Matpic
mathematics, in pictures
Series

Sensitive dependence — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two orbits of the logistic map at 3.9, started 0.0001 apart. Two sequences from almost the same starting point, plotted together against the step number.

    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.

    part 1 · dynamics
  2. The Lyapunov exponent, 2.8 to 4. The average rate at which nearby orbits separate, plotted against the parameter.

    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.

    part 2 · dynamics
  3. A periodic point and a wandering one, both near 0.3, parted by step 4. The distance between the orbit of a periodic point and the orbit of a point from a dense orbit, both starting in the same small interval, plotted against the step until the wandering orbit nears the point farthest from the periodic one.

    Sensitivity comes free

    The standard definition of chaos asks for three things: an orbit that goes everywhere, periodic orbits everywhere, and sensitive dependence on the starting point. The third, the one the word chaos is usually taken to mean, turns out to follow from the other two. A periodic point and a wandering point that start side by side must eventually part, because the wanderer has to visit places the periodic orbit never goes.

    part 3 · dynamics
  4. What the tent of slope 3 keeps: 32 pieces after 5 steps. Rows showing the parts of the unit interval that remain inside it for 0 to 5 steps of the open tent map of slope 3, halving into a Cantor set.

    Chaos on a set nobody lands on

    Stretch the interval by three and fold it, and a third of it lands outside. Almost every starting point wanders chaotically for a few steps and then leaves for good; the points that never leave form a Cantor set of no length, on which the map is as chaotic as any map can be. How fast points escape, how fast they are stretched, and how thin the surviving set is are three numbers tied by one equation: the dimension is one minus their ratio.

    part 4 · dynamics

All series