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Published byAsher Morgan Modified over 9 years ago
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Chaos and Order (2)
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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)
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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)
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Rabbits Time
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Rabbits
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Value of r Final Population
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Value of r Final Population 1.8
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Value of r Final Population 1.82.3
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Value of r Final Population 1.82.32.5
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Value of r Final Population 1.82.32.53.0
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Value of r Final Population
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Divergence of initially close points: Definition: dx(n)=2 n dx(0) where is the Lyapunov exponent dx(3) dx(0)
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Lyapunov exponent for the Verhulst process
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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.
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Chaos can arise in turbulent fluid flow…
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…and in orbital dynamics…
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And in biological systems:
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Brassica Romanesco
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When can chaos arise? In the iterated flow of raindrops down a slope:
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The shapes making up eroded landscapes and coastlines are known as `fractals’.
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If Log(Coastline_length) grows with (1-D)log(ruler_length) + b, then the coastline has fractal dimension D
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When can chaos arise? In the motion of a double pendulum:
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When can chaos arise? Trying to get two non-linear programs to converge: x y
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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.
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65,536 random binary digits.
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Reflective and Rotational Symmetries Reduce Information Content
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Rotational and Reflective Symmetries reduce information content
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Six axes of Reflected Symmetry
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What is the Information Content of a Fractal Image?
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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.
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Conclusions Any process involving non-linear feedback may become chaotic. The output of a chaotic process may appear random, but has a hidden order.
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