Strange attractor — the series

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.

part 1 · dynamics

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.

part 2 · dynamics

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.

part 3 · dynamics

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.

part 4 · dynamics

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.

part 5 · dynamics

Where one Hénon orbit spends a million steps. A density map of 1000000 steps of one Hénon orbit counted into 420 × 140 cells: 2542 cells visited, half the time spent in 608 of them.

Where the time goes on an attractor

A chaotic orbit's position is unpredictable within a few dozen steps. How it divides its time is not — start anywhere, follow long enough, and the fraction of time spent in each region comes out the same. That distribution lives on a set of no area, and among the infinitely many ways an orbit could spend its time, typical starts pick exactly one.

part 6 · dynamics

A quadrilateral the Hénon map carries strictly inside itself. Hénon's trapping quadrilateral with corners (-1.33, 0.42), (1.32, 0.133), (1.245, -0.14), (-1.06, -0.5), the image of its boundary under the Hénon map lying strictly inside it, and an orbit on the attractor.

A region the orbit cannot leave

That the Hénon map and the Lorenz flow have attractors takes two lines of arithmetic each — a region mapped strictly inside itself, a quantity that falls outside an ellipsoid. That the attractors are strange took a computer and twenty years for Lorenz, and for Hénon's classical parameters it has never been done. The difference between the two questions is the difference between a region and what lives in it.

part 7 · dynamics

Two stable cycles a hair from the classical Hénon attractor. The Hénon attractor at a = 1.4 with the certified stable period-13 orbit at a = 1.39945 and period-19 orbit at a = 1.4001616 drawn over it.

Stable cycles a hair from Hénon's chaos

Nobody knows whether the Hénon map at a = 1.4 is chaotic, because a stable cycle in a window too thin to see would look exactly the same. So hunt the windows: a stable orbit of period 19 at a = 1.4001616, proved to exist and to attract by interval arithmetic, in a window 15 hundred-millionths wide and 0.00016 from the classical value — and nothing nearer, down to a spacing of a ten-billionth.

part 8 · dynamics

An attractor with corners. One orbit of the Lozi map at a = 1.7, b = 0.3: 7000 points on the attractor; it lies in the region where a strange attractor is proved.

Replace the parabola with a corner

Nobody can prove Hénon's map is chaotic at its classical values, since a stable cycle might hide in a window too thin to find. Replace x² in it by |x| and the whole difficulty disappears: one cone of directions is stretched by every step, in three lines of arithmetic, and across 301 parameters the Lyapunov exponent is a straight rising line with no windows at all.

part 9 · dynamics