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 it becomes unstable.

This stable 2-cycle gives rise to a stable 4-cycle just as it becomes unstable.

This stable 4-cycle gives rise to a stable 8-cycle just as it 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 the proof 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.

Return to Logistic Map Cycles.