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HONR 300/CMSC 491 Chaos ( Flake, Ch. 10; Mitchell, Ch. 2)

Explore chaos theory through visualizations of time series and state space graphs. Learn about fixed points, limit cycles, and quasiperiodic regimes, as well as the properties of chaos like determinism, sensitivity, and ergodicity.

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HONR 300/CMSC 491 Chaos ( Flake, Ch. 10; Mitchell, Ch. 2)

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  1. HONR 300/CMSC 491Chaos (Flake, Ch. 10; Mitchell, Ch. 2) Prof. Marie desJardins, March 1, 2016

  2. Key Ideas • Visualizing chaos • Time series graph • State space graph • Types of regimes: • Fixed points • Limit cycles • Quasiperiodic • Chaos • Properties of chaos • Determinism • Sensitivity • Ergodicity

  3. As Randall Munroe Sees It xkcd.com

  4. Fixed Point Behavior

  5. Limit Cycle

  6. NetLogo Model • (Note: Flake uses the form of the logistic map based on 4r whereas the .nlogo model uses r. So you have to multiply his values of r by 4 to get the same behavior in the .nlogo model.)

  7. Chaos

  8. Complex Limit Cycle

  9. Bifurcations of the Logistic Map

  10. Bfucation Diagram Region of stability! Region of bifurcation Chaotic regime

  11. Properties of Chaos • Determinism • Sensitivity to initial conditions • Ergodicity

  12. If you don’t believe in “the butterfly effect” or the conclusion of The Sound of Thunder, consider these thoughts: • The Game of Life is much simpler than the world. Yet changing the state of just one cell in the initial configuration can lead to a completely different state of the world in a very small number of iterations. • Millions of years is a really long time for the “compound interest” to take effect. • And remember the old saying: For want of a nail the shoe was lost.For want of a shoe the horse was lost.For want of a horse the rider was lost.For want of a rider the battle was lost.For want of a battle the kingdom was lost.And all for the want of a horseshoe nail.

  13. Hang onto your hats... • Next week we’re going to see the fractal dimension rear its Hydra-like head again!

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