Deterministic Chaos

Logistic Map Cycles

As s increases from 0 to 4, we observe
* Each stable cycle goes through an infinite sequence of period-doublings.
    For example, the stable fixed point gives rise to a stable 2-cycle just as the fixed point becomes unstable.
    This stable 2-cycle gives rise to a stable 4-cycle just as the 2-cycle becomes unstable.
    This stable 4-cycle gives rise to a stable 8-cycle just as the 4-cycle becomes unstable.
In general, any stable n-cycle gives rise to a family
2n-cycle → 4n-cycle → 8n-cycle → 16n-cycle → ...
* Each tangent bifurcation produces a stable cycle, that then gives rise to a cascade of period-doubling bifurcations.
* For each s-value, there is at most one stable cycle, and for many there are none.
* If there is a stable cycle, the iterates of 1/2 converge to the cycle. This was proved by Fatou and Julia. (What's special about x = 1/2 is that it is the critical point, the point at which the logistic map's derivative is 0.)
* Call the range of s-values of a stable cycle and all its stable period-doubling descendants a periodic window.
    The order in which these periodic windows arise is a bit complicated, but completely understood.
    This was discovered by Metropolis, Stein, and Stein, and is called the U-sequence. (Sometimes this is called the MSS-sequence.)
    They showed every periodic window contains an s-value for which x = 1/2 belongs to the cycle. These cycles are called superstable.
    With each superstable cycle, Metropolis, Stein, and Stein associated a sequence of symbols L and R, denoting whether a point of the cycle falls to the Left or Right of 1/2.
    The two 4-cycles shown below have sequences RLR and RLL. (Four-cycles containing x = 1/2 have only three symbols, because the cycle is always understood to start with x = 1/2.)
    The heart of understanding the ordering of the periodic windows is finding how the symbol sequences of any two superstable cycles are related to the order of the s-values at which they occur.
* For some s-values there is no stable cycle at all, and the dynamics are chaotic. Jakobsen proved the chaotic s-values are a non-negligible portion of the range 3.6 ≤ s ≤ 4, but the amount of chaos is difficult to estimate.

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