BY: GOKUL RAMADOSS AND FIONA ZHANG feat. Mr. Olsson and Sunny Chaos!!ASKJFHAL BY: GOKUL RAMADOSS AND FIONA ZHANG feat. Mr. Olsson and Sunny

What is chaos theory?

Background Determinism Sensitivity to initial conditions (a.k.a. butterfly effect)

Determinism Seemingly random transformations do have patterns Future states can be predicted from initial state In principle, BUT not in practice!

Sensitivity to Initial Conditions Minor errors increase exponentially until meaning is lost http://www.youtube.com/watch?v=8qD6RWlDeuY

Lorenz Attractor Weather Slight changes have dramatic effects

Lorenz Equation

Solving the ODE

Arnold’s Cat Map Transformation Maps S onto S Preserves Area No overlap

Arnold’s Cat Map A. Multiplication B. (x,y) mod 1

Arnold’s Cat Map Iterations

Arnold’s Cat Map Determinism p x p pixel grid coordinates of form (m/p , n/p) So, when a coordinate is transformed...

Arnold’s Cat Map Determinism Returns to initial within p^2 iterations (n) n = period, Π = period of pixel map Set of iterates = n-cycle

Arnold’s Cat Map Mathematics Multiplication by A dilates the image by a factor of the eigenvalue in the direction of the eigenvector

Arnold’s Cat Map Sensitivity Even an insignificant error in measurement has a significant effect. n iterations => A^n Makes true determinism impossible

Fun with Cat Maps http://www.jasondavies.com/catmap/#http:// www.mychandlerschools.org/cms/lib6/AZ01001 175/Centricity/Domain/2157/ad.jpg http://www.jasondavies.com/catmap/#https:// scontent-a-sjc.xx.fbcdn.net/hphotos- ash2/p206x206/313425_280175245349513_1416 451222_n.jpg

3DDD CAT MAPSSS Find the 3x3 equivalent

Chaotic Iterations in the Real World Image encryption Weather forecasting http://www.youtube.com/watch?v=AYcuLZ1OW D8

Image Encryption Use an Arnold Transformation x number of times on a picture. Produce a key with certain numbers that can be inserted into an equation to yield x. To decrypt just invert the transformation x number of times.

Cool Example of Encryption -A lot of times high security files are encrypted in this manner.

Why is this so secretive? Encrypted Without Key With Key