Mathematical Chaos
In our experiments, we saw that when a = 4, the population sequence (xn) is chaotic and ergodic. We defined a chaotic system to be one where two different, but very close, seeds result in wildly different population sequences. Thus, the smallest experimental error will lead to totally erroneous results. While an ergodic system is one where the sequence (xn) will approach every possible point arbitrarily closely (more formally, for every pair of elements of [0,1], there is an xn between them). Thus, an ergodic system is one which is `trying to converge to everything'.
Fix a = ng to converge to everything'.
Fix a = 4. Set xk = sin2 pyk (for each k Î N), and substitute in the logistic equation, xn+1 = axn (1-xn).
Task 7. Show that this simplifies to sin2 pyn+1 = sin2 2 pyn.
Observe that, because sin2 pt = sin2 p(t+1), the
preceding equation is equivalent to
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Just as any real number, x say, in [0,1] can be written as a decimal expansion, x = decimal expansion, x = 0.1234567890123 ¼, for example; so x can be written with a binary expansion, x = 0.11001100110101¼. And just as multiplying 0.12345¼ by 10 and taking the result mod 1, gives 0.2345¼; multiplying the binary expansion of x, 0.11001100¼, by 2 and taking the results mod 1, gives 0.1001100¼.
Thus we see that, given y1s binary expansion, to calculate yn it is sufficient to shift the binary string to the left n places, and drop the integer part.
Task 7 (continued). Use the above to show that yn,
and hence xn, is chaotic - if y1 and y¢1 are in [0,1],
then yn - y¢n = 2n (y1 - y¢1) mod 1.
[Further, show that given a and b in [0,1], with a < b, then, provided the seed y1 has an infinite binary expansion, there is an N such that a < yN < b. (Hint: write a and b in their binary expansion, and note that a < b, means that the binary digits of a and b are equal up to some nth term, where a has binary digit 0 and b has binary digit 1.)]