eventually
escapes.
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Mandelbrot Set
Prisoners black
Escapees use this
How can we make the pictures?
A screen is an array of pixels
For each of the pixels, calculate whether it
escapes and if so how many steps it takes to
reach e.g. 2. Color the pixel according to table
above. Note Color assignment is of course
arbitrary.
Im Maximum
Im Mininum
Re Mininum
Re Maximum
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Mandelbrot Set
The Heart
Intersects real axis from -0.75 0.25.
Julia set c 0 0i
Associated with a Julia set that has a period 1
attractor.
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Mandelbrot Set
The Buds
Period 4
Period 5
Associated with Julia sets that have higher
period attractors.
Period 2
c -1 0i
P.3
c -0.134 0.742i
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Mandelbrot Set
Some Features
The Mandelbrot Set is connected The boundary is a
fractal and infinitely long The dimension is 2 It
is quasi-self-similar
And some more pictures
Its better to use the applet though
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Key Points of the Day

• Fractal Dimension.
• Julia Set
• Mandelbrot Set

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Think about it!

• What is our dimension?

Julia, Romeo, Singing, Dire Straits
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References
http//mathworld.wolfram.com/
Dave Shorts course on complex numbers
Peter Alfelds Mandelbrot Applet