Generator

Two orbits of the logistic map at 3.9, started 0.0001 apart

A generator in the dynamics library, called 23 times across 7 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

divergence is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

Two orbits of the logistic map at 3.9, started 0.0001 apart. Two sequences from almost the same starting point, plotted together against the step number.

The Lyapunov exponent, 2.8 to 4

What the tent of slope 3 keeps: 32 pieces after 5 steps

Survivors of open tents of slope 3 and 4, step by step

Orbits that wander and then escape, under the tent of slope 3

Escape, stretching and dimension: 1 − κ/λ on four repellers

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

doubling 1 times returns 0/1 to itself ×120
of total length (2/3)^0 ×11
after 0 steps there are 2 to the 0 surviving pieces ×6
slope 3: box-counting dimension equals 1 − κ/λ ×4
the number of steps is a whole number between 10 and 200 ×3
a periodic point lies within the interval ×1
a random start survives s/(s − 2) steps on average ×1
a rational turn has only its periodic points ×1
an irrational turn is transitive and nothing else ×1
and at the step z nears q, z and p are at least half the far distance apart ×1
and below the onset of chaos it is negative ×1
at r = 4 the exponent is log 2, which is exactly computable ×1
doubling on the line is sensitive and nothing else ×1
the column count is a whole number between 40 and 1200 ×1
the dense orbit enters the interval ×1
the dense orbit later comes near the far point ×1
the doubling map has all three ×1
the largest period is a whole number between 2 and 8 ×1
the lower parameter is between 0 and 4 ×1
the measured escape rate is log(s/2) ×1
the number of strips is a whole number between 4 and 64 ×1
the orbit enters every one of the 16 strips within 400 steps ×1
the orbits separate exactly when the parameter is past the onset of chaos ×1
the parameter is between 0 and 4 ×1
the range runs upward and is wide enough to draw ×1
the slope is above 2, so the middle escapes ×1
the starting point is between 0 and 1 ×1
the two orbits begin the stated distance apart ×1
the two starts are close together ×1
the upper parameter is between 0 and 4 ×1
the view is one of lyapunov, dense, periodic, banks, table, cantor, survival, transient, kg ×1
z and p start inside the same small interval ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Dynamics

A bounce is a fold of the table

Reflect the room instead of the ball and every bounce disappears — the trajectory becomes a straight line through a tiled plane, and questions about what a ball does forever become questions about the slope of that line.

Dynamics

A difference too small to draw

Two starting points a ten-thousandth apart, under the same rule, with nothing random anywhere. Within forty steps they have nothing in common — and the rule was not doing anything to them that it does not do to everything.

Dynamics

Chaos on a set nobody lands on

Stretch the interval by three and fold it, and a third of it lands outside. Almost every starting point wanders chaotically for a few steps and then leaves for good; the points that never leave form a Cantor set of no length, on which the map is as chaotic as any map can be. How fast points escape, how fast they are stretched, and how thin the surviving set is are three numbers tied by one equation: the dimension is one minus their ratio.

Dynamics

How fast two orbits part

The word "sensitive" is an adjective. Averaging the logarithm of one derivative along an orbit turns it into a number — one that says how many steps of prediction the map allows, and whose sign says whether it allows any.

Dynamics

Sensitivity comes free

The standard definition of chaos asks for three things: an orbit that goes everywhere, periodic orbits everywhere, and sensitive dependence on the starting point. The third, the one the word chaos is usually taken to mean, turns out to follow from the other two. A periodic point and a wandering point that start side by side must eventually part, because the wanderer has to visit places the periodic orbit never goes.

Dynamics

The obstacle that makes a table chaotic

Put one round post in the middle of a square table and every trace of order goes. Two paths that start a hundred-thousandth of a degree apart end up on opposite sides of the table, and the reason is that a wall curving outwards multiplies a gap where a flat one only adds to it.

Dynamics

The same map in different coordinates

The tent map and the logistic map at four look nothing alike and are the same map, carried onto each other by a change of variable. Everything either one does the other does, and the change of variable is a sine squared.

The whole library · What the figures prove