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