Concept

Feigenbaum constant

The number 4.6692…, the limiting ratio between successive intervals of a period-doubling cascade. It is the same for every smooth one-humped map with a quadratic top, a universality explained by a renormalisation that rescales the map onto itself.

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

Where the period doubles, and by how much the gaps shrink. The parameters at which the period doubles, with the ratio of consecutive gaps beside them.

A constant that does not care which map

The gaps between successive period doublings shrink by a factor. Measure that factor for the logistic map and you get 4.669. Measure it for a completely different map and you get 4.669, and nobody expected that.

dynamics · Period-doubling

The turning point's orbit over the bifurcation diagram, 3.55 to 4. The bifurcation diagram of the logistic map from 3.55 to 4 with the curves f(1/2), f²(1/2), … up to the 6th image drawn over it. The first two bound the attractor and the rest trace the dark lines inside it.

The dark lines are one point's orbit

Past the end of the period-doubling cascade the bifurcation diagram turns into grey bands crossed by darker curves. Every one of those curves is the orbit of a single point — the top of the hump — and the places where they meet are exactly where the bands merge, in a second cascade that runs backwards at the same rate.

dynamics · Period-doubling

Named alongside it

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

Bifurcation Logistic map Period-doubling Attractor Chaos Convergence Critical point Invariant density Renormalisation Scaling Self-similarity Universality

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