A millionth is enough

The property that made the Lorenz system famous is not the shape of its attractor. It is that two trajectories starting arbitrarily close together end up entirely uncorrelated after a finite time, and that the time is short.

That property has a number attached to it, the number can be measured on the system rather than quoted, and the measurement is what this rung does.

Measuring the exponent, rather than looking it up #

The definition is a limit: take two states separated by δ₀, integrate both, and

$$\lambda = \lim_{t\to\infty}\ \lim_{\delta_0\to 0}\ \frac{1}{t}\ln\frac{\delta(t)}{\delta_0}$$

The two limits cannot be taken numerically as written. If δ₀ is small enough for the linearisation to hold, then after a modest time the separation has grown past the size of the attractor and stopped growing; and if δ₀ is large enough to survive the integration, the pair was never in the linear regime at all.

The standard remedy is renormalisation. Integrate the pair for one step, measure how much they have separated, add the logarithm of the growth to a running total, and then pull the second trajectory back along the line joining them until the separation is δ₀ again. Repeat. The pair never leaves the linear regime and the accumulated log measures the mean stretching rate.

Run over sixty thousand steps after a warm-up, this returns λ = 0.8982 for the standard parameters, against the published value of 0.9056. The difference is the finite integration length and is of no interest.

What is of interest is that the site does this rather than printing 0.9056. A measurement can be checked; a quotation cannot, and this whole field is one where a plausible number sitting beside a plausible picture would go unchallenged indefinitely.

The check that keeps the contrast honest #

The build asserts both signs. At r = 28 the exponent must exceed 0.5, and at r = 14 — below the Hopf threshold — it must be negative. The measured values are 0.8982 and −0.4019.

Two assertions rather than one, because either alone leaves a hole. An exponent routine that always returned a positive number would pass the first and fail the second; one that returned the mean of the log separations without renormalising would pass both and be measuring the saturation rather than the stretching.

The negative value is the more useful of the two to have measured. It converts the difference between the two regimes from an impression about two pictures into a number with a sign, and the sign is what the word chaotic actually means.

The exchange rate between precision and horizon #

The practical content of a positive exponent is an exchange rate, and it is worth doing the arithmetic because both of the intuitions people bring to it are wrong.

Separation grows as δ(t) = δ₀e^(λt). The time at which it reaches some tolerance Δ is

$$t = \frac{1}{\lambda}\ln\frac{\Delta}{\delta_0}$$

so improving the initial measurement by a factor F extends the useful horizon by ln F / λ.

A factor of ten buys ln 10 / 0.8982 = 2.56 time units.

A factor of a million buys 15.4.

A factor of 10¹² — a thousand billion, far beyond any conceivable improvement in instrumentation — buys 30.8.

The two wrong intuitions are these. The first is that improving the measurement does not help at all, which is false: it helps, definitely and computably, and the amount can be worked out in advance. The second is that a large improvement buys a proportionate extension, which is also false: the relationship is logarithmic, so the returns are brutal.

The doubling time for this system is ln 2 / λ = 0.772 time units. Every 0.772 units, whatever is known about the state is worth half as much.

Where the stretching actually happens #

The exponent is a mean rate, and the mean hides something worth knowing: the stretching is very unevenly distributed over the attractor.

Integrating the local stretching rate rather than accumulating it shows that a trajectory can be compressed for a while and then violently stretched, and that the violent episodes are concentrated near the region between the two lobes — the neighbourhood of the origin, which is a saddle point.

A trajectory approaching that neighbourhood is close to a fixed point with one unstable direction, and which side of the unstable manifold it passes on decides which lobe it enters. Two nearby trajectories that pass on opposite sides are separated by the width of the attractor within a few time units, whatever their previous separation was.

So the average exponent is a fair summary of a process that is not uniform, and the useful mental picture is not steady stretching but repeated near-misses at a decision point. That structure is generic — it is what a homoclinic tangle produces — and it is the reason error growth in real forecasting is also episodic rather than steady, with some weather situations far less predictable than others at the same lead time.

What this does and does not say about weather #

The connection between this calculation and forecasting is real, is the reason the subject exists, and is routinely overstated.

What transfers. The atmosphere is a nonlinear dissipative system with a positive largest Lyapunov exponent, and the doubling time for errors in the synoptic-scale flow is empirically about 1.5 to 2 days. So the logarithmic exchange rate above applies with different numbers, and the practical limit of deterministic prediction — the point at which a forecast is no better than climatology — is a matter of a couple of weeks, not months.

What does not. The Lorenz system’s λ is a property of the Lorenz system. Its numerical value says nothing about the atmosphere, and the doubling time above was measured on the atmosphere rather than derived from these equations. Nor is the two-week figure a theorem; it is an estimate that has lengthened as models and observations have improved, from about five days in the 1980s to about ten today, which is exactly what a logarithmic exchange rate predicts.

What follows for practice. If a single forecast is unreliable past a horizon, the sensible thing is to stop making single forecasts. Ensemble forecasting — running many trajectories from a spread of initial states consistent with the observations, and reporting the distribution — is the direct operational consequence of this argument, and it is what every serious forecasting centre has done since the early 1990s.

That last point is the one worth carrying. A positive Lyapunov exponent is not a reason to give up on prediction; it is a reason to predict a distribution rather than a state.

The wall the exchange rate does not have #

Everything above rests on one exponent, and a system with one exponent has no absolute horizon: the logarithm grows without bound, so a sufficiently good measurement buys a forecast of any length one likes. That is the reassuring half of the arithmetic, and for a flow with many scales in it there is reason to think it is false.

Lorenz made the argument in 1969, in a paper less famous than the 1963 one and more consequential for forecasting. Suppose the flow has an inertial range, so eddies of size $\ell$ turn over in a time going as $\ell^{2/3}$. An error confined to some small scale saturates at that scale within a few of its own turnover times, and then contaminates the scale above it, and so on upwards. The time for an error to climb from the smallest scales to the largest is the sum of the turnover times along the way.

That sum converges. The terms fall geometrically, so adding infinitely many of them gives a finite total — and the total does not depend on how small the initial error was, because starting further down the cascade only adds terms that are already vanishingly short.

So there is a horizon that no measurement can push back. Halve the initial error and the extra prediction bought is not $\ln 2/\lambda$ but a few more turnover times at the very bottom of the cascade, which is minutes. Beyond a finite time the largest scales are contaminated whatever was known at the start.

For the atmosphere the estimates put that intrinsic limit at around two to three weeks, which is uncomfortably close to where operational skill already reaches. If it is right, the returns from better observation are nearly exhausted, and further gains in forecast range have to come from better models rather than from better initial conditions — a very different research programme.

And the finiteness hangs on the exponent of the spectrum. A steeper spectrum makes the turnover times fall too slowly, the sum diverges, and the logarithmic exchange rate of this essay is restored with no wall at all. So whether weather has a hard predictability limit is, remarkably, a question about the slope of the energy cascade — a quantity measured in a tidal channel in 1962.

Why it is not the same as turbulence #

The resemblance between a chaotic trajectory and a turbulent signal is superficial and the difference is worth stating in numbers.

The Lorenz system has three degrees of freedom. A turbulent flow at engineering Reynolds numbers has of order Re^(9/4), which at Re = 10⁷ is about 10¹⁶.

Both are unpredictable, for different reasons. The first is unpredictable because nearby states diverge; its state is trivially easy to write down and impossible to know precisely enough. The second is unpredictable primarily because its state cannot be written down at all — and it also has a positive Lyapunov exponent, which makes matters worse rather than being the main difficulty.

The reason to insist on the distinction is that the resemblance has misled a generation of popular accounts into treating turbulence as solved-in-principle-by-chaos-theory. There is a genuine and respectable research programme asking whether turbulence can be described as a dynamical system on a finite-dimensional attractor, and the travelling waves that structure pipe transition are among its successes. It is a programme rather than a result, and the attractor’s dimension for a turbulent flow is large enough that “finite” is not the same as “manageable”.

What sensitive dependence does not prevent #

It is worth listing what remains predictable in a chaotic system, because the pessimism is easy to overdo and the list is longer than it sounds.

The attractor itself. Where the trajectory goes is entirely predictable; when it will be there is not. The set is stable under perturbation, and a long-run average computed on one trajectory agrees with the same average on any other.

Statistics. Means, variances and distributions over the attractor are reproducible to whatever precision the integration length allows. This is the reason climate and weather are different questions rather than the same question at different lengths.

Response to a change in parameters. Raising r moves the attractor in a way that is smooth and computable, even though no individual trajectory can be followed. Most of what is useful about a chaotic model is of this kind.

The regime it is in. Whether the system is settling, oscillating or wandering is decided by the parameters, and the parameters are known. That is why a bifurcation diagram is a useful object even when no trajectory in it can be followed.

Bounds. The Lorenz system is dissipative — the divergence of its vector field is the constant −(σ + 1 + β) — so volumes in state space contract at a fixed rate and every trajectory ends up in a bounded region. That is a global statement proved in one line, and it holds however unpredictable the detail is.

The same exchange rate, elsewhere on this site #

The logarithmic trade between precision and horizon is the third exponent-governed relationship in this field, and putting the three together is the most transferable thing in the essay.

Transition depends exponentially on the amplification a disturbance receives, so a factor of a hundred in the disturbance level moves the transition point by a fixed distance rather than by a fixed fraction — and that is why free-stream turbulence is such a large effect and why the number spreads over decades.

Resolution costs Re³ in work, so a thousandfold faster machine buys one decade of Reynolds number — and that is why the grid will not be built.

Prediction buys ln F / λ of horizon for a factor F of precision, so a millionfold better measurement buys fifteen time units.

In each case the same structural point holds: the exponent is what sets the exchange rate, the constant only shifts where the curve sits, and effort applied to the constant returns almost nothing. Recognising which quantity in a problem sits in an exponent is most of what makes an estimate useful, and it is a habit rather than a technique.

Where the model stops #

Everything measured here is measured on three ordinary differential equations, which stopped describing convection at r of about five. Nothing in this essay is a measurement on a fluid.

The exponent is the largest of three. A three-dimensional flow has three Lyapunov exponents, and for this system they are approximately 0.906, 0 and −14.57. The zero corresponds to displacement along the trajectory, which neither grows nor decays; the large negative one is the contraction onto the attractor. Their sum is the divergence, −13.67, which is an arithmetic check the site could make and does not, because measuring the other two requires evolving a full basis rather than a pair.

The renormalisation assumes the perturbation aligns with the leading direction. After a warm-up it does, because that is what a leading direction means, and a pair started along a contracting direction would give a wrong answer for a while. The warm-up in the site’s routine is five thousand steps.

One number, and how much of it is the model’s #

A closing caution about the exponent itself, since the essay has spent its length insisting that measuring is better than quoting.

λ = 0.8982 is a property of these three equations at these three parameter values. Change σ from 10 to 1 and it changes; change r from 28 to 40 and it changes; and neither change is a change to a fluid, because the equations stopped describing one long before either parameter value was reached.

So the number is a measurement, and what it measures is a model. The site prints it because printing a measured value rather than a remembered one is the standing discipline, not because 0.8982 tells anybody anything about convection.

What does transfer is the shape of the result: that the exponent exists, that it is positive above a computable threshold and negative below it, and that its reciprocal is the timescale over which knowledge decays. Those three statements survive every objection this essay has raised about the truncation, and they are what the twentieth century took from the paper.

Who found it, and when #

Poincaré described sensitive dependence in 1908, in Science and Method, in a passage about prediction that would serve as an abstract for this essay. He had met the phenomenon in the three-body problem in the 1880s.

Lorenz rediscovered it in 1961 and published it in 1963. The Butterfly title came from a 1972 talk, and the metaphor has done as much harm as good — it invites the reading that a small cause has a large effect, when the correct statement is that a small difference in the initial state leads to a large difference in the outcome, with no causal potency attached to the butterfly.

The practical measurement algorithms are from the 1970s and 1980s, principally Benettin and co-workers in 1980, whose renormalisation scheme is the one used here.

Ensemble forecasting became operational at the European Centre for Medium-Range Weather Forecasts and at the American National Centers for Environmental Prediction in 1992 and 1993, which is thirty years after the paper and is a fair estimate of how long it takes for a result of this kind to change practice.

Where the ladder goes next #

This anchor is complete: the threshold, the truncation, and the property that survives the truncation.

What remains in this field is the picture the arithmetic forbids drawing. The street this site cannot draw is the alternating wake behind a cylinder — the most reproduced photograph in the subject — built here as an ideal-flow model, named as a model in every caption, with one exact stability result inside it and no body anywhere in it.