24.

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Page 122
2.1.4—
How to Tell Randomness from Chaos
If the data generated by random and deterministic mechanisms look so much alike, how can we tell which mechanism actually generated the data?
Although the sequence of values of these two data sets looks similar, a transformation of these data looks different for random and deterministic mechanisms. This transformation is a plot called a phase space and the points in it are called the phase space set. Thus the phase space set can be used to determine if the sequence of values was generated by a random or a deterministic mechanism.
The phase space set is constructed in the following way:
We take the first two values x(1) and x(2) and make believe that these two values are the X and Y coordinates of a point. That is, we set X = x(l) and Y = x(2), and plot a point at those coordinates.
Then we take the next two values x(2) and x(3), and again make believe that these two values are the X and Y coordinates of a point. Thus we now plot a point at coordinates X=x(2), Y=x(3).
Then we take the next two values x(3) and x(4), and again make believe that these two values are the X and Y coordinates of a point. Thus we now plot a point at coordinates X=x(3), Y=x(4).
And so on.
1—
Left:
Random
The phase space set of points produced from the random data fills up the 2-dimensional space.
2—
Right:
Deterministic Chaos
The phase space set of points from the chaotic data does not fill up the 2-dimensional space. It forms a 1-dimensional object called a strange attractor.
The parabola seen in the phase space set reveals to us the equation x(n+1) = 3.95 x(n) [1-x(n)] that was used to generate the data.

 
[Cover] [Abbreviated Contents] [Contents] [Index]


Fractals and Chaos Simplified for the Life Sciences
Fractals and Chaos Simplified for the Life Sciences
ISBN: 0195120248
EAN: 2147483647
Year: 2005
Pages: 261

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