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Chaos can be found even in very simple systems; you can even try this
one out on your calculator. Choose a number between –2 and 2; call
this ‘initial condition’ X1. Calculate the square of X1 and
subtract 2; call the result X2. Now do the same thing to X2 (square it
and subtract 2) to get X3. Continue to apply the same rule, or algorithm,
to generate a long sequence of numbers (we can write this process down
as an equation Xn+1 = Xn -2). This system is deterministic – if you
start from exactly the same initial condition you will always get exactly
the same answer. But because we calculate X-squared it is also non-linear.
This is one of the necessary conditions for chaos.
The blue, solid line shows the sequence of numbers X1, X2,
X3, etc., up to X40 starting with the initial condition X1 = 0.4. Now
suppose we had made a small error in the initial condition and started
with X1 = 0.4001. The red, dotted line shows what happens. For the first
10 or so steps all seems well – a small initial error doesn’t
affect the result. But then the two sequences of numbers become quite
different. This is chaos – small initial differences can lead to
completely different predictions.
The formula in this example is a version of an equation, used in the study
of population growth, called the logistic equation.
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