Dynamics — page 3

The window that opens with a stutter

The period-three window does not fade in. At r = 1 + √8 a cycle of three appears out of nothing, and just before it does, the chaotic orbit keeps imitating the cycle that is not there yet — for twenty steps, then fifty, then hundreds, in quiet stretches whose length grows as one over the square root of the distance to the window.

A road where nobody overtakes

Rule 184 moves every 1 one cell to the right whenever the cell ahead is empty. It is one of only five elementary rules that never change the number of 1s, and that single property turns it into a model of traffic with an exact transition: below half density every jam dissolves, above it jams can never all clear and drift backwards against the flow.

No local rule can count the votes

A ring of cells, each holding 0 or 1, has to agree on whichever value is in the majority — every cell seeing only its neighbours. The best-known rule gets it right most of the time and wrong near a tie; no rule of any radius gets it right always. Yet two rules run one after the other do, on every ring, and the first of them is the traffic rule.

Almost every orbit is fair

Double a number and keep the fractional part, and do it again, and again. Where the orbit goes is written in the number's binary digits, and for almost every starting point it spends exactly a quarter of its time in each quarter of the interval. The proof is a short argument about Fourier coefficients being pushed to infinity — and it leaves room for a set of exceptions with no length at all, which includes every fraction and, for all anyone can prove, every number anyone has ever named.

A solvable chaos of every degree

The logistic map at four is chaotic and, through a change of coordinates, completely solvable: its orbits are cosines of doubling angles. The trick is not a one-off. For every whole number n there is a polynomial of degree n that multiplies angles by n instead of 2, and every one of them is exactly as solvable, has exactly nᵏ points of period k, and preserves the same distribution — and any two of them commute, which almost no two polynomials do.

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.

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.

A double root halves the error instead of squaring it

Near an ordinary root, Newton's method squares its error at every step and a handful of steps reach full precision. Near a double root it only halves the error — twenty steps where five would do, and a ceiling of about eight correct digits that no number of steps can break through. Doubling the step repairs the double root and ruins the simple one.

A cubic method that is Newton's in disguise

Halley's method, from 1694, uses the second derivative as well as the first and cubes the error at every step where Newton's squares it. It is also, exactly, Newton's method applied to a different function — p divided by the square root of p′ — and that single fact explains why its basins are calmer, why it walks out of the trap that holds Newton for ever, and why its boundaries are still fractal.

The ball that stays outside the table

Turn billiards inside out. A point outside a convex table looks at the corner on its right, jumps straight through it, and lands as far beyond as it started before. Round a square every orbit closes; round a circle every orbit keeps to its own circle; round a regular pentagon the orbits form islands with a fractal between them. Whether some table lets a point wander off to infinity was Moser's question, and the answer — yes, for a kite — took until 2007.

The word a straight line spells

A ball rolling across a square table, forever, hits walls in some order: V for a side wall, H for the end wall. Unfold the table and the ball becomes a straight line across a grid, and the order of walls becomes a word — VHVHVVHVHVVHV… for the golden slope. That word has exactly n + 1 different blocks of every length n, the fewest any word that never repeats can have; every stretch of it holds its fair share of H's to within one; and its blocks occur with at most three different frequencies.

A whole interval of speeds

Below the critical line every orbit of the circle map goes round at the same average speed. Above it the map folds back on itself, and the speed depends on where the orbit starts — not a few different values but a whole interval of them, every fraction in it the speed of some periodic orbit, and almost none of them ever seen by an orbit started at random.

The last circle to break

Kick a spinning rotor once a turn and most of its motions stay on curves that wind round forever, walls no orbit can cross. As the kick grows the walls break one by one, and the last to go is the one whose winding is the golden ratio — at a kick of 0.9716, found by watching a sequence of periodic orbits approximate it and asking whether they are stable.

Counting in a base that is not a whole number

Multiply by β and keep the fractional part, over and over, and the whole parts you throw away are the digits of the starting number in base β — even when β is the golden ratio. Which digit strings can ever appear is decided by one string alone: the way the number 1 is written in that base. In base φ it is .11, so 11 is the only thing forbidden; in base 1.8 it never ends, and no finite list of rules describes what is allowed.

Where a base-β orbit spends its time

Follow x ↦ βx mod 1 for a million steps and count how often the orbit visits each part of the interval. For the doubling map the answer is evenly; for the golden ratio it is a staircase with one step, spending 1.17 times the average near 0 and 0.72 times it near 1. The step sits exactly where the orbit of the number 1 lands, and in a base whose 1 never has a finite expansion the staircase has infinitely many steps.

Where the Collatz map is a coin

Extend the Collatz map from the whole numbers to the 2-adic integers — binary strings that run on for ever to the left — and it stops being mysterious. It becomes, after a change of coordinates, the simplest chaotic system there is: shifting a string of coin tosses one place. Everything about it is then known, and none of it says anything about the whole numbers, which is the most instructive failure in the whole story.

Fourteen fractions that list the primes

Change the Collatz rule so that the multiplier depends on the remainder modulo some other number than two, and the resulting maps can compute anything a computer can. John Conway proved it in 1972, which means no method can decide, for every such map, whether its orbits come down — and fourteen fractions, applied in order, turn out to be enough to print every prime.

Where the time goes on an attractor

A chaotic orbit's position is unpredictable within a few dozen steps. How it divides its time is not — start anywhere, follow long enough, and the fraction of time spent in each region comes out the same. That distribution lives on a set of no area, and among the infinitely many ways an orbit could spend its time, typical starts pick exactly one.

A region the orbit cannot leave

That the Hénon map and the Lorenz flow have attractors takes two lines of arithmetic each — a region mapped strictly inside itself, a quantity that falls outside an ellipsoid. That the attractors are strange took a computer and twenty years for Lorenz, and for Hénon's classical parameters it has never been done. The difference between the two questions is the difference between a region and what lives in it.

Every window is the whole diagram again

Zoom into one strand of the period-three window of the logistic map and the whole bifurcation diagram is there again — a single value, a fork, a cascade of doublings, chaos, bands and windows — upside down and ninety-seven times narrower. The doublings inside it shrink by Feigenbaum's 4.669, exactly as the originals do, because near the top of its hump the map's third iterate is itself a one-humped map, turned over.

Windows counted like necklaces

The logistic map has one window of period three, two of period four, three of period five, five of period six, nine of period seven — and the sine map, which shares no algebra with it, has exactly the same numbers, in exactly the same order along the parameter. The counts are 1, 1, 1, 2, 3, 5, 9, 16, 28, 51, and they are the number of ways to thread a necklace of beads in two colours.

A jam that comes from nowhere

Give cars on a ring road a top speed of five cells a step, let each slow to the gap ahead, and add one more rule: now and then, at random, a driver eases off by one. That is the whole of the Nagel–Schreckenberg model, and it produces what the exactly solvable rule 184 could not — jams that form in free traffic with no obstacle, drift backwards against the flow, and cost the road more than a third of its capacity. Set the top speed to one and remove the chance, and it is rule 184 again, cell for cell.

A rule that remembers one row back

Only six of the 256 elementary cellular automata can be run backwards, and all six are trivial: shifts, copies and complements. Every interesting rule forgets. But make the new row depend on the row before the current one as well — combine the rule's output with it cell by cell — and every rule, rule 30 included, becomes exactly reversible: run forwards, swap the last two rows, run the same rule again, and the starting row comes back cell for cell. Nothing is lost, and yet the patterns still look as disordered as ever.

How smooth the disguise is

An unlocked circle map is a rigid rotation in different coordinates, and whether those coordinates are smooth is decided by arithmetic. Solving for the change of coordinates means dividing, harmonic by harmonic, by numbers that come arbitrarily close to nought — and the golden rotation number keeps them far enough away that the result is not merely smooth but analytic, while a number lying close to a fraction makes it buckle.