Transcript x n+1

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)
Time
Rabbits
Rabbits
Rabbits
Rabbits
Rabbit Multiplication, r=1.8
1.80
1.60
1.40
1.20
Next Year's Rabbits
1.00
0.80
x1 to x2
x2 to x1
0.60
0.40
0.20
0.00
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-0.20
-0.40
Number of rabbits (000) this year
1.1
1.2
1.3
1.4
1.5
1.6
Rabbit Multiplication, r=2.3
2.00
1.50
Next Year's Rabbits
1.00
x1 to x2
0.50
x2 to x1
0.00
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-0.50
-1.00
Number of rabbits (000) this year
1.1
1.2
1.3
1.4
1.5
1.6
Rabbit Multiplication, r=2.5
2.00
1.50
Next Year's Rabbits
1.00
x1 to x2
0.50
x2 to x1
0.00
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-0.50
-1.00
Number of rabbits (000) this year
1.1
1.2
1.3
1.4
1.5
1.6
Rabbit Multiplication, r=3
2.00
1.50
Next Year's Rabbits
1.00
0.50
x1 to x2
x2 to x1
0.00
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-0.50
-1.00
-1.50
Number of rabbits (000) this year
1.1
1.2
1.3
1.4
1.5
1.6
Value of r
Final Population
1.8
Value of r
Final Population
Final Population
1.8
2.3
Value of r
Final Population
1.8
2.3 2.5
Value of r
Final Population
1.8
2.3 2.5
3.0
Value of r
Value of r
Final Population
Divergence of initially close points:
Definition:
dx(n)=2ndx(0)
where  is the
Lyapunov exponent
dx(3)
dx(0)
Lyapunov exponent
for the Verhulst
process
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.
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 and
coastlines 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 + iy
Begin 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.