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Chaotic behavior is surprisingly simple to generate. The equation:

x[n+1] = alpha * x[n] * (x[n] - 1)

is chaotic for alpha > 3.56. In other words, if you run this Python code:

   def chaos(alpha):
       x = 0.5 # The "initial condition"
       l = []
       for i in range(100):
           x = alpha * x * (1 - x)  # The logistic equation
           l.append(x)
       return l
then

chaos(2.6) yields [...., 0.615, 0.615, 0.615, 0.615, 0.615, 0.615, ....]

chaos(3.2) yields [...., 0.513, 0.799, 0.513, 0.799, 0.513, 0.799, ....] (a 2-periodic sequence)

chaos(3.8) yields [...., 0.649, 0.865, 0.443, 0.937, 0.221, 0.656, ...] (non-periodic --- chaos!)

When we say that chaotic systems are unpredictable, it means that small differences in the initial conditions get amplified. So, if I replace x = 0.5 with x = 0.50001, then the result of chaos(2.6) will be (almost) the same because it's not chaotic, but the result of chaos(3.8) will be completely different.

The diagram at http://en.wikipedia.org/wiki/File:LogisticMap_BifurcationDia... is drawn entirely using the equation above, and it's the simplest example of funky mathematics that I know of. See also http://en.wikipedia.org/wiki/Logistic_map




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