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Concept

Tent map

A map of an interval to itself made of two straight pieces meeting at a peak, both steeper than the diagonal. It is the simplest chaotic map, and the return map of the Lorenz system is close to one.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The tent map and the logistic map, joined by a change of coordinate. Two cobweb diagrams side by side — the tent map at slope two and the logistic map at four — with the orbit of one carried to the orbit of the other by a curve drawn between them.

The same map in different coordinates

The tent map and the logistic map at four look nothing alike and are the same map, carried onto each other by a change of variable. Everything either one does the other does, and the change of variable is a sine squared.

dynamics · Iteration The flow, reduced to one dimension. A scatter of 2395 points: each successive maximum of the Lorenz trajectory's third coordinate against the one before it. The points lie along a single curve with a sharp peak, which is the one-dimensional map the flow induces.

The flow that is really a map

A trajectory wandering through three dimensions is hard to reason about. Record only the successive maxima of one coordinate and the wandering collapses onto a curve — a map of an interval to itself, with a corner in the middle, which is a thing the theory can handle.

dynamics · Strange attractor What the tent of slope 3 keeps: 32 pieces after 5 steps. Rows showing the parts of the unit interval that remain inside it for 0 to 5 steps of the open tent map of slope 3, halving into a Cantor set.

Chaos on a set nobody lands on

Stretch the interval by three and fold it, and a third of it lands outside. Almost every starting point wanders chaotically for a few steps and then leaves for good; the points that never leave form a Cantor set of no length, on which the map is as chaotic as any map can be. How fast points escape, how fast they are stretched, and how thin the surviving set is are three numbers tied by one equation: the dimension is one minus their ratio.

dynamics · Sensitive dependence

Named alongside it

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

ChaosIterationLyapunov exponentCantor setChange of variableConjugacyDimensionFixed pointFractal dimensionHomeomorphismLogistic mapOrbit

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