Matpic
mathematics, in pictures
Theme

Sensitive to everything

Systems where a difference too small to draw becomes the whole difference, and the reason that is a property of the rule rather than of the measurement.
The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points. Dynamics

The road paved with doublings

Turn one dial slowly and watch what a map settles into. It settles on a point, then on two points, then four, then eight — faster and faster, and the doublings run out at a parameter that is finite.

 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. 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.

 The Lyapunov exponent, 2.8 to 4. The average rate at which nearby orbits separate, plotted against the parameter. 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.

 The Mandelbrot set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region. Dynamics

The shape in every picture of itself

One line of arithmetic, repeated, with a single complex number as its only input. Sort the numbers by whether the result stays bounded and the boundary between the two answers is the most complicated object anyone draws from a rule this short.

 A Julia set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region. Dynamics

One c, one picture

The same iteration, with the parameter held still and the starting point varied instead. Every complex number gets its own picture, and moving the parameter a hair can shatter it into dust.

 The basins of Newton's method on z³ = 1. The complex plane coloured by which cube root of one Newton's method converges to from each starting point. Dynamics

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.

 The Lorenz attractor at ρ = 28. A trajectory of the Lorenz equations, projected onto two of its three coordinates. Dynamics

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 link that makes every traveller later. Four nodes and two routes, with the equilibrium flow and travel time before a zero-cost link is added between A and B and after. The travel time rises from 10 to 12. Applied

The road that makes everyone later

An equilibrium is a state nobody can improve alone, which is a much weaker thing than a state anybody would choose. Adding a link that costs nothing to use makes every traveller in this network strictly slower, and the arithmetic says by exactly how much.

 The orbit a computer draws, and the orbit. Two orbits of the tent map from the same starting fraction plotted against the step number — one computed exactly in whole-number arithmetic and periodic, one computed in double precision and reaching zero. Dynamics

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.

 One disc, and two paths that stop being near each other. Two nearly identical billiard paths drawn on an empty square and on a square with a circular obstacle, with the separation between them plotted against distance travelled. 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.

 The flow, reduced to one dimension. A scatter of 2395 points: each successive maximum of the Lorenz trajectory's third coordinate against the one before it. The points lie along a single curve with a sharp peak, which is the one-dimensional map the flow induces. Dynamics

The flow that is really a map

A trajectory wandering through three dimensions is hard to reason about. Record only the successive maxima of one coordinate and the wandering collapses onto a curve — a map of an interval to itself, with a corner in the middle, which is a thing the theory can handle.

 A closer start buys time and nothing else. The logarithm of the separation between two Lorenz trajectories plotted against time, for three different initial separations. The three curves are straight and parallel over most of their length, with the same fitted slope. Dynamics

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.

 Two equilibria, and two tests that disagree. The row chooser's expected payoff from each option against the column chooser's behaviour, for a joint effort worth more than a safe one. The lines cross at 0.750, which is the mixed equilibrium and the boundary between the two basins. Applied

Two equilibria and no way to choose

A game can have two states nobody wants to leave, one paying more than the other, and the definition of an equilibrium has nothing to say about which happens. The two standard tie-breakers disagree, and the one that wins is usually the worse.

 A dimension of 1.2576, from two stretching rates. The running averages of the Hénon map's two Lyapunov exponents, settling at 0.4177 and -1.6217. Kaplan and Yorke's formula turns them into a dimension of 1.2576 without counting a single box. Dynamics

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.

 A population that settles at 2/3. A contest over a prize worth 4 that costs 6 to fight for. Left: the growth rate of the share playing Hawk against that share, which is nought at 0, at 2/3 and at 1. Right: the share over time from 5 starting points, all converging on 2/3. Applied

A mixture that is a population

A mixed equilibrium between two choosers is a knife-edge nobody has a reason to stand on. Read the same mixture as a population whose shares grow with how well they do, and it becomes a point every population is carried to — or one every population circles for ever without arriving.

 The logistic map applied 6 times at 3.5, 3.83, 4, and its folds. Side-by-side graphs of the logistic map composed with itself, one per parameter, each labelled with how many monotone pieces it has. Dynamics

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.

 An area of starting points from which z³ − 2z + 2 is never solved. The complex plane coloured by which root of z³ − 2z + 2 Newton's method reaches from each starting point, with the points that reach no root left uncoloured. Dynamics

An area that never finishes

Newton's method's famous failure is a boundary, and a boundary has no area — a random start misses it with probability one. The real failure is different in kind: a polynomial with small whole-number coefficients whose method has a region of starting points, with area, from which it provably never terminates.

 Intermittency at r = 1 + √8 − 0.0003. A time series of 600 steps of the logistic map just below the period-three window. Long stretches that look like a cycle of three, shaded, alternate with irregular bursts; there are 6 such stretches here. Dynamics

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 periodic point and a wandering one, both near 0.3, parted by step 4. The distance between the orbit of a periodic point and the orbit of a point from a dense orbit, both starting in the same small interval, plotted against the step until the wandering orbit nears the point farthest from the periodic one. 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.

 What the tent of slope 3 keeps: 32 pieces after 5 steps. Rows showing the parts of the unit interval that remain inside it for 0 to 5 steps of the open tent map of slope 3, halving into a Cantor set. 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.

 How many steps Newton's method needs on (z − 1)²(z + 1). A square of the complex plane shaded in bands by the number of iterations a root-finding method needs from each starting point, darker meaning slower, with the roots marked. Dynamics

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.

 Newton's and Halley's basins for z³ − 1. Two squares of the complex plane side by side, each coloured by which root a starting point converges to, the left under Newton's method and the right under Halley's, with non-converging starts marked. Dynamics

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.

All themes