Graphics Programs Reference
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(5) There Is a Value u
4 Beyond Which Chaos Ensues
It is possible to prove that the sequence
u
k
tends to a limit
u
∞
. The value of
u
∞
, sometimes called the
Feigenbaum parameter
, is aproximately 3.56994... .
Let's see what happens if we use a value of
u
between the Feigenbaum
parameter and 4.
<
X = itseq(f, 0.75, 100, 3.7); plot(X)
1
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This is an example of what mathematicians call a
chaotic
phenomenon! It
is not random — the sequence was generated by a precise, fixed
mathematical procedure — but the results manifest no predictible pattern.
Chaotic phenomena are unpredictable, but with modern methods (including
computer analysis), mathematicians have been able to identify certain
patterns of behavior in chaotic phenomena. For example, the last figure
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