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<font face="Georgia, Times New Roman, Times, serif" size="2" color="#CCCCCC">The
Lyapunov exponent describe how quick a pertubation increase or decrease in
a dynamic system. It serve as indicator for the difference between chaos and
regularity. A dynamic system is examined at successive times; At time (n),
the system is described with a variable X(n) and one admits that a specific
law exists that allows to derive X(n) from the preceding X(n-1). The object
of the theory is to analyse the behavior of the successive X's when n grows
to infinity.</font>
<font face="Georgia, Times New Roman, Times, serif" size="2" color="#CCCCCC">Real
dynamic systems are to complex to describe with a few mathematical functions
(ex: animal, cell populations). But we can define an abstract system with
a simple function like: <br>
<span>f(x)=b*sin(x+r)</span><br>
what have a more complex behaviour like it looks.</font>
<font face="Georgia, Times New Roman, Times, serif" size="2" color="#CCCCCC">This
picture show a rectangle of the A,B area.<br>
For each point of the rectangle the behaviour of the iteration<br>
<span>x(n+1)=b*sin(x(n)+r)</span><br>
is analysed. The variable b have a fixed value. It was the idea of Markus
Marion (Scientific American article) to let the variable r follow a pattern
like (ABABAB) or (AAABBB) were A and B are two predeterminated values of the
A B area. Each pixel in the image corresponds to a particular choice of A
and B, whereas the different sequences lead to the differnt types of images.
Thus the picture represent a view in to the AB plane and the colors represents
the value of the Lyapunov exponent at this point. Positive Lyapunov exponent
result means chaos is present, a negative exponent means the sequence of numbers
is regular and periodic.</font>
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