HONR 300/CMSC 491 Chaos (Flake, Ch. 10; Mitchell, Ch. 2) Prof. Marie desJardins, March 1, Chaos 3/1/16

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 Chaos 3/1/16 2

As Randall Munroe Sees It Chaos 3/1/16 3 xkcd.com

Fixed Point Behavior Chaos 3/1/16 4

Limit Cycle Chaos 3/1/16 5

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.) Chaos 3/5/12 6

Chaos Chaos 3/1/16 7

Complex Limit Cycle Chaos 3/1/16 8

Bifurcations of the Logistic Map Chaos 3/1/16 9

10 Bfucation Diagram Region ofbifurcationRegion ofbifurcation Chaotic regime Region of stability!

Properties of Chaos Determinism Sensitivity to initial conditions Ergodicity Chaos 3/1/16 11

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. Chaos 3/1/16 12

Hang onto your hats... Next week we’re going to see the fractal dimension rear its Hydra- like head again! Chaos 3/1/16 13