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Concept

Invariant density

A density on the states of a dynamical system that the map carries to itself. When it exists it gives the long-run fraction of time a typical orbit spends in each region, and for the logistic map at its top parameter it is the arcsine law.

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

Where a long orbit of the logistic map spends its time. A histogram of 60000 iterates in 32 bins, with the density the map preserves drawn over it as the exact share each bin should hold.

The histogram an orbit leaves

When no single step of an orbit is worth reporting, what is left is where it spends its time. That distribution is not uniform, it does not depend on where the orbit started, and it can be computed in closed form.

dynamics · Iteration 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.

Arcsine distributionAttractorBifurcationChaosConjugacyCritical pointErgodicityFeigenbaum constantIterationLogistic mapMeasurePeriod-doubling

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