Chapter 6 · The Lorenz (1963) system: the butterfly

Overview

In 1963 Edward Lorenz truncated the equations for thermal convection to three ordinary differential equations, integrated them on an LGP-30, and found that a run restarted from a printout rounded to three decimals — $0.506$ instead of $0.506127$ — diverged completely from the original. That is the origin of the two-week forecast limit, and of the phrase that came from the title of his 1972 talk.

This chapter is the hinge of the book. Chapter 4 established that chaos needs a phase space of at least three dimensions; Lorenz 63 is the smallest system built from atmospheric physics that has one. Everything in Parts III to V — Lyapunov exponents, error growth, ensembles, adjoints, data assimilation — is developed on this system before being trusted on anything larger.

The notebook works through three things in order:

1. The attractor, and the bifurcation that makes it. Vary $\sigma$, $\rho$ and $\beta$ and the initial state, and watch the long-term behaviour change qualitatively: decay to the origin, a spiral onto one of the convective rolls $C^\pm$, the strange attractor, and the periodic window near $\rho \approx 100$. A live readout classifies the regime and fits the leading finite-time Lyapunov exponent.
2. Sensitive dependence. Two trajectories from initial states differing by $\delta_0$, with $\ln\|\delta\|$ plotted against time so the exponential stretch, and the saturation that ends it, are both visible. This single twin pair is the experiment chapter 7 opens by taking apart, so it is worth running before reading that one.
3. What a model time unit is worth. $\lambda \approx 0.9$ MTU$^{-1}$ here against $\approx 0.35$ day$^{-1}$ for the atmosphere — and the demonstration that no single MTU-to-days conversion reconciles the two. Match the growth rates and the doubling times disagree; match the doubling times and 1 MTU becomes 2.6 days rather than 5. What transfers is the exponential law, not the constant in front of it.

What this chapter deliberately leaves to others. Measuring $\lambda_1$ properly rather than from one twin pair is chapter 7; ensemble construction, spread and calibration are chapter 17; the forecast-skill record is chapter 22; and the logarithmic return on better observations, $\Delta t = \ln 10/\lambda$, is measured in chapter 20. This chapter is the system itself.

The model

$X$ measures the intensity of the convective overturning, $Y$ the temperature difference between ascending and descending branches, and $Z$ the departure of the vertical temperature profile from linear. Symbols follow the notation page.

The origin is always a fixed point; for $\rho > 1$ a symmetric pair appears at

the steady clockwise and counter-clockwise rolls, and these lose stability in a Hopf bifurcation at

for the classical $\sigma = 10$, $\beta = 8/3$. Above $\rho_H$ neither roll is stable and the trajectory wanders between them forever without repeating.

All of this comes from chaoslib.systems and chaoslib.lyapunov. Two identities are worth knowing because they make the implementation checkable rather than merely plausible: the Jacobian’s trace is $-(\sigma+1+\beta)$ at every state, so the three Lyapunov exponents must sum to exactly that; and the tangent-linear propagator must satisfy $\det\mathbf{M} = e^{\tau\,\mathrm{tr}\mathbf{J}}$. Both are asserted in the test suite, and the first is displayed live in the notebook.

For the record, the values the library reproduces: $\lambda_1 = 0.9056$, $\sum_i\lambda_i = -13.6667$, and a Kaplan–Yorke dimension of $2.06$ — the attractor is more than a surface and less than a volume.

A note on the figures

Phase space is drawn as static 2-D projections — $x$–$z$, $x$–$y$, and $x(t)$ — rather than a rotatable 3-D scene. The $x$–$z$ projection is the image everyone recognises as the butterfly; it is the same every time; and it does not ask the reader to drag the plot into a usable viewpoint before they can see the two lobes. The parameters remain fully interactive: every slider still redraws the figure.

Exercises

Analytic. Derive $\rho_H$ by linearising about $C^+$ and finding where a complex pair of eigenvalues crosses the imaginary axis. Then explain why the formula requires $\sigma > \beta + 1$, and what happens physically when it does not hold.

Computational. Fix $\rho = 28$ and measure the leading finite-time Lyapunov exponent at twenty different points on the attractor, using a 0.5 MTU window. Plot the distribution. The asymptotic value is 0.9056 — how wide is the spread around it, and what does its width imply for the claim “the atmosphere has a two-week limit”?

Exploratory. Reduce $\delta_0$ by a factor of ten and measure how much extra lead time you gain before the error reaches half of saturation. Repeat twice more. The gain per decade should be roughly constant, at $\ln 10/\lambda$. Now argue from that number about how much forecast improvement a tenfold increase in observing-system accuracy could deliver — and compare your answer to the historical record in Chapter 22.

Further reading

• Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141. Still the clearest statement of the result.
• Lorenz, E. N. (1993). The Essence of Chaos — the discovery told by the person who made it.
• Palmer & Hagedorn (2006), Predictability of Weather and Climate, ch. 2 [citation needed: pages].
• Kalnay (2003), Atmospheric Modeling, Data Assimilation and Predictability, §6.1.