Chaos on a set nobody lands on

Every chaotic map considered so far sends its interval into itself. Orbits have nowhere to go but round, and the conditions that make them chaotic — an orbit that goes everywhere, periodic orbits everywhere — describe how they go round. Many real systems are not closed like that. A marble on a table of bumpers, a speck of dye in a stream, a comet passing through a planetary system: each may behave irregularly for a while and then leave the region where the irregularity happens, never to return.

The simplest model of such a system is a map that sends part of its interval outside it.

The map is a tent with slope three: $x \mapsto 3x$ on the left half of the interval and $x \mapsto 3(1 - x)$ on the right. Each half is stretched to length one and a half, so the middle third of the interval — the points from one third to two thirds — is sent above 1, out of the interval, and never comes back. After one step two thirds of the interval remain, in two pieces. After two steps each piece has lost its own middle third, leaving four pieces and four ninths of the length. After $k$ steps there are $2^k$ pieces, each $3^{-k}$ long, and $(2/3)^k$ of the original length.

The points that never leave are the points in every one of those stages: exactly the middle-thirds Cantor set, the set that has no length and uncountably many points.

Almost everything leaves #

The surviving set has length zero, so a starting point chosen at random lies outside it with probability one. It survives for a while and then leaves.

The fraction of random starts still inside after $k$ steps falls by a factor of two thirds at every step, and on a logarithmic scale that is a straight line. Its slope is the escape rate, written $\kappa$: the survivors decay like $e^{-\kappa k}$, and for the tent of slope $s$ the exact value is $\kappa = \log(s/2)$ — $0.405$ for slope 3 and $0.693$ for slope 4. The simulation measures $0.396$ and $0.704$ from the starts it follows, and at the later steps the steeper tent’s line starts to wobble, because only a few hundred of its forty thousand starts are left to count.

The geometric decay is the signature of transient chaos, and it is different in kind from the behaviour of a map that merely takes a long time to settle. In an ordinary transient the orbit approaches something — a fixed point, a cycle — and the approach is visible. Here there is nothing to approach. The orbit is chaotic until the moment it leaves, and whether it leaves at step 3 or step 30 is as unpredictable as everything else about it. The only thing that is predictable is the statistics: out of a large number of starts, the fraction still present after each step, which is a number the map fixes exactly and which an experiment can measure with nothing more than a clock and a count. That is why the escape rate is the quantity physicists reach for first when a system is suspected of transient chaos. It needs no knowledge of the equations and no picture of the chaotic set — only the observation that survivors decay by the same factor, step after step.

Wandering, then leaving #

The individual orbits look exactly like chaotic orbits until they stop.

A random start has a two-thirds chance of surviving each step, so its lifetime — the number of steps up to and including the one that throws it out — has average $1 + \tfrac23 + \tfrac49 + \cdots = 3$. Most starts are gone within a few steps. The ones drawn in the figure were selected as the longest-lived of four thousand, and the longest lasted 22 steps, which is what it takes when a starting point happens to lie within about $3^{-21}$ of the surviving Cantor set.

That is the whole mechanism. A long life means a start very close to the survivors, and the orbit of such a start follows the orbit of a nearby survivor — which is chaotic — for as long as the two stay close. Sensitive dependence pulls them apart at a rate of $\log 3$ per step, the stretching of the tent, so the start that was $3^{-21}$ away shadows its survivor for about 21 steps and then is thrown into the middle and out. The lifetime of a transient is the number of steps sensitivity needs to reveal that the start was not quite on the repeller.

A lifetime is a count of digits #

For the tent of slope three the lifetime of a start can be read off its expansion in base three. The survivors are the middle-thirds Cantor set, which is the set of numbers whose base-three digits are all 0 or 2 — the digit 1 is exactly what puts a number in a removed middle third. A start whose first base-three digit is 1 lies in the middle third and leaves at once; one whose first digit is 0 or 2 survives the first step.

The map then acts on the digits almost as the doubling map acts on binary ones. On the left half, multiplying by three shifts the base-three expansion one place; on the right half, $3(1 - x)$ shifts it and replaces each digit $d$ by $2 - d$, which swaps 0 and 2 and leaves 1 alone. Either way, the digits that are 1 stay 1 and move one place forward. A start survives exactly as many steps as there are digits before the first 1 in its base-three expansion, and it leaves on the step that brings that 1 to the front.

So the lifetime distribution is the distribution of the position of the first 1 in a random string of base-three digits. Each digit avoids 1 with probability two thirds, independently, and the first 1 arrives at position $k$ with probability $(2/3)^{k-1}(1/3)$ — the geometric law whose average is three, and whose tail the survival figure plots. The 22-step orbit in the figure above began with 21 base-three digits of 0s and 2s: agreement with a point of the Cantor set to within about $3^{-21}$, exactly the shadowing the previous section described. The same kind of digit bookkeeping gives the sum $1 + \tfrac23 + \tfrac49 + \cdots$ its value, as the geometric series that fits in one square does for halves.

Chaos on the survivors #

On the Cantor set itself nothing escapes, and the map there is fully chaotic.

Write a point of the surviving set by recording, at each step, whether its orbit is in the left or the right piece: an infinite string of two letters. Every string occurs, since at each stage both pieces survive and the choice at each step is free; and the map shifts the string one letter along, because a point’s next position is determined by its remaining choices. That is the doubling map in disguise, on a different set: the shift on strings of two letters, which is what symbolic dynamics reads off any such map.

So all three of Devaney’s conditions hold on the Cantor set, exactly as they hold for the doubling map on the interval. There is an orbit that goes everywhere on it (the string listing every finite word), periodic orbits are dense (the periodic strings), and by the argument of the essay on where sensitivity comes from, sensitivity follows. The map is as chaotic on the repeller as any map can be, and the repeller is a set that a randomly chosen point lands on with probability zero.

In base three the whole structure is visible at once. A point of the repeller is a string of 0s and 2s, the map shifts the string and possibly swaps its two letters, and an orbit that goes everywhere on the repeller is a string of 0s and 2s containing every finite string of 0s and 2s. The periodic points are the repeating strings — $1/4$ is $0.020202\ldots$ in base three and lies on the repeller, returning to itself after two steps. Everything that made the doubling map chaotic has been carried onto a set of dimension $0.631$ by writing binary digits as 0s and 2s instead of 0s and 1s.

Three numbers and one equation #

The open tent has three numbers attached to it, each measured in a different way. The stretching rate $\lambda$, the Lyapunov exponent, is how fast nearby orbits on the repeller separate: $\log s$ for the tent of slope $s$. The escape rate $\kappa$ is how fast random starts leave: $\log(s/2)$. And the dimension $D$ of the repeller is how thin the surviving set is: by counting boxes, $2^k$ boxes of side $s^{-k}$ cover it, so $D = \log 2 / \log s$.

The three are tied by a single equation, which Holger Kantz and Peter Grassberger found in 1985:

$$D = 1 - \frac{\kappa}{\lambda}.$$

For slope 3 it reads $1 - 0.405/1.099 = 0.631$, which is $\log 2/\log 3$; for slope 4, $1 - 0.693/1.386 = 0.5$. The reason is a balance. Each step stretches every surviving piece by $e^\lambda$ and throws away enough of it that the total surviving length falls by $e^{-\kappa}$. The survivors after $k$ steps are pieces of length $e^{-\lambda k}$ totalling $e^{-\kappa k}$, so there are $e^{(\lambda - \kappa)k}$ of them, and a set covered by that many pieces of that size has dimension $(\lambda - \kappa)/\lambda$.

The tent of slope four loses half the interval at every step instead of a third, and stretches harder, by four instead of three. Both changes thin the survivors: the pieces shrink faster and there are no more of them, two per piece per step either way. The dimension falls from $0.631$ to exactly one half, and the equation accounts for it — the escape rate doubles, from $\log 1.5$ to $\log 2$, while the stretching rate grows only from $\log 3$ to $\log 4$.

How thin the chaotic set is, is decided by how fast points leave compared with how fast they are stretched. A map that stretches hard and loses little has a repeller of dimension nearly one, almost the whole interval; a map that loses nearly everything at each step has a repeller of dimension nearly zero, a sparse dust. When nothing escapes, $\kappa = 0$ and $D = 1$: the closed chaotic maps are the case in which the repeller fills the interval.

The same thing in two dimensions, and in the laboratory #

The equation generalises. For a two-dimensional map with one stretching and one contracting direction, the repeller’s dimension along each direction is fixed by the escape rate and the corresponding exponent, and the partial dimensions add. This is the same bookkeeping that ties a dimension to the stretching rates on an attractor — the Kaplan–Yorke formula — with the escape rate added for the points that are lost.

Transient chaos is common in physical systems precisely because most regions of interesting dynamics are open. A particle scattered off three hard discs bounces chaotically among them and then leaves; the set of trajectories that bounce for ever is a fractal, and its dimension governs how the scattering angle depends on the incoming path. Dye injected into a flow past an obstacle is stretched and folded in the wake and then carried downstream; the fractal filaments it leaves behind trace the unstable directions of the chaotic set that the fluid passes near. And a system approaching an attractor can wander for a long time near a chaotic repeller first — the transients that precede settling are, very often, this.

The logistic family shows the same thing at its edge. Past its top parameter, $x \mapsto rx(1-x)$ with $r$ above 4 sends a middle interval above 1 and out, exactly as the tent does, and the points that never leave form a Cantor set on which the map is conjugate to the same shift on two letters. As $r$ comes down to 4 the escaping middle shrinks to a single point, the escape rate falls to zero, and by the equation the dimension rises to one: the repeller grows until it fills the interval and becomes the closed chaotic map of the essay on orbits that part. Transient chaos is the closed kind with a leak, and the leak’s size sets the dimension.

What the figures cannot show #

The Cantor figure draws five stages. The surviving set is their intersection, and no finite stage contains it; the dimension is a statement about the limit, and the figure shows the halving and shrinking from which it is computed.

The survival and lifetime figures are simulations with fixed seeds, forty thousand and four thousand starts. Their measured rates carry sampling error of a few percent, visible in the way the steeper line wobbles at the end, and the exact values in the notes are the ones the argument gives. The longest-lived orbits were chosen by selection, so they are not typical: a typical orbit lasts three steps and looks like nothing.

And the Kantz–Grassberger table is for the symmetric tent, where the stretching is the same at every point and all three numbers are exact. For maps whose stretching varies, the equation still holds with the exponent and the dimension taken with respect to the natural measure on the repeller — the distribution of long-lived orbits — and its proof then needs the machinery of that measure, which the uniform tent lets the argument avoid.

The question it leaves: what one measurement reveals #

The three numbers each ask for a different experiment. The escape rate needs only a stopwatch: release many particles, count the survivors, fit a line. The stretching rate needs pairs of nearby orbits followed while they survive. The dimension needs a picture of the survivors at fine resolution. The equation says any two determine the third, so the easiest two can stand in for the hardest.

That is how the equation is used. In scattering experiments the escape rate and the exponent are the accessible quantities and the fractal dimension of the chaotic set is inferred, not imaged. The question the tent leaves is how much of a chaotic system’s structure can be recovered from such indirect measurements in general — which of its invariants are tied together, and which are genuinely independent — and for systems with several stretching directions and non-uniform escape the full answer is still being assembled.

The tent is also a warning about what a simulation sees. A computer following one start of the open tent will almost always report that it leaves within a few steps, and will never report the Cantor set, because the set has no length and a random start misses it. The chaotic set is found by asking the opposite question — which starts survive $k$ steps — and letting $k$ grow, which is what the first figure does. Chaos that nobody lands on has to be looked for backwards, from the survivors, and not forwards from any particular start.

A dust that runs the show #

The picture that comes out of the open tent is of chaos concentrated on a set nobody visits. Almost every starting point escapes, usually within three steps, and the ones that last long do so only because they began extraordinarily close to a Cantor set of dimension $0.631$. On that set the map is fully chaotic, and every long-lived orbit is a copy of one of its orbits, borrowed for as many steps as sensitivity allows.

The rates at which starting points leave and at which nearby orbits separate are properties anyone can measure without seeing the set. The equation $D = 1 - \kappa/\lambda$ says those two measurements already contain the set’s shape: the thinner the dust, the faster points leave, and the ratio is exact.

Set beside a set with no interval in it and length to spare, the repeller is the opposite kind of Cantor set: it has no length at all, and its size is measured only by its dimension. Removing the middle quarter and then shrinking pieces would leave length behind; removing a fixed fraction at every scale, as the open tent does, leaves none. Which of the two a chaotic system produces is decided by the balance of escape and stretching, and the tent is the case in which that balance can be written down in one line.