Systems that are Sensitive to Initial Conditions

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Systems that are sensitive to initial conditions are those that whose trajectories diverge in ways that are not predictable. One of those systems is the chaotic Lorenz attractor, but even simpler systems can show dependence on the initial conditions.

In this example the diverging behaviour is obtained by the totally deterministic algorithm 2 times modulo 1 or (2x%1) that you can even try out in your calculator.

  1. take a random number between 0 and 1 (e.g. 0.823)
  2. multiply that number by 2 (e.g. 2*0.823=1.646)
  3. calculate the modulo 1 (%) of this value (e.g. 1.646 % 1=0.646). modulo 1 corresponds in this case to make the integer part of the number 0.
  4. use the result obtained in 3. and repeat from 2.

Now start with a second number very similar to the first (e.g. 0.825) and repeat the process (in the animation the initial difference is just 0.01% between the two values).

You’ll see that for the first iterations the calculations (the trajectory) are similar but suddenly they jump all around. After some iterations you can’t predict the behavior of the second trajectory, even if you know the first trajectory. This clearly shows that the system is sensitive to initial conditions. A very simple and strange system indeed.