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.

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.

* 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.