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Concept

Interval arithmetic

Computation in which each number is replaced by an interval guaranteed to contain it, with every result rounded outward. A computer can then prove inequalities despite rounding error, which is how computer-assisted proofs in dynamics are made rigorous.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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

Named alongside it

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

Computer-assisted proofHenon mapStrange attractorBifurcationCertificateLorenz systemLyapunov exponentLyapunov functionNewtons methodPeriodic orbitTrapping region

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