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

2. What is chaos theory?

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

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

5. Sensitivity to Initial Conditions
Minor errors increase exponentially until meaning is lost

6. Lorenz Attractor
Weather
Slight changes have dramatic effects

7. Lorenz Equation

8. Solving the ODE

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

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

11. Arnold’s Cat Map Iterations

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

13. 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

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

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

16. Fun with Cat Maps
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17. 3DDD CAT MAPSSS
Find the 3x3 equivalent

18. Chaotic Iterations in the Real World
Image encryption
Weather forecasting D8

19. 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.

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

21. Why is this so secretive?
Encrypted
Without Key
With Key