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Ladder

Strange attractor — the ladder

5 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. The Lorenz attractor at ρ = 28. A trajectory of the Lorenz equations, projected onto two of its three coordinates.

    Two lobes and no cycle

    Three equations, three variables, and a trajectory that never crosses itself, never repeats, and never leaves a region of zero volume. The set it settles onto is not a point, not a loop, and not a surface.

    rung 1 · dynamics
  2. The flow, reduced to one dimension. A scatter of 2395 points: each successive maximum of the Lorenz trajectory's third coordinate against the one before it. The points lie along a single curve with a sharp peak, which is the one-dimensional map the flow induces.

    The flow that is really a map

    A trajectory wandering through three dimensions is hard to reason about. Record only the successive maxima of one coordinate and the wandering collapses onto a curve — a map of an interval to itself, with a corner in the middle, which is a thing the theory can handle.

    rung 2 · dynamics
  3. A closer start buys time and nothing else. The logarithm of the separation between two Lorenz trajectories plotted against time, for three different initial separations. The three curves are straight and parallel over most of their length, with the same fitted slope.

    A closer start buys only time

    Two trajectories from almost the same place separate exponentially, and the rate does not depend on how close they began. Halving the initial error buys one fixed interval of extra agreement, and no amount of precision buys more than a fixed number of those.

    rung 3 · dynamics
  4. Stretch, fold, and what is left. 5 stages of the horseshoe map's surviving set: one square, then two strips, then four, up to 16, each narrower than the last by a factor of 3.

    Stretch, fold, and what is left

    A system that pushes every pair of nearby points apart and keeps them all inside a bounded region has only one option, and it is the one a baker uses. Stretching and folding is the mechanism, and what survives infinitely many folds is a Cantor set.

    rung 4 · dynamics
  5. The same banding at every magnification. The Hénon attractor drawn from 26000 points, followed by 2 magnifications of one part of it. Each magnification resolves what looked like a single curve into several parallel ones.

    Neither a surface nor a solid

    The attractor has no volume, because the flow shrinks volumes at a rate that can be read off the equations. It is also not a surface, because a surface cannot carry chaotic dynamics. What is left is an object of dimension a little over two, and that number is measurable.

    rung 5 · dynamics

All ladders