Notation

The symbols and sign conventions here are mandatory across every chapter and every chaoslib docstring. A chapter that needs a new symbol adds it here in the same commit. Where the two course texts disagree, this book follows Kalnay (2003) for data assimilation and Palmer & Hagedorn (2006) for predictability.

Units are SI unless a standard domain alternative is noted.

Dynamical systems

Symbol	Meaning	Notes
$x$	state vector	$x \in \mathbb{R}^n$ ; $n$ is the state dimension
$\dot x = f(x)$	the dynamics	autonomous unless stated
$\mathcal{M}$	the nonlinear model map	$x(t+\tau) = \mathcal{M}(x(t))$
$t$	time	MTU (model time units) for L63; time units for L96
$\tau$	lead time / optimisation window	always positive
$\mathbf{J}$	Jacobian $\partial f_i/\partial x_j$	of the continuous dynamics
$\mathbf{M}$	tangent linear propagator	of the discrete map over $\tau$
$\mathbf{M}^{\!\top}$	the adjoint	transpose under the Euclidean inner product
$\delta x$	perturbation / error	$\delta_0 = \\|\delta x(0)\\|$
$\lambda_i$	Lyapunov exponents	ordered $\lambda_1 \ge \lambda_2 \ge \dots$ , units of inverse time
$\lambda(x,\tau)$	finite-time (local) exponent	a property of a state, not the attractor
$\sigma_i$	singular values of $\mathbf{M}$	ordered descending; $\sigma_1$ is optimal growth
$h_{KS}$	Kolmogorov–Sinai entropy	$\sum_{\lambda_i>0}\lambda_i$ , nats per unit time
$D_{KY}$	Kaplan–Yorke dimension	from the spectrum
$D_2$	correlation dimension	from a sampled trajectory
$T_d$	error-doubling time	$\ln 2/\lambda_1$
$\mathcal{K}^\tau$	Koopman operator	$(\mathcal{K}^\tau g)(x) = g(\mathcal{M}^\tau(x))$ ; linear on observables
$g$	an observable, or a dictionary of them	a function of the state, not a state
$\mathbf{K}$	the finite-dictionary compression of $\mathcal{K}^\tau$	what EDMD fits
$\phi$	a Koopman eigenfunction	$\mathcal{K}^\tau\phi = e^{\lambda\tau}\phi$
$\nu$	invariant (natural) measure	“climate” as a distribution on state space
$\langle A\rangle_\nu$	ensemble average	$\int A\,d\nu$ ; the object a climatology estimates
$\bar A_T$	time average over a window $T$	the estimator a control run provides
$\tau_{\rm int}$	integrated autocorrelation time	$\int\rho(s)\,ds$ ; a time, not a lag count
$\tau_{\rm erg}$	ergodic time	how long to sample the whole attractor, not just its fast part
$W_1$	1-Wasserstein distance between measures	carries the units of the variable
$D_N^*$	star discrepancy of a sample	sampling error of the worst indicator observable

Time-unit convention. For Lorenz 63, 1 MTU is read as $\approx 5$ atmospheric days; for Lorenz 96 with $F=8$ , 1 time unit is read as 5 days. These are interpretive conventions, not derivations — state them as such.

Specific systems

Symbol	Meaning
$\sigma, \rho, \beta$	Lorenz 63 parameters (Prandtl, Rayleigh, geometry)
$\rho_H$	Hopf threshold, $\sigma(\sigma+\beta+3)/(\sigma-\beta-1)$
$C^\pm$	the non-trivial Lorenz 63 fixed points
$N, F$	Lorenz 96 site count and forcing
$r$	logistic-map parameter
$\mu$	double-well tilt; the fold is at $\mu_c = 2/3\sqrt3$
$\theta, \omega$	pendulum angle and angular velocity
$L, g, m$	pendulum length, gravity, mass

Note the collisions, all of them deliberate rather than tidied away, because each symbol is standard in its own literature. $\mathbf{K}$ is the Kalman gain of chapters 18 to 20 and the finite-dictionary Koopman operator of chapter 31; the two never appear in the same chapter, and chapter 31 says so where it introduces it. $g$ is gravity for the pendulum and an observable for the Koopman operator. $\mu$ is the double-well tilt of chapters 27 and 30 and is the conventional symbol for an invariant measure; this book writes the measure $\nu$ throughout and reserves $\mu$ for the tilt. And $\sigma$ is the Lorenz 63 Prandtl number, the noise amplitude of a stochastic run, and the conventional symbol for a singular value. Chapters using both write singular values as $\sigma_i$ with an explicit subscript, and say so in the text.

Predictability and error growth

Symbol	Meaning
$E(t)$	forecast error (RMS unless stated)
$E_\infty$	saturation error — the climatological level
$\Delta t$	gain in forecast horizon
$D(p\\|q)$	relative entropy (Kullback–Leibler), nats
$I(X;Y)$	mutual information, nats
$H(p)$	Shannon entropy, nats

Information units are nats (natural logarithm) everywhere. Divide by $\ln 2$ for bits; never mix the two in one figure.

Data assimilation (following Kalnay 2003)

Symbol	Meaning
$x^b$	background (prior) state
$x^a$	analysis (posterior) state
$x^t$	true state
$y$	observations
$\mathbf{H}$	observation operator
$\mathbf{B}$	background error covariance (fixed, as in 3D-Var)
$\mathbf{P}^b, \mathbf{P}^a$	background / analysis error covariance (evolving)
$\mathbf{R}$	observation error covariance
$\mathbf{K}$	Kalman gain
$J(x)$	variational cost function
$N$	ensemble size

Sign convention for innovations: $d = y - \mathbf{H}x^b$ , observation minus background. The analysis increment is $\mathbf{K}d$ , so a positive innovation moves the analysis towards the observation.

Ensembles and verification

Symbol	Meaning
$\bar x$	ensemble mean
$s$	ensemble spread (RMS about the mean, $N-1$ normalisation)
CRPS	continuous ranked probability score
BS	Brier score
ACC	anomaly correlation coefficient

Calibration identity: a reliable ensemble satisfies RMS spread $=$ RMS error of the ensemble mean. State any departure from it as under- or over-dispersion, not as “error”.