Chaos and Order (2). Rabbits If there are x n rabbits in the n-th generation, then in the n+1-th...
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Transcript of Chaos and Order (2). Rabbits If there are x n rabbits in the n-th generation, then in the n+1-th...
Rabbits
If there are xn rabbits in the n-th generation, then in the n+1-th generation, there will be
xn+1=(1+r)xn
(only works if xn > 2)
Rabbits and Foxes
If foxes increase when rabbits become plentiful, this equation becomes:
xn+1=(1+r)xn - rxn2
(still only works if xn > 2)
Divergence of initially close points:
Definition:
dx(n)=2ndx(0)
where is theLyapunov exponent
dx(3)
dx(0)
Characteristics of Chaos
Two ingredients-- non-linearity and feedback --can give rise to chaos.
Chaos is governed by deterministic rules, yetproduces results that can be very hard to predict.
Images of chaotic processes can display a highlevel of order, characterised by self-similarity.
If Log(Coastline_length) grows with (1-D)log(ruler_length) + b,then the coastline has fractal dimension D
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.
Formula for the Mandelbrot Set
For each (x,y) in [(Xmin, Xmax), (Ymin, Ymax)],Define z0 = x + iyBegin loop with j = 1 to Maximum_Iterations
{zj = zj-1 * zj-1+ z0;if |zj|>2, leave loop
}
Colour the point (x,y) with colour(j)
Total information content: 120 characters, 256 possibilities for each; hence, 960 bits.