1. Chaos and Order (2)

3. Rabbits If there are x n rabbits in the n-th generation, then in the n+1-th generation, there will be x n+1= (1+r)x n (only works if x n > 2)

6. Rabbits and Foxes If foxes increase when rabbits become plentiful, this equation becomes: x n+1= (1+r)x n - rx n 2 (still only works if x n > 2)

7. Rabbits Time

8. Rabbits

15. Value of r Final Population

16. Value of r Final Population 1.8

17. Value of r Final Population 1.82.3

18. Value of r Final Population 1.82.32.5

19. Value of r Final Population 1.82.32.53.0

20. Value of r Final Population

21. Divergence of initially close points: Definition: dx(n)=2 n dx(0) where is the Lyapunov exponent dx(3) dx(0)

22. Lyapunov exponent for the Verhulst process

24. Characteristics of Chaos Two ingredients-- non-linearity and feedback -- can give rise to chaos. Chaos is governed by deterministic rules, yet produces results that can be very hard to predict. Images of chaotic processes can display a high level of order, characterised by self-similarity.

25. Chaos can arise in turbulent fluid flow…

27. …and in orbital dynamics…

28. And in biological systems:

29. Brassica Romanesco

30. When can chaos arise? In the iterated flow of raindrops down a slope:

34. The shapes making up eroded landscapes and coastlines are known as `fractals’.

35. If Log(Coastline_length) grows with (1-D)log(ruler_length) + b, then the coastline has fractal dimension D

36. When can chaos arise? In the motion of a double pendulum:

37. When can chaos arise? Trying to get two non-linear programs to converge: x y

38. Randomness, Chaos and Order We saw in last Friday’s lecture that a random image has maximal information content. If an image has less than maximal information content, it displays order.

39. 65,536 random binary digits.

40. Reflective and Rotational Symmetries Reduce Information Content

41. Rotational and Reflective Symmetries reduce information content

42. Six axes of Reflected Symmetry

43. What is the Information Content of a Fractal Image?

44. Formula for the Mandelbrot Set For each (x,y) in [(X min, X max ), (Y min, Y max )], Define z 0 = x + iy Begin loop with j = 1 to Maximum_Iterations { z j = z j-1 * z j-1 + z 0 ; if |z j |>2, leave loop } Colour the point (x,y) with colour(j) Total information content: 120 characters, 256 possibilities for each; hence, 960 bits.

45. Conclusions Any process involving non-linear feedback may become chaotic. The output of a chaotic process may appear random, but has a hidden order.