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

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 !!

The Koch Snowflake Note here N=4 & r=3 Hence i.e.

The Sierpinski Gasket Here N=3, r=2 Using We have

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!

Credits: Thank you wikipedia contributors for many of the figures Questions?? Credits: Thank you wikipedia contributors for many of the figures