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Concept

Henon map

A map of the plane that stretches, folds and squeezes, sending (x, y) to (1 − ax² + y, bx). Iterated at its classic settings it traces a strange attractor, and it is the simplest invertible map known to have one.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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.

dynamics · Strange attractor 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.

dynamics · Strange attractor 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.

dynamics · Strange attractor 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.

dynamics · Strange attractor

Named alongside it

The objects these essays reach for when they reach for this one.

Strange attractorPeriodic orbitBifurcationComputer-assisted proofInterval arithmeticLorenz systemLyapunov exponentCertificateCounterexampleErgodic theoremFractalInvariant measure

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