Sensitive dependence
Named by 16 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
Where Newton's method goes instead
An algorithm designed to find roots, run from every starting point at once. Three roots, three basins, and a boundary at which all three are arbitrarily close — so a rule with no randomness in it has starting points whose answer cannot be predicted.
Two lobes and no cycle
Three equations, three variables, and a trajectory that never crosses itself, never repeats, and never leaves a region of zero volume. The set it settles onto is not a point, not a loop, and not a surface.
The orbit a computer draws
A chaotic orbit computed in floating point is not the orbit of the point it started from. Sometimes it is the true orbit of a nearby point, which is enough; sometimes the arithmetic simply runs out, and the picture is of the rounding.
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.
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 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.
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.
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.
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.
Lyapunov exponentChaosOrbitIterationPeriodic orbitLogistic mapAttractorBasin of attractionBifurcationBinary expansionDerivativeDeterminism