Ladder

Sensitive dependence — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. step 14two orbits started 0.0001 apart, which is a distance no drawing can showthey are visibly apart by step 14, and by the end share nothing but their interval

    A difference too small to draw

    Two starting points a ten-thousandth apart, under the same rule, with nothing random anywhere. Within forty steps they have nothing in common — and the rule was not doing anything to them that it does not do to everything.

    rung 1 · dynamics
  2. 0rthe average of log |f′| along the orbit — positive means nearby orbits separateat r = 4 it is log 2 = 0.6931, which is the one value here that can be checked exactly

    How fast two orbits part

    The word "sensitive" is an adjective. Averaging the logarithm of one derivative along an orbit turns it into a number — one that says how many steps of prediction the map allows, and whose sign says whether it allows any.

    rung 2 · dynamics

All ladders