Two orbits of the logistic map at 3.9, started 0.0001 apart
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.
At its defaults
show: "lyapunov"
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.
- and below the onset of chaos it is negative ×1
- at r = 4 the exponent is log 2, which is exactly computable ×1
- the column count is a whole number between 40 and 1200 ×1
- the lower parameter is between 0 and 4 ×1
- the number of steps is a whole number between 10 and 200 ×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 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
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
DynamicsA 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.
DynamicsHow 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.
DynamicsThe 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.
DynamicsThe 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.