Matpic
mathematics, in pictures
Concept

Stability

Whether a small disturbance to a state dies away or grows, decided by the rule's slope there.

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

r = 2.6slope in (−1, 1) — attractingr = 3.3slope outside (−1, 1) — repellingat r = 2.6 the slope at the crossing is -0.60 and the staircase walks inat r = 3.3 it is -1.30 and the staircase walks out — the crossing has not moved, its steepness has

A point that pulls, and a point that pushes

Every crossing of a curve with the diagonal is a value the rule leaves alone. Whether anything ever arrives there is decided by one number — the slope at the crossing — and the picture makes the reason obvious.

dynamics · fixed points 10rthe logistic map's attractor at 460 parameters between 2.4 and 4one column per parameter, and the number of points in a column is the period there

The road paved with doublings

Turn one dial slowly and watch what a map settles into. It settles on a point, then on two points, then four, then eight — faster and faster, and the doublings run out at a parameter that is finite.

dynamics · period doubling

Named alongside it

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

Logistic mapAttractorBifurcationChaosCobwebContractionConvergenceDerivativeFixed pointIterationLinearisationOrbit

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