Matpic
mathematics, in pictures
Concept

Stability

Whether a small disturbance to a state dies away or grows, decided by the rule's slope there. It is decided by whether the map's slope at the fixed point is under one in size, and it is what separates an attracting point from a repelling one.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

A fixed point that attracts, and one that does not. The same map at two parameters, with the staircase walking towards the crossing in one and away in the other.

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 The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points.

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 x ↦ cos x: two starts, one destination. A map whose graph is nowhere steeper than a fixed factor under one, with staircases from two different starting points converging on the same crossing, and the distance to it falling under a geometric bound.

A map that shrinks everything

One extra hypothesis — that every distance is shortened by at least a fixed factor — turns the existence of a fixed point into its uniqueness, an algorithm for finding it, and a bound on the error after any number of steps.

analysis · Fixed points The splits no group can beat, for three partners. The triangle of ways to split a fixed total between three players, with each coalition's demand drawn as a straight cut across it, and the region surviving every cut shaded.

A split nobody can walk away from

Every way of dividing what a group earns is a point of a triangle, and every coalition's threat to leave cuts a straight line across it. What survives all the cuts is the set of stable divisions — and for one three-player game there is nothing left.

applied · The core The rotation number against the parameter at K = 1. The measured rotation number of a circle map plotted against its parameter, forming a staircase that is flat over an interval at each simple rational.

The staircase that is flat almost everywhere

A map of the circle advances by an average amount each step. Plot that average against the parameter driving it and the graph is flat over an interval at every rational, rises only on a set of measure zero, and still climbs from nothing to one.

dynamics · Mode locking A path of 4 bounces that closes, in a triangle of 100°, 40°, 40°. A triangular billiard table with a periodic path found by an exhaustive sweep of starting positions and directions.

The triangle nobody can settle

Does every triangular billiard table have a path that closes on itself? Acute triangles do, right triangles do, triangles with rational angles do — and for the rest the question has been open since it was asked.

dynamics · Billiards

Named alongside it

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

Periodic orbitBifurcationDerivativeExhaustive searchFixed pointIterationLogistic mapOrbitSelf-similarityAllocationApproximationAttractor

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