Chaos and Order (2). 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.

Chaos and Order (2)Chaos and Order (2)

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)

Rab

bits

Rab

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Time

Rab

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Rab

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1.8

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1.8 2.3

Value of r

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1.8 2.3 2.5

Value of r

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1.8 2.3 2.5 3.0

Value of r

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Divergence of initially close points:

Definition:

dx(n)=2ndx(0)

where is theLyapunov exponent

dx(3)

dx(0)

Lyapunov exponentfor the Verhulstprocess

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.

Chaos can arise in turbulent fluid flow…

…and in orbital dynamics…

And in biological systems:

Brassica Romanesco

When can chaos arise?

In the iterated flow of raindrops down a slope:

The shapes making up eroded landscapes andcoastlines are known as `fractals’.

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

When can chaos arise?

In the motion of a double pendulum:

When can chaos arise?

Trying to get two non-linear programs to converge:

x

y

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.

65,536 random binary digits.

Reflective and Rotational Symmetries Reduce Information Content

Rotational and Reflective Symmetries reduce information content

Six axes of Reflected Symmetry

What is the Information Content of a Fractal Image?

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.

Conclusions

Any process involving non-linear feedback

may become chaotic.

The output of a chaotic process may appear random, but has a hidden order.