The flow that is really a map

The picture of the Lorenz attractor is a tangle. It is a beautiful tangle and it resists every question anybody wants to ask of it: whether the trajectory is periodic, how long it stays on one lobe, whether two nearby starts stay together. Those are questions about a curve in three dimensions, and there is very little machinery for curves in three dimensions.

There is a great deal of machinery for maps of an interval to itself. The whole of this field’s first several essays is about them — cobwebs, fixed points, periods, the way a parameter turns a settled orbit into a wandering one. If the flow could be turned into such a map, all of it would apply.

Lorenz did exactly that in the paper that introduced the system, and the reduction is one sentence long.

The reduction #

Watch the third coordinate, and note the value each time it reaches a local maximum — which happens once per loop round a lobe. That gives a sequence of numbers $z_1, z_2, z_3, \ldots$, and the question is whether $z_{n+1}$ is determined by $z_n$.

There is no reason it should be. The state of the system is three numbers, and knowing one of them at one moment is knowing a third of the state. Two trajectories passing through the same maximum from different directions could perfectly well go on to different next maxima.

To see how strong the claim is, consider what the alternative would look like. If the next maximum depended on the two coordinates the record throws away, the scatter would fill a band whose width measured how much those coordinates mattered. A band of half the range would say the reduction is useless; a band of five per cent would say it is a rough model. What the figure measures is a band of well under one per cent, which says the discarded information is, at this resolution, not information.

They do not. Plotting $z_{n+1}$ against $z_n$ over thousands of returns gives a curve, thin enough that the departure from it is a fraction of a per cent — which says that the missing two-thirds of the state is, to that accuracy, determined by the third that was recorded.

That is a fact about the system rather than a general truth, and it is worth being clear about why it holds here. The Lorenz flow contracts volume rapidly, so a blob of starting conditions is squashed flat almost immediately. After a few loops every trajectory lies within a very thin sheet, and on a thin sheet one coordinate does determine the rest. The thinness of the return map is the volume contraction, measured.

What the curve says #

The curve has a shape, and every feature of it means something.

It has a corner. The map rises steeply, turns at a peak, and falls steeply. That is a tent-like map, and tent maps are the standard example of a chaotic interval map — the branch slopes exceed one in absolute value everywhere, so every distance is expanded at every step.

Its slope is everywhere greater than one in magnitude. That is what makes the dynamics chaotic rather than merely complicated: two nearby values of $z$ are pushed apart at every return, and there is no interval on which the map is a contraction. A map with an interval of contraction somewhere would have a stable periodic orbit hiding in it, and long computations would eventually fall into that orbit and stay — which is the behaviour a great many nonlinear systems have and this one does not.

It has no attracting fixed point. The map crosses the diagonal, so there is a fixed point — a periodic orbit of the flow — but the slope there is greater than one, so it repels. The trajectory approaches it, is thrown off, comes back, and is thrown off differently. Every periodic orbit of the Lorenz system is unstable in exactly this way, which is why the flow never settles onto any of them. And there are infinitely many of them: a map whose slope exceeds one everywhere has a periodic point of every period, by an argument about how many times the $k$-th iterate crosses the diagonal, so the attractor is threaded with unstable closed orbits of every length. The orbits are dense in it and every one of them is avoided.

Reading the branches #

The curve has two branches meeting at the corner, and which branch a return lands on is not an arbitrary label — it is which lobe the trajectory has just been round.

A maximum of $z$ that is small comes from a tight loop close to a fixed point; a large maximum comes from a wide excursion that swings over to the other lobe. So the position along the horizontal axis records the size of the last loop, and the branch records which side of the corner it fell on, which is where the trajectory goes next.

That gives a coding. Label each return $L$ or $R$ according to its branch, and the orbit becomes an infinite word in two letters. Two orbits with the same word are near each other and stay near each other for as long as the words agree; two orbits whose words first differ at the $k$-th letter separate after $k$ returns. The map’s slope being everywhere above one is exactly the statement that every word occurs, so the orbits of the flow are in correspondence with the sequences of two symbols — which is the symbolic dynamics of the first essays of this field, arriving in a system nobody set up to be symbolic.

That correspondence is the practical payoff of the reduction and is worth stating as a count. There are $2^{20}$ words of length twenty, so there are about a million distinguishable behaviours over twenty returns, and a computation that tracked the flow directly would have to resolve all of them in three continuous coordinates. Tracking the word instead is exact bookkeeping on a finite alphabet.

Why this is the standard move #

The construction has a name and it predates Lorenz by seventy years: the Poincaré section.

The idea is to replace a continuous flow by the map it induces on a surface that the flow crosses. Choose a surface transverse to the trajectories; follow a trajectory from a point on the surface until it returns; record where it lands. The flow in $d$ dimensions becomes a map in $d - 1$, and everything periodic about the flow becomes something fixed about the map.

A closed orbit of the flow is a fixed point of the section map. A period-doubled orbit is a period-two point. A quasi-periodic torus is an invariant circle. Every question about the flow’s long-term behaviour becomes a question about the map’s, and one dimension has been removed for free.

Lorenz’s reduction is a section in this sense with an extra piece of luck. The section is the surface where $\dot z = 0$ and $z$ is at a maximum, which reduces three dimensions to two; and the volume contraction then reduces two to one, because the attractor’s intersection with that surface is nearly a curve rather than a region. Two reductions, and only the first is a general technique.

Running the construction at a second parameter is what distinguishes a property of the system from a property of one picture. The attractor changes size, the range of maxima changes, and the reduction survives — which says that the thinness is caused by the contraction rather than by a lucky choice of $\rho$.

What it costs #

The reduction throws away time, and that is the whole of the price.

The return map says nothing about how long a return takes. Two successive maxima might be separated by half a time unit or by two, depending on whether the trajectory made a wide loop or a tight one. Recovering the flow from the map needs a second function — the return time — and every statement about frequencies, periods in seconds, or resonances lives there and not in the map.

The section itself makes a choice. Taking maxima of $z$ is one surface; taking crossings of the plane $z = \rho - 1$, or of $x = 0$, gives different sections and different maps. The maps are conjugate — they describe the same dynamics — and they do not look alike, so a picture of “the” return map is a picture of one choice. Lorenz’s is the one that comes out thinnest, which is why it is the one everybody draws.

And the map is not exactly one-dimensional. The scatter is thin and not zero. That residual thickness is where the genuinely three-dimensional behaviour hides, and it matters for questions of the finest resolution: the true set of periodic orbits, and the precise dimension of the attractor. The one-dimensional model answers coarse questions exactly and fine questions approximately, and the honest statement of the reduction includes which is which.

What the curve is not #

Two readings of the figure are tempting and wrong, and both are worth stating.

It is not a graph of an exactly defined function. No formula produces it. The curve is a property of a solution of a differential equation, computed by integrating and recording, and the sense in which $z_{n+1}$ depends on $z_n$ is empirical to within the measured thickness. Writing down a tent map and studying that is a model of the Lorenz system, not a reduction of it, and results proved about the model transfer only as well as the model fits.

And the corner is not a corner. At the resolution drawn the map appears to have a sharp peak. It does not: the true map has a cusp with infinite slope on one side, and no finite computation resolves the difference. The mathematical work of Tucker in 1998 — which proved that the Lorenz attractor exists, rigorously, with computer assistance — needed exactly this kind of care about what the picture is and is not showing.

That last figure is the argument for the whole rung stated as an observation. Nobody looking at three tangles at three parameters would notice that one of them has an extra fold; the difference is invisible in the projection and obvious in the return map, because the return map is a graph and a graph can be compared with another graph.

The general lesson about dimension #

Reducing three dimensions to one looks like a large loss and is nearly free here, and the reason generalises.

A dissipative system contracts volume. After a while every trajectory lies on an attracting set whose dimension is much lower than the space’s, and the effective number of variables is the dimension of that set rather than the number of coordinates. The state space is three-dimensional and the dynamics is not.

That is why weather models with millions of variables can have low-dimensional attractors and why the whole subject of reducing high-dimensional dynamics to low-dimensional models is a subject. Lorenz’s own system is already a reduction of that kind — it comes from truncating a partial differential equation for convection to three modes — so the return map is a reduction of a reduction, and the fact that both work is the reason the system is famous.

What Lorenz did with it #

The 1963 paper is worth reading for what it does with the reduction, because the modern account inverts the emphasis.

Lorenz introduces the three equations, integrates them, notices the aperiodicity, and then — in the last third of the paper — constructs the return map by hand from a table of computed maxima. He plots it, observes it is close to a curve, and immediately draws the conclusion: the map has slope greater than one everywhere, so all its periodic points are unstable, so the flow has no stable periodic orbit and no computation will ever find one settling down.

That argument is the paper’s actual theorem, and it is a theorem about a one-dimensional map. The famous picture of the attractor is a plot in the paper; the reasoning is entirely about the return map, because that is the only object the available theory could say anything about.

The care he took is worth recording too. He notes that the map is approximate, that the true section is two-dimensional, and that his conclusion is therefore about the model rather than proved about the system. Establishing that the Lorenz attractor exists rigorously took another thirty-five years, and the gap between what the picture showed and what could be proved was open the whole time.

What the pictures cannot show #

The scatter is a finite sample of an infinite orbit. Two thousand returns is enough to see a curve and not enough to establish that the curve is what the orbit fills out for ever. The thinness the figure measures is the thinness of what was computed.

The integration is numerical and the trajectories are wrong. Every computed Lorenz trajectory diverges from the true one with the same initial condition, exponentially, for exactly the reason the system is interesting. What justifies the picture is shadowing — the computed orbit is close to some true orbit, not to the intended one — and that is a theorem rather than something the drawing displays.

The corner’s true shape is beyond any drawing. The map is drawn from a scatter, and near the cusp the scatter thins because few returns land there — which is a fact about how the orbit spends its time, not about the map. Where the picture is least informative is exactly where the interesting behaviour is, and no amount of computation changes that: a cusp with infinite slope is approached by finitely many points however many are taken.

And the section is invisible. The construction says “record the value at each maximum”, which is a surface in the three-dimensional space and an operation on a trajectory. Neither is drawn: the figures show the input, a tangle, and the output, a scatter, and the step between them is prose.

Where the ladder goes next #

The next rung measures what the return map’s slope predicts: two starts a billionth apart separate at a rate that does not care how close they began. After that the mechanism producing that separation is drawn — stretching and folding — and the last rung asks what kind of set the folding leaves behind.

Also named as a debt, since this rung raises it and does not settle it: the return time, the second function needed to rebuild the flow from the map, whose distribution is what turns a statement about returns into a statement about seconds. Nothing above draws it, and the closely related question of how a system’s timing is set is a different anchor’s.

Sideways, the interval maps this reduction produces are the ones the symbolic-dynamics essay codes as words, the divergence of nearby trajectories is the observation Lorenz actually made, and the volume contraction that makes the section thin is what distinguishes this from a conservative system.

What is worth carrying away #

A reduction that loses two thirds of the state and predicts the next value to within a fraction of a per cent is not a coincidence; it is a measurement of how flat the system’s attractor is.

The Lorenz return map works because the flow squashes everything onto a thin sheet before it does anything else, and a thin sheet is nearly one-dimensional. The reduction is therefore not a modelling assumption but a consequence of dissipation, and the residual thickness is a quantitative statement of how much dissipation there was.

The habit worth taking is to ask, of any low-dimensional model, what was contracted away. If the answer is “a direction the dynamics squashes exponentially”, the model is a reduction. If the answer is “a direction nobody measured”, it is a guess.