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Concept

Return map

A return map records where a flow next crosses a chosen surface, as a function of where it crossed last. When the flow contracts strongly, its crossings fall on a curve and the map is a function of one variable, so the theory of iterated maps of an interval applies to the flow.

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

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 Where the Rössler flow's peaks fall, as one parameter changes. Peak values of x against c from 2.5 to 6.2 for the Rössler system with a = b = 0.2: period 1, 2 and 4 at c = 2.5, 3.5 and 4, chaos beyond about 4.2, a period-3 window near 5.3.

A flow that doubles like a parabola

Otto Rössler's three equations have one nonlinear term and a single knob. Turn it and the orbit closes after one loop, then two, then four, then never — the logistic map's cascade, in a flow in three dimensions. Record each peak of the orbit against the one before and the reason appears: the points lie on a single rounded hump. The gaps between doublings shrink by ratios 3.49, 4.54, 4.59, climbing towards Feigenbaum's 4.669, and the chaos contains a period-3 window that repeats the whole cascade.

dynamics · Period-doubling

Named alongside it

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

Strange attractorBifurcationChaosDifferential equationDimensionFeigenbaum constantFixed pointIterationPeriod-doublingPoincare sectionTent mapUniversality

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