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Period-doubling — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points.

    The road paved with doublings

    Turn one dial slowly and watch what a map settles into. It settles on a point, then on two points, then four, then eight — faster and faster, and the doublings run out at a parameter that is finite.

    part 1 · dynamics
  2. Where the period doubles, and by how much the gaps shrink. The parameters at which the period doubles, with the ratio of consecutive gaps beside them.

    A constant that does not care which map

    The gaps between successive period doublings shrink by a factor. Measure that factor for the logistic map and you get 4.669. Measure it for a completely different map and you get 4.669, and nobody expected that.

    part 2 · dynamics
  3. The turning point's orbit over the bifurcation diagram, 3.55 to 4. The bifurcation diagram of the logistic map from 3.55 to 4 with the curves f(1/2), f²(1/2), … up to the 6th image drawn over it. The first two bound the attractor and the rest trace the dark lines inside it.

    The dark lines are one point's orbit

    Past the end of the period-doubling cascade the bifurcation diagram turns into grey bands crossed by darker curves. Every one of those curves is the orbit of a single point — the top of the hump — and the places where they meet are exactly where the bands merge, in a second cascade that runs backwards at the same rate.

    part 3 · dynamics
  4. Intermittency at r = 1 + √8 − 0.0003. A time series of 600 steps of the logistic map just below the period-three window. Long stretches that look like a cycle of three, shaded, alternate with irregular bursts; there are 6 such stretches here.

    The window that opens with a stutter

    The period-three window does not fade in. At r = 1 + √8 a cycle of three appears out of nothing, and just before it does, the chaotic orbit keeps imitating the cycle that is not there yet — for twenty steps, then fifty, then hundreds, in quiet stretches whose length grows as one over the square root of the distance to the window.

    part 4 · dynamics

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