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Ladder

Period-doubling — the ladder

2 distinct arguments against 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.

    rung 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.

    rung 2 · dynamics

All ladders