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HONR 300/CMSC 491 Chaos (Flake, Ch. 8)

Discover the fascinating world of chaos theory and the properties of chaos through visualizations such as time series graphs and state space graphs. Learn about different types of regimes including fixed points, limit cycles, quasiperiodic behavior, and chaos. Explore the key ideas of determinism, sensitivity to initial conditions, and ergodicity. Prepare to be amazed by how small changes can lead to vastly different outcomes in complex systems.

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HONR 300/CMSC 491 Chaos (Flake, Ch. 8)

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  1. HONR 300/CMSC 491Chaos (Flake, Ch. 8) Prof. Marie desJardins, March 2, 2011

  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. Complex Limit Cycle

  7. Chaos

  8. Bifurcations of the Logistic Map

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

  10. 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.

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

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