fractional dimensions, strange attractors & chaos
TRANSCRIPT
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Fractional Dimensions, Strange Attractors & Chaos
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Old Familiar Faces
Dimensions of some Familiar Figures
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‘Weird’ Objects
What about these objects?
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How to ‘Measure’ dimensions?
One gets N copies if one scales by a factor r The dimension ‘d’ is given by OR drN
239
155
3464
)log(
)log(
r
Nd
)5log(
)5log(d
)3log(
)9log(d
)4log(
)64log(d
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The ‘Cantor Set’
This is the 1/3 Cantor Set Note here N=2 & r=3
Hence
i.e. Cantor Set is 0.63 dimensional !!
)log(
)log(
r
Nd
63.0)3log(
)2log(d
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The Koch Snowflake
Note here N=4 & r=3
Hence
i.e.
)log(
)log(
r
Nd
26.1)3log(
)4log(d
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The Sierpinski Gasket
Here N=3, r=2
Using
We have
)log(
)log(
r
Nd
58.1)2log(
)3log(d
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Fractals in Nature
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Computer Generated Fractals I
The ‘Julia Set’
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Computer Generated Fractals II
The ‘Mandelbrot Set’
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The Butterfly Effect
Flap of a butterfly’s wing in Rio de Janeiro causes a hurricane in Lahore
Mathematically sensitivity of
a system on initial conditions
Think Billiards
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The Logistic Map
Very simple system exhibiting ‘chaos’
Can be a model for bacterial population
‘r’ can be thought of as net growth rate
As ‘r’ varies one sees a drastic changes in behavior
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As were increase r ……..
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…. and …finally ……CHAOS
Note sensitivity on IC
System does NOT ‘settle down’
Unpredictable!!
Where are the fractals?
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The ‘Parameter Picture’
Choose different IC Run the system for long times Plot long time behavior for different ‘r’ The resulting picture
has fractal structure!!
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Lorenz System (Butterfly Effect)
A simplified Weather Model
For certain values of parameters is chaotic
Q: Is our weather unpredictable?
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What should you take away?
Fractals are all around us
There is an intrinsic link between chaotic systems and fractals
Fractals can be generated easily on a computer
Butterfly Effect was a cool movie!
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Questions??
Credits: Thank you wikipedia contributors for many of the figures