Matpic
mathematics, in pictures
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. 10rthe logistic map's attractor at 460 parameters between 2.4 and 4one column per parameter, and the number of points in a column is the period there

    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. perioddoubles at r =gap ratio22.998304—43.4488464.743183.5438344.6385163.5643124.6464323.568719—each doubling is found by bisection, and each ratio is measured from the two gaps beside itthe last one is 4.646; Feigenbaum's constant is 4.6692, and it is the same for any map with a smooth hump

    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