Post on 05-Jan-2016
Fractional Dimensions, Strange Attractors & Chaos
Old Familiar Faces
Dimensions of some Familiar Figures
‘Weird’ Objects
What about these objects?
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
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
The Koch Snowflake
Note here N=4 & r=3
Hence
i.e.
)log(
)log(
r
Nd
26.1)3log(
)4log(d
The Sierpinski Gasket
Here N=3, r=2
Using
We have
)log(
)log(
r
Nd
58.1)2log(
)3log(d
Fractals in Nature
Computer Generated Fractals I
The ‘Julia Set’
Computer Generated Fractals II
The ‘Mandelbrot Set’
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
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
As were increase r ……..
…. and …finally ……CHAOS
Note sensitivity on IC
System does NOT ‘settle down’
Unpredictable!!
Where are the fractals?
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!!
Lorenz System (Butterfly Effect)
A simplified Weather Model
For certain values of parameters is chaotic
Q: Is our weather unpredictable?
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!
Questions??
Credits: Thank you wikipedia contributors for many of the figures