Consider a normal function:
y = 2x + 3
This simply means: take a number (x), multiply it by 2, add 3, and you will get the result (y). We can now work out that:
If x = 0 then y = 3
If x = 1 then y = 5
Iteration is a process by which a function is used repeatedly, with the output from the function put back into the function as the input. So, still using the function y = 2x + 3, we could start off with 0, and find that f(0) = 3. Then we feed the result back into the function, to get f(3) = 9. f(9) = 21, f(21) = 45, and so on.
In this case, the iteration will diverge to infinity - the terms will keep on getting bigger and bigger. This process of using the results as the input for the next stage is known as mathematical feedback.
To indicate that this is an iteration, we can use the following notation:
x(n+1) = 2x(n) + 3
This indicates that the (n+1)th term is found by multiplying the nth term by two and adding three.
The function which we have been using as an example, y = 2x + 3, is a linear function. This means that a graph of y against x will give a straight line. This particular function will not give rise to chaos, because it is linear. To find a chaotic function, we need to find a non-linear function - one which produces a curve when plotted on a graph.
Non-linear functions are produced by having not just x's, but x raised to a power - for example, x to the power two, usually called x squared.
Consider the function:
y = x^2
If we perform an iteration of this function, the value will quickly diverge to infinity if the initital value of x is greater than 1 - if we start with x = 2, the sequence will be 2, 4, 16, and so on. If the value of x is 1, the value will always be 1, as 12 = 1. If x is 0 or higher but less than one, it will converge to 0. If the value of x is negative, the same will happen as would have happened to a positive number of the same value.
This function, therefore, is not chaotic - we can tell the pattern of its iteration no matter what its starting value.
Now let's look at the equation y = 2x(x-1), which can be expanded and written y = 2x^2 - 2x.
It is not at all easy to predict the behaviour of this equation when iterated. To help us study the pattern, we can plot a graph, with the number of iterations along the x-axis (horizontal), and the function value along the y-axis (vertical). Using this graph, we can then see how the value of the function has changed as the iterations have proceeded.
If this is done (these graphs can be seen in the main InterFACE project site), the graphs obtained with very similar initial values of x look very different.
So, we have finally found chaos! What we see in the both of the two graphs is not convergance to a single value, or divergance to infinity - the function is chaotic.
But the other important feature to notice about the two graphs is that, although the initial conditions were very similar, the graphs produced are radically different. The difference in the initial condition - the starting value of x - between the two graphs was just 0.001, or 0.1%. And yet, because the system is sensitive to inital conditions, this has had a great impact.