Neither a surface nor a solid

Two facts about the Lorenz attractor are easy to establish and, taken together, leave very little room for what it can be.

It has no volume. The divergence of the Lorenz vector field is a negative constant, so a blob shrinks at a rate that does not depend on where it is. The divergence is $-(\sigma + 1 + \beta)$, so any blob of initial conditions has its volume multiplied by $e^{-(\sigma+1+\beta)t}$. At the standard parameters that is a factor of about $e^{-13.7}$ per unit of time — a millionth every second unit — and the attractor, being the limit of everything, has volume zero.

And it is not a surface. A flow on a two-dimensional surface cannot be chaotic: the Poincaré–Bendixson theorem says a bounded orbit in the plane must converge to a fixed point or to a closed orbit, and the same holds on any surface of genus zero. There is no room for two trajectories to separate on a surface without one of them crossing the other, and trajectories cannot cross.

So the object is squeezed from both sides: too thin to be solid, too tangled to be flat.

Those two arguments are worth contrasting, because they are of completely different kinds. The first is a one-line computation with the equations in front of it, and it would work equally for any dissipative system. The second is a topological theorem about what a plane allows, and it uses nothing about the Lorenz equations at all — it would forbid chaos in any two-dimensional flow whatever. One argument is about these equations and the other is about the number two.

The volume argument, done properly #

The claim that volumes shrink is worth deriving rather than quoting, because it is the one piece of exact arithmetic in the subject.

Take a small blob of initial conditions of volume $V$. As it flows, its volume changes at a rate given by the divergence of the field integrated over the blob — that is the statement of the divergence theorem applied to the flow. For the Lorenz equations,

$$\nabla \cdot F = \frac{\partial}{\partial x}\big(\sigma(y-x)\big) + \frac{\partial}{\partial y}\big(x(\rho - z) - y\big) + \frac{\partial}{\partial z}\big(xy - \beta z\big) = -\sigma - 1 - \beta.$$

That is a constant, independent of position, which is unusual and makes the conclusion exact: $\dot V = -(\sigma+1+\beta)V$, so $V(t) = V(0)e^{-13.7t}$ at the standard parameters.

Any set that the flow maps into itself and that is invariant must therefore have zero volume, since its volume is both constant and shrinking. That includes the attractor.

Two things follow. The dimension is at most three and cannot be three. And the shrinking is fast: a cube of side one collapses to a millionth of its volume in half a time unit, which is why every computed trajectory lands on the attractor almost immediately and why the return map is thin.

Why a surface will not do #

The other side of the squeeze is a theorem rather than a computation, and it is worth stating carefully because it is the reason three dimensions are the minimum for this behaviour.

In the plane, a bounded trajectory that does not converge to a fixed point must approach a closed orbit. The proof is essentially the Jordan curve theorem: a trajectory divides the plane, and once it has crossed a transversal in one direction it can never get back to the other side. That constraint is topological and has nothing to do with the equations.

So chaos needs three dimensions for a flow, and it needs only two for a map — because a map is allowed to jump, and the horseshoe is a map. That is the sense in which the return map is the flow’s real content: a two-dimensional section of a three-dimensional flow is exactly the setting where the horseshoe lives.

The Hénon map in the figures is a two-dimensional map for exactly this reason. It is not the Lorenz system and it is not a section of it; it is the simplest map with the same two properties — area contraction and a fold — and it exhibits the same layered structure in a place where the layers can be drawn.

That figure is the reason the rest of this rung is necessary. Nothing about it suggests a fractional dimension; it looks like a folded ribbon, and for thirty-five years the question of what it actually is stayed open partly because the picture is so persuasive. A convincing drawing of the wrong thing is harder to argue with than no drawing at all.

Locally a surface times a dust #

What the object actually is can be said precisely.

Take a small piece of the attractor and cut it with a plane transverse to the flow. The cross-section is not an interval and is not a point: it is a Cantor set — a dust of leaves at every scale, exactly as the horseshoe’s surviving strips predict.

So the attractor is locally a two-dimensional sheet in the flow’s two spanning directions, times a Cantor set across them. Its dimension is $2 + d$ where $d$ is the dimension of that Cantor set, and $d$ is small — about $0.06$ for the Lorenz attractor.

That small fraction is the whole difference between a surface and this, and it is why every drawing looks like a folded sheet. The extra six hundredths of a dimension is not visible in any picture and is the reason the picture is not the truth.

The number has a formula. The Kaplan–Yorke conjecture computes the dimension from the Lyapunov exponents, by finding how many of them can be added before the sum turns negative and interpolating:

$$D = k + \frac{\lambda_1 + \cdots + \lambda_k}{|\lambda_{k+1}|}.$$

With the Lorenz exponents $0.9$, $0$ and $-14.6$, the first two sum to $0.9$ and the third is $-14.6$, so $D = 2 + 0.9/14.6 \approx 2.06$. The dimension is computed from the stretching rates, which is the connection between the previous rung’s measurement and this one’s geometry.

What the number is and is not #

Dimension is not a single notion above the whole numbers, and the differences matter here.

Box-counting dimension covers the set with boxes of side $\varepsilon$ and asks how the count grows — the definition the fractal-dimension anchor develops. It is what a computation measures.

Hausdorff dimension is finer, defined by covers of varying size, and is the one theorems are stated about. The two agree for self-similar sets and can differ for others.

Correlation dimension is a fourth notion, measured from a trajectory by counting how many pairs of points fall within a distance $\varepsilon$ of each other and fitting a power law. It is what an experimentalist can compute from a measured time series without knowing the equations, and it was the tool that established fractional dimensions in real data — in fluid experiments and in physiological recordings — during the 1980s.

And the Kaplan–Yorke dimension is neither: it is a formula in the exponents, conjectured to equal the others for typical attractors, and proved only in special cases. So the number $2.06$ is a prediction of a formula that agrees with measurements, and is not a theorem about the Lorenz attractor.

That is worth being precise about because the situation is common. A quantity that three definitions agree on for the examples anybody checks, and that nobody can prove they agree on in general, is a normal state of affairs, and the honest report is the agreement rather than a claim of proof.

Reading the formula #

The Kaplan–Yorke expression rewards being read rather than applied, because it says something intelligible about what a dimension is.

The sum $\lambda_1 + \cdots + \lambda_k$ is the rate at which a $k$-dimensional volume grows under the flow — a $k$-dimensional patch is stretched along its $k$ most-expanding directions, and the logarithms add. So the question “how many dimensions can the flow support without shrinking them away?” is answered by the largest $k$ with a non-negative sum.

For the Lorenz system: one dimension grows at $0.9$; two dimensions grow at $0.9 + 0 = 0.9$; three dimensions shrink at $0.9 + 0 - 14.6$. So two-dimensional area is preserved-and-then-some and three-dimensional volume is destroyed, and the dimension is between two and three.

The fraction interpolates. Adding a fraction $f$ of the third direction gives a growth rate $0.9 - 14.6f$, and setting that to zero gives $f = 0.9/14.6 \approx 0.06$. So the dimension is the amount of the contracting direction that the expansion can just afford to keep.

That reading makes the formula’s conjectural status intelligible too. It computes the dimension of a set on which volume is exactly preserved in the relevant sense, and whether the attractor is that set is the part nobody can prove in general.

What it costs to have no volume #

Almost every point is not on it. The attractor has measure zero, so a randomly chosen state is not on it, and neither is any state a computer can represent exactly. That is not a problem — everything converges to it — and it means every statement about “typical behaviour” concerns the approach as well as the limit.

Integrals over it need a different measure. Asking for the average value of a coordinate over the attractor cannot use ordinary volume, since the volume is zero, and cannot use surface area either, since the attractor is not a surface. What is used is the physical measure: the long-run fraction of time an orbit spends in each region, which exists for the Lorenz system by a theorem of Tucker and is what every numerical average is really computing.

The dimension is not stable under changing the parameters. It varies continuously in places and jumps in others, because the attractor itself appears and disappears as $\rho$ moves. So a number like $2.06$ describes one point of parameter space, and the family of attractors across the parameter range is a subject in its own right that nothing here touches.

And “the attractor” is doing work as a definite article. Establishing that the Lorenz system has an attractor, in the technical sense, and that it is what the pictures show, was open from 1963 until 1998 — it was Smale’s fourteenth problem for the century, settled by Tucker with a computer-assisted proof. Everything above was believed on the strength of computations for thirty-five years.

What was actually open, and what closing it took #

The gap between “everybody has seen the pictures” and “it is a theorem” was thirty-five years, and it is worth knowing what was in it.

The difficulty is that the Lorenz equations have no closed-form solutions and the attractor is defined as a limit of numerical trajectories. Every argument about it therefore starts from a computation, and a computation of a chaotic system is wrong in the specific way the previous rung measures: the computed orbit is not the intended one.

The repair is to compute with sets rather than with points. Interval arithmetic tracks a box guaranteed to contain the true state, and a box grows as the integration proceeds. Tucker’s 1998 proof integrates a covering of the section forward with rigorous bounds, shows that the covering maps into itself, and extracts from that a proof that the flow has a genuine attractor with the expected structure.

The technique is now standard and the shift it represents is worth noticing: a computer-assisted proof is not a computation trusted, it is a computation that carries its own error bound, so the trust is in a few hundred lines of interval arithmetic rather than in a floating-point trajectory. That is the same distinction that separates a numerical experiment from a proof anywhere else, and it is why the four-colour theorem and this belong to one category.

Smale put the question on his list of problems for the twenty-first century as problem fourteen, and it was answered before the century began — which is unusual and says something about how ripe the technique was rather than how easy the problem was.

What the pictures cannot show #

Six hundredths of a dimension is invisible. Every drawing of the Lorenz attractor is a drawing of a folded surface, and the fraction that makes it not a surface would need magnifications far beyond what the layering resolves. The Hénon pictures stand in for it because in two dimensions the fractional part is large — about $0.26$ — and can be seen.

The Hénon pictures are of a different system. They are used because the phenomenon is the same and the fraction is large enough to see; they are not the Lorenz attractor and no claim about the Lorenz attractor’s numbers follows from them. Substituting a visible instance for an invisible one is a legitimate move and it has to be said out loud.

Magnification is limited by the number of points. Each zoom in the figures has fewer orbit points in it than the last, because a smaller window catches a smaller share of the orbit. Three magnifications is roughly where a computation of this size runs out, and the claim is about infinitely many.

And zero volume cannot be drawn at all. A line drawn on a screen has width; a set of zero volume drawn on a screen is indistinguishable from a thin solid one. The volume argument is exact and lives entirely in the divergence calculation, which is three derivatives and a sum.

Where the ladder goes next #

This rung closes the ladder’s first pass: the flow has been reduced to a map, its sensitivity measured, its mechanism identified, and its geometry described.

Named here as debts. The physical measure — the distribution of time over the attractor — which every average above quietly uses and which nothing here constructs. And Tucker’s proof, whose method (rigorous interval arithmetic on a computed flow) is a different subject from anything on this ladder and is what turned all of this from a strongly supported belief into mathematics.

Sideways, the counting of boxes at successive scales is the fractal-dimension anchor’s subject, the impossibility of chaos in the plane is the Jordan curve theorem doing work in an unexpected place, and the exponents that produce the dimension are the previous rung’s measurement.

Reading that figure a third time, with the dimension in hand, changes what it shows. The first rung read it as a reduction; the second as a source of the exponent; here it is the measurement of the fractional dimension, because the width of the scatter is exactly the extent of the Cantor set across the sheets. The same picture answers three questions and looks identical in all three, which is a fair summary of why the ladder is worth climbing rather than the first rung being enough.

What is worth carrying away #

When two constraints exclude every ordinary answer, the answer is an object of a kind that had to be invented.

Nothing with volume can be invariant under a volume-shrinking flow, and nothing flat can carry chaotic dynamics. Between those two exclusions there is no smooth object at all, and what fits is a sheet times a dust — an object with a dimension that is not a whole number, which is a notion nobody would have defined without needing it.

There is a second thing worth keeping, about the order in which the subject arrived. The pictures came in 1963, the mechanism in 1967, the dimension formula in 1979, and the proof that any of it described a real object in 1998. Every step was believed for years on the strength of computations before it was established, and every step turned out to be right. That is a good record for a subject whose central objects nobody could draw exactly, and it is worth remembering when a computation is all that is available.

The habit worth taking is to compute what an object cannot be before asking what it is. Here two short arguments — one a derivative, one a topological theorem — narrow the possibilities to a single kind of set, and the measurement of exactly which one is the easy part afterwards.