Chaos
Chaos
(A) Logistic Mapping:
The iterative process yn+1 = kxn(1 - xn), which is referred to as a logistic mapping, illustrates many interesting features of Chaos Theory.
The value of k is taken to be between 0 and 4, and the stages of the iterative process are illustrated using the graphs of:
y = kx(1 - x) and y = x.
These are illustrated in this diagram:
Fig (i) |
The stages of the iterations are indicated by the horizontal and vertical lines:
x0 is the starting value (anywhere between 0 and 1). The value of kx0(1 - x0) is then calculated , represented by the vertical green line, this is then 'transferred to the x-axis, by going horizontally to the line y = x, and then down to the x-axis, giving the next value, x1.
The process is then repeated, starting with x1 instead of x0, represented by the cyan lines leading to x2.
The process then continues indefinitely, leading to a sequence whose behaviour depends on the value chosen for k.
It either
- converges on the point of intersection of the graphs
- eventually oscillates between 2 or more values
- produces a chaotic sequence of values.
The 'interesting' values of k lie between 3.5 and 4, and you can experiment for yourself if you download chaos.zip.
(B) Sierpinski Sieve
In 1915 the Polish mathematician Vaclav Sierpinski produced the so-called Sierpinski Sieve.
Think of a solid equilateral triangle. If you join up the mid-points of each of the sides you divide your original triangle into 4 smaller equilateral triangles. Now remove the middle one, and you are left with 3 triangles and a ‘hole’.
Now do the same process with the 3 remaining triangles (you now have 9 triangles and 4 ‘holes’)....... and then do the same with the remaining 9 triangles............then with the remaining 27 triangles ..................
The process can continue indefinitely and produces a fractal diagram
.
The same result can be produced by the following iterative process:
- Draw an equilateral triangle and a random starting point (anywhere)
- Choose a vertex of the equilateral triangle at random.
- To find the next point: take the mid-point of the line joining your previous point to the vertex you chose in step (b).
- Repeat steps (b) and (c) indefinitely.
The program chaos.zip allows you to see this happening ........ but can you see why it should produce the same result?
You can also see what happens if:
- you start with any regular polygon (instead of just a triangle) or
- you don’t necessarily take the mid-point in step (c), but go some chosen fraction of the way from the previous point to the selected vertex.
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Created: 21 January 1999
Last Updated: 21 January 1999
