Chapter 7 · Lyapunov exponents and doubling times

Part III — Quantifying chaos and predictability Knob: ρ, integration length $T$, window $\tau$ live notebook

Overview

Chapter 6 showed two nearly identical Lorenz trajectories pulling apart and fitted a growth rate to the separation. Run that experiment again from a different point on the attractor and you get a different number — 0.6 here, 1.1 there. So which is the rate? Is there one at all? And if the answer depends on where you start, what does a forecast centre mean when it says the atmosphere has a two-week limit?

This chapter turns “chaotic” into a number and is careful about which number. There are three, they are routinely confused, and they answer different questions: the spectrum $\lambda_1 \ge \lambda_2 \ge \lambda_3$ , which describes the attractor and depends on nothing else; the finite-time exponent $\lambda(x,\tau)$ , which describes one state at one lead time; and a twin-trajectory fit, which describes one realisation and depends partly on luck. Chapter 6 measured the third. This chapter computes the first, shows how widely the second varies, and ends with the case where all of them mislead.

The model

Oseledets’ multiplicative ergodic theorem guarantees that

\lambda_i = \lim_{T\to\infty}\frac{1}{T}\ln \sigma_i\!\left(\mathbf{M}(x_0,T)\right)

exists and is independent of $x_0$ for almost every starting point — which is what makes these properties of the system rather than of an experiment. Computing them directly fails numerically, because every column of $\mathbf{M}$ grows like $e^{\lambda_1 T}$ and the subdominant directions are lost within a few Lyapunov times. The Benettin algorithm carries an orthonormal frame along the trajectory, re-orthonormalises by QR at every step, and accumulates $\ln|R_{ii}|$ ; nothing overflows and every direction stays resolved. chaoslib.lyapunov implements it.

The check that makes the result trustworthy is exact. For Lorenz 63 the divergence of the flow is state-independent, so

\sum_i \lambda_i = -(\sigma + 1 + \beta) = -13.6\overline{6}

holds at any integration length, independent of the trajectory. That tests the implementation rather than its convergence — a sign error or a mis-ordered QR fails it immediately, while the leading exponent alone would still look plausible. It is asserted in the test suite and displayed live in the notebook.

What is computed live, and what is not

The spectrum at the reader’s chosen ρ is computed live (about ten seconds). Two figures are precomputed by scripts/generate_rho_sweep.py, because each needs many Benettin runs and neither has a knob: the $\lambda_1(\rho)$ curve across the whole transition, and the transient-chaos measurement in Section 5. Computing those live would cost roughly 80 seconds and four minutes respectively in the browser, for a result identical for every reader.

Prefer to run it yourself? Download as a Jupyter notebook — it installs its own dependencies in the first cell. The sliders are static in Jupyter; edit value= and re-run to change a parameter.

Exercises

Analytic. Show that $\sum_i \lambda_i$ equals the time-averaged divergence $\langle \nabla\!\cdot\! f\rangle$ , and hence that it is exactly $-(\sigma+1+\beta)$ for Lorenz 63. Then explain why $\lambda_2 \approx 0$ is structural for any continuous-time flow on a bounded attractor that is not a fixed point.

Computational. Verify that the doubling time and the Lyapunov time differ by $\ln 2$ , and that $D_{KY} = 2 + \lambda_1/|\lambda_3|$ reproduces the value chapter 8 obtains from a sampled trajectory with no reference to the dynamics.

Exploratory. Set ρ = 22.7 and vary the integration length. The exponent decays from +0.73 towards zero. Before reading Section 5, work out what could make a positive exponent shrink as you measure it for longer — then check your answer against the trajectory itself.

A caution this chapter is built around

Below the transition, Lorenz 63 has a chaotic saddle: an invariant set that is genuinely chaotic but is not an attractor. A trajectory near it separates exponentially, looks chaotic in every diagnostic, and then falls off and settles onto a fixed point. At ρ = 22.7 a 900-MTU run ends with the state completely motionless — a range of $2\times10^{-13}$ — while $\lambda_1$ measured over $T = 50$ comes out at $+0.73$ .

A positive finite- $T$ exponent is not proof of a chaotic attractor. It is evidence of exponential separation over the window you measured, and the gap between those two claims is the gap between $\lim_{T\to\infty}$ in Oseledets’ theorem and the $T$ you could afford.