Lorenz system
Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
Every loop of the butterfly links every other
The Lorenz equations have infinitely many periodic orbits, each a closed loop winding round the two lobes in its own order. Name each loop by that order — LR, LLR, LLRLR — and the linking number of any two can be read off the names, without solving anything: it is half the number of times a left-going strand passes a right-going one. Every such crossing has the same sign, so every pair of loops links, positively. Computing forty-five pairs from the equations themselves, the Gauss integral lands on the predicted whole number every time.
Named alongside it
The objects these essays reach for when they reach for this one.
Strange attractorHenon mapPeriodic orbitAttractorBifurcationComputer-assisted proofDissipationErgodic theoremFixed pointFractalFractal dimensionGenus