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Fractional Dimensions, Strange Attractors & Chaos

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.

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Fractional Dimensions, Strange Attractors & Chaos

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  1. Fractional Dimensions, Strange Attractors & Chaos

  2. Old Familiar Faces • Dimensions of some Familiar Figures

  3. ‘Weird’ Objects • What about these objects?

  4. How to ‘Measure’ dimensions? • One gets N copies if one scales by a factor r • The dimension ‘d’ is given by OR

  5. 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 !!

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

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

  8. Fractals in Nature

  9. Computer Generated Fractals I • The ‘Julia Set’

  10. Computer Generated Fractals II • The ‘Mandelbrot Set’

  11. 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

  12. 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

  13. As were increase r ……..

  14. …. and …finally ……CHAOS • Note sensitivity on IC • System does NOT ‘settle down’ • Unpredictable!! • Where are the fractals?

  15. 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!!

  16. Lorenz System (Butterfly Effect) • A simplified Weather Model • For certain values of parameters is chaotic • Q: Is our weather unpredictable?

  17. 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!

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

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