Lyapunov exponent
Named by 12 essays across one field — each of them below, with the objects they name alongside it.
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.
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 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.
A closer start buys only time
Two trajectories from almost the same place separate exponentially, and the rate does not depend on how close they began. Halving the initial error buys one fixed interval of extra agreement, and no amount of precision buys more than a fixed number of those.
A dimension from the stretching rates
An attractor has no construction rule, so its dimension has to be counted — which was the whole case for defining dimension by counting. Kaplan and Yorke's formula computes it instead, from two numbers that describe the map and never look at the set.
The folds that measure chaos
Apply the logistic map six times and its graph goes up and down 38 times at r = 3.5 and 64 times at r = 4. How fast that number of folds multiplies with each further step is the map's topological entropy: zero through the whole cascade of period doublings, log of the golden ratio in the window of three, log 2 at the top — and it never decreases as r rises.
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 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 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.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Sensitive dependenceChaosLogistic mapGolden ratioPeriodic orbitEigenvalueFibonacci numbersIterationLogarithmOrbitSimulationStrange attractor