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 2 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

Named alongside it

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

Lorenz system Strange attractor Computer-assisted proof Ergodic theorem Fractal Interval arithmetic Invariant measure Lyapunov exponent Lyapunov function Periodic orbit Time average Trapping region

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