Area-preserving map
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Doubling without an attractor
Feigenbaum's 4.669 holds for every map that folds and loses area. A map that keeps area — a frictionless pendulum sampled once a swing — has no attractors at all, yet its periodic orbits double just the same: a period-1 orbit at a = 3, period 2 at a = 4, then 4.1205, 4.1344, 4.1360 and on. The intervals shrink by 8.721 each time, not 4.669, and the orbits by 4.018, not 2.503. Let the map leak a little area and the cascade starts with one constant and finishes with the other.
An orbit stuck to an island
A chaotic orbit in the standard map stretches at 0.44 a step, and then for 19,000 steps it almost stops stretching at all. It has wandered into the fringe of an island of regular motion and been caught. So gaps between visits to a box do not die away exponentially as they would in a uniformly chaotic map; they fall as a power, and an orbit of 400 million steps still has not seen the longest.
Named alongside it
The objects these essays reach for when they reach for this one.
Periodic orbitChaosFeigenbaum constantNewtons methodPeriod-doublingRenormalisationRotation numberSimulationStabilityUniversality