Disclaimer
First let me state, that I am not a mathematician.Second the way things are presented here is a simplified version.
As always the whole truth is more complex.
If you are interested in the details and mathematical exactness, please consult Wikipedia.
Banach Fixed Point Theorem (Simplified)
- a set of elements, where you are able to measure the distance between the elements, is called metric space.- a fixed point is an element of a function, which gets mapped to itself: f(z) = z e.g. f(z) = z2 then f(1) = 1
- now we choose a function f(z) with a "contracting" property:
distance of f(z1) and f(z2) < distance of z1 and z2 for all z1, z2 of the metric space.
i.e. if you take two elements out of the metric space and calculate the function of each element,
the distance of the two calculated elements will be smaller than the distance of z1 and z2.
The Banach fixed point theorem now says:
1) A contracting function in a metric space always has a fixed point.
2) You can calculate a fixed point for a start value s out of the metric space by doing an iteration:
z0 = s
z1 = f(z0)
z2 = f(z1)
zn+1 = f(zn)
Explanation: because f(z) is contracting, the consecutive elements z0, z1, z2, z3 get closer and closer.
Such a sequence is called Cauchy sequence.
Example: 0.1, 0.12, 0.127, 0.1274, 0.12746, 0.127465, 0.1274651, 0.12746516.
If you go on with this example forever, you see the example converges to a value round about 0.127465....
Dynamical Systems
A dynamical system is a system, which state changes over time. An example is the weather.As very simplified example take the sun, which heats up water on the sea. The sea weather changes state,
while getting heated by the sun.
Mathematically the sun is a function changing the state of the sea weather. Again this can be described with an iteration:
z0 = s
zn+1 = f(zn)
Will the sea weather reach a stable state, where the sun shines for two weaks? Or will there be a storm on the sea?
Stable state is just another word for a fixed point: f(z) = z.
In a very simplified model of the weather the weather will reach a stable state,
if the sun function is contracting. (Banach fixed point theorem)
Obstacles
Another example for a dynamical system is the flow of water.What happens, if you put a rough and edgy obstacle into the water?
Around the obstacle you have turbulence, water spirals and such.
For certain obstacles a small change in the obstacle will result in great changes of the turbulence.
This is called the butterfly effect.
Now let's put a mathematical obstacle called c into the iteration:
The function now is: f(z) + c, or more precise: f(z,c)
z0 = s
zn+1 = f(zn,c)
Here we are :-) This iteration is the one for calculating fractal images.
The Mandelbrot style shows for which obstacles c and a certain start value s the sequences will get a fixed point, diverge or oscillate.
The Julia style shows for which start values s and a certain obstacle c the sequences will get a fixed point, diverge or oscillate.