Sensitive to everything — page 2
Dynamics
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.
![A strand of the logistic map's diagram between r = 3.8284 and 3.8572, upside down. The bifurcation diagram of the logistic map cropped to r in [3.8284, 3.8572] and x in [0.44, 0.565], with the value axis inverted.](/f/a-strand-of-the-logistic-map-s-diagram-between-r-3-8284-36b84e6c.svg)
Dynamics
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.
![What a rule collects when every value comes from the same distribution. uniform on [0, 1]: best rule 0.995, one threshold 0.635 of E[max] at n = 200; exponential: best rule 0.905, one threshold 0.678 of E[max] at n = 200; Pareto, tail exponent 2: best rule 0.802, one threshold 0.714 of E[max] at n = 200; Pareto, tail exponent 1.2: best rule 0.802, one threshold 0.682 of E[max] at n = 200.](/f/what-a-rule-collects-when-every-value-comes-from-the-same-distribution-cdb6768b.svg)
Probability
When every value comes from the same hat
A rule that sees values one at a time and must keep or discard each on the spot can guarantee half of what a prophet collects, and no more, when the values come from different distributions. When they all come from the same one, the guarantee rises to 0.745 — and a single fixed threshold, set so that each value crosses it with chance 1/n, already secures 1 − 1/e. For bounded values the best rule collects nearly everything; only a heavy tail, where one enormous value carries the prize, keeps the gap open.
Probability
The rank that remembers every value
Values arrive one at a time, each must be kept or discarded on the spot, and the aim is to keep one whose rank among all of them is low on average. Told only who is leading, the best rule gets 3.87. Shown the values, a rule gets below 2.33 — and how much lower the best possible rule goes is not known, because the rank of what is kept depends on every value seen, and the best rule may need to remember all of them.
Dynamics
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.
Geometry
A skeleton that ignores small teeth
The medial axis of a shape grows a branch to every bump on its outline, however small, so a scanned shape has a skeleton made mostly of noise. Keep only the points of the axis whose nearest boundary points are spread across a circle of radius at least λ, and the branches to bumps smaller than about 2λ disappear while the real skeleton stays. That λ-medial axis moves only a little when the outline wobbles by much less than λ — and the price is λ itself, a size below which a feature is declared not to exist, which can even cut a shape's skeleton in two.
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
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.
Dynamics
Chaos on the line between two roots
Newton's method for a quadratic is the one case solved completely: every starting point goes to the nearer root, and the plane splits into two half-planes. Cayley did it in 1879. What his solution also contains is a line on which the method never settles — and a change of variable shows that on that line Newton's method is exactly the doubling map, the simplest chaos there is. Run on the real line for x² + 1, which has no real root to find, it becomes a source of Cauchy-distributed randomness.
Dynamics
The cosines of the powers of two
Start at cos 1 and apply x ↦ 2x² − 1 over and over. The map is chaotic and exactly solvable: the k-th point is cos(2ᵏ), the cosine of two to the k radians. Iterate it in floating point and the computed orbit leaves the true one after 51 steps. Read the binary digits of 1/(2π) instead and the true orbit can be followed for as long as digits can be computed — here 262,080 steps — and at that length it is as fair as any orbit can look. Whether it stays fair is a question about those digits, and nobody can answer it.
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.
![Where the secant method goes, from every starting pair. x^3−x, 180×180 on [−2,2]²: to −1 10164, to 0 11954, to 1 10164, none 118.](/f/where-the-secant-method-goes-from-every-starting-pair-f444ff40.svg)
Dynamics
Newton's method without a derivative
Replace Newton's tangent by the chord through the last two guesses and the method no longer needs a derivative — but it needs two starting values, so its basins live in a plane of pairs instead of on a line. The errors multiply instead of squaring, their logarithms add like Fibonacci numbers, and the digits gained grow by the golden ratio: 1, 3, 5, 9, 15, 25, 41, 67, 109.
Dynamics
Two humps and one entropy
A map of an interval with one hump has its chaos measured by one number, the entropy, and that number never decreases as the hump rises. Give the map two turning points — a hump and a dip — and the parameter space becomes a plane. The entropy is no longer a function of one setting, and the question becomes whether the maps of a given entropy form one connected curve or several. John Milnor conjectured one. Counting the laps of every cubic's iterates exactly, over the whole plane of hump heights and dip depths, draws the curves — and each comes out whole.
Algebra
A disc that turns for ever
At each fraction on the Mandelbrot set's cardioid a bulb is born. At the irrational numbers between them something rarer can happen: a disc round the fixed point rotates rigidly for ever, with no point falling in or flying out. Whether it exists depends on nothing but the digits of one continued fraction, and on one sum of their logarithms.
Dynamics
Replace the parabola with a corner
Nobody can prove Hénon's map is chaotic at its classical values, since a stable cycle might hide in a window too thin to find. Replace x² in it by |x| and the whole difficulty disappears: one cone of directions is stretched by every step, in three lines of arithmetic, and across 301 parameters the Lyapunov exponent is a straight rising line with no windows at all.
Dynamics
An orbit stuck to an island
A chaotic orbit in the standard map stretches at 0.44 a step, and then for 19,000 steps it almost stops stretching at all. It has wandered into the fringe of an island of regular motion and been caught. So gaps between visits to a box do not die away exponentially as they would in a uniformly chaotic map; they fall as a power, and an orbit of 400 million steps still has not seen the longest.
Analysis
A ball rebuilt everywhere but its centre
Rebuild a solid ball from its frequencies, keeping those below a cut-off, and read the result at the centre, where the ball is perfectly smooth. In the plane the answer settles on 1. In space it swings between 0.36 and 1.64 for ever, because the whole surface of the ball is the same distance from the centre and its jump arrives there in step.
Dynamics
A conserved quantity that is not there
In 1964 Michel Hénon and Carl Heiles asked whether a star in a galaxy obeys a third conservation law besides energy and angular momentum. They cut its orbit by a plane and looked. At low energy every orbit drew a clean curve, as if the law held; at higher energy the curves dissolved into dust. Integrating their model again shows the share of chaotic orbits climbing from nothing at energy 1/12 to almost all at 1/6.