Generator

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

A generator in the dynamics library, called 60 times across 14 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.

Scrambled in three steps, restored in thirty

A difference that grows by φ² a step

No grid takes longer than three times its size

Every pixel returns on a cycle that divides the period

Halfway round, upside down

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
at r = 4, ε = 0.00 the measured exponent is λ + ln|1 − 2ε| ×74
at β = 0.30 the root-mean-square grows faster than a typical run ×17
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 one-way pull of 0.35 fails and 0.6 succeeds ×1
a periodic point lies within the interval ×1
a perturbed q-twisted state returns to itself exactly when cos(2πq/N) > 0 ×1
a random start settles into a twisted state with neighbouring phases differing by the same amount all round ×1
a random start survives s/(s − 2) steps on average ×1
a rational turn has only its periodic points ×1
a run of a million steps gives Viswanath's constant ×1
a tenth of plus signs already makes it grow ×1
a twisted state's average phase arrow has length nought ×1
above the locking value the chain is one cluster ×1
after 57 steps every picture is turned through half a turn about the corner ×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
and the centre settles on Viswanath's constant ×1
and the centre stays put within a spread ×1
apart, bursting, and locked ×1
as the coupling falls the clusters only multiply ×1
at full chaos λ is ln 2 ×1
at r = 4 the exponent is log 2, which is exactly computable ×1
at twenty steps the sampled mean square matches the exact one ×1
at ε = 0.776 the ring of five locks and the ring of six does not ×1
below the threshold the residual order falls like one over the square root of N ×1
below the threshold the runs shrink, above it they grow, at it they do neither ×1
doubling on the line is sensitive and nothing else ×1
each run of a thousand steps grows at close to Viswanath's rate ×1
every chain locks 3 per cent above max |Sₖ| and fails to 3 per cent below ×1
every pixel's cycle length divides the period ×1
every run settles into a twisted state ×1
Gaussian frequencies: incoherent at weak coupling, ordered at strong ×1
just above the locking coupling every oscillator keeps one frequency; just below, the chain splits ×1
more plus signs never slow it down ×1
mutual coupling locks from a quarter ×1
on a 61 × 61 grid the cat map returns every picture after 30 steps ×1
on a 74 × 74 grid the period is 114 ×1
one-way coupling locks from a half ×1
order appears near Kuramoto's threshold 2/(π g(0)) ×1
quadrupling the steps doubles the spread ×1
repeating the plus step alone grows fastest ×1
rings of up to five can lock and rings of six or more cannot ×1
the cat map's period is half the Fibonacci numbers' Pisano period ×1
the column count is a whole number between 40 and 1200 ×1
the cycles account for every pixel ×1
the dense orbit enters the interval ×1
the dense orbit later comes near the far point ×1
the direction piles up sharply rather than spreading evenly ×1
the doubling map has all three ×1
the empirical law of Kₗ/√N at a thousand oscillators is within 0.05 of Kolmogorov's everywhere ×1
the exact mean square has grown ×1
the first coupling is below the threshold of a quarter and the second above it ×1
the largest period is a whole number between 2 and 8 ×1
the locking coupling over the square root of N approaches √(π/2) ln 2 ×1
the lower parameter is between 0 and 4 ×1
the mean square grows like φ, so its root grows like √φ ×1
the measured escape rate is log(s/2) ×1
the median run has shrunk ×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 period equals 3N exactly at N = 10, 50 and 250 in this range ×1
the period never exceeds three times the grid size ×1
the picture comes back at step 30 and not at the steps drawn before it ×1
the range runs upward and is wide enough to draw ×1
the separation grows by φ² = 2.618 at every step until it saturates ×1
the share of locked oscillators is the share with |ω| below K r ×1
the slope is above 2, so the middle escapes ×1
the spread of the estimate narrows as the runs lengthen ×1
the starting point is between 0 and 1 ×1
the three couplings sit below the average threshold, just above it, and above the fixed point's ×1
the truth lies strictly between the naive guesses ×1
the two orbits begin the stated distance apart ×1
the two starts are close together ×1
the typical rate crosses one at the threshold ×1
the upper parameter is between 0 and 4 ×1
the view is one of lyapunov, dense, periodic, banks, table, cantor, survival, transient, kg, randfib, randfibrate, randfibnaive, randfibangle, randfibbias, betaruns, betarate, betamean, betaspread, betapatterns, syncpair, syncsweep, syncdrive, syncbubble, syncring, kurhero, kurcurve, kurfinite, kurtime, kurlocked, kurgauss, kchero, kcsums, kctest, kcsqrt, kckolm, kcclusters, krhero, krbasins, krstable, krreach, krslips ×1
the weak pair stays apart and the strong pair locks ×1
the winding number stops changing early in the run ×1
the winding number's variance grows in proportion to the ring's length ×1
weak coupling leaves the phases scattered and strong coupling gathers them ×1
well above the threshold the measured order matches √(1 − 2γ/K) ×1
with enough neighbours every start synchronises; with two, under three in five do ×1
with minus signs only the sequence cycles and does not grow ×1
with plus signs only it grows like the golden ratio ×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 chaotic map that always comes back

Arnold's cat map stretches a square by the golden ratio squared and wraps it round, and any two nearby points fly apart: a difference of a million-millionth fills the square in twenty-seven steps. Apply the same map to a picture made of pixels and it dissolves into noise within a few steps — and then, after exactly thirty steps on a 61 × 61 grid, every pixel is back where it started. Chaos and exact return are both true, and the Fibonacci numbers decide how long the return takes.

Dynamics

A crowd of clocks that falls into step

Give a thousand oscillators a thousand different natural rhythms and let each be pulled towards the average phase. Below a definite pull nothing happens; above it a cluster forms and grows, and Kuramoto's 1975 calculation says exactly where: at twice the spread of the frequencies, with the order growing as the square root of one minus the threshold over the pull. A simulation lands on that curve to the third decimal.

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

A growth rate no step contains

Start with 1, 1 and at every step either add the last two numbers or subtract them, choosing by a coin. The sequence wanders, shrinks, even returns near zero — and grows, almost surely, like 1.13198824 to the power of the step. That number is a Lyapunov exponent of a product of random matrices, and it cannot be found by averaging anything about a single step: the two steps do not commute, and the rate depends on which way the vector is pointing when each one is applied.

Dynamics

A long enough chain never locks

Line oscillators with different natural rhythms up in a row and let each feel only its two neighbours. The chain locks into one frequency exactly when the coupling exceeds the largest running total of their frequency differences — a random walk tied down at both ends — so the coupling needed grows like √N, and its law is Kolmogorov's, the same distribution that tests whether a sample fits a curve.

Dynamics

A ring of clocks settles into a twist

Put identical oscillators round a ring, each pulled only by its neighbours, and start them at random. They always lock — every one ends at the same frequency — but most of the time not in step: the angles wind round the ring a whole number of times and stay wound. The winding number a random start chooses is spread like a bell whose width grows like the square root of the ring, and only coupling to about a third of the ring makes full synchrony certain.

Dynamics

A threshold no average can see

Damp the random Fibonacci sequence — add or subtract β times the term before last, by a coin — and somewhere between β = 0.6 and β = 0.8 almost every run stops shrinking and starts growing. The switch is at β ≈ 0.70258. The average of the squares, which can be computed exactly, grows for every β there is, so the one number that decides the sequence's fate is invisible to the one number anybody can compute.

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.

Dynamics

Two chaotic orbits made to agree

Chaos means two copies of a system started a hair apart soon have nothing in common. Couple them — let each take a fraction ε of its next step from the other — and above a definite strength they agree to every digit for ever, while each stays exactly as chaotic as before. The strength needed is read off the same number that measured the chaos: the Lyapunov exponent.

The whole library · What the figures prove