Deterministic does not mean predictable.
Explore how rounded initial conditions, unstable dynamics, and the prediction target determine how long a detailed forecast remains useful.
This Exploration develops the complete argument while letting you change the assumptions and watch what survives.
Watch first · short video
Charlotte Moser introduces why deterministic dynamics can still lose predictability.
An exact rule is not the same as an exactly known starting state. Round the state, restart the same equations, and follow what happens.
The first number echoes the reported rounding example. The three-component Lorenz-63 state is a modern illustration, not Lorenz's historical state.
Same equations. Nearly the same initial state. Detailed trajectory information has a finite lifetime.
A small initial error does not only have a size—it also has a direction. In an unstable system, perturbations pointing in different directions can grow at very different rates.
δx(t + Δt) ≈ Aδx(t)
The matrix A is a local approximation of the dynamics. It tells us what happens to a very small perturbation over a short interval: the perturbation may rotate, stretch, or compress.
Both perturbations begin with exactly the same size. One points along a direction strongly amplified by the dynamics; the other points along a direction that is temporarily compressed.
δx(t + Δt) ≈ Aδx(t)
The matrix A is a local approximation of the dynamics. It tells us what happens to a very small perturbation over a short interval: the perturbation may rotate, stretch, or compress.
Knowing only the size of the initial error is therefore not enough. Its direction in state space matters too. Forecast uncertainty can grow much faster in some directions than in others.
Nearby paths can lose detailed agreement while remaining organized by the same dynamical structure.
Detailed trajectory agreement can be lost while the structure organizing the motion persists.
A forecast horizon depends on the initial error, how errors grow, the prediction target, and the tolerance chosen for that target.
error(t) ≈ initial error × exp(λt)
Tp ≈ (1/λ) log(tolerance / initial error)
A 1-fold reduction in initial error changes forecast time logarithmically. For 100×, ΔTₚ = (1/λ) log(100), not 100× the time.
A predictability horizon is chosen relative to a target and tolerance. It is not a universal moment when all prediction becomes impossible.
An ensemble begins from a family of plausible states. What it represents depends on which uncertainties the experiment includes.
More members sample the uncertainties already included here. They do not automatically repair a missing source of uncertainty.
Ensemble spread becomes a probability statement only after the forecasting procedure has been evaluated against outcomes.
An ensemble is a scientific object, not a decorative fan of lines.
These are two different kinds of perfectly valid models. The difference is not that one model is correct and the other is noisy or incorrect. The difference is where uncertainty enters the dynamics.
Run 1: exact overlap with the reference trajectory.
A deterministic model contains no random input. Once the equations and the initial condition are fixed, the future trajectory is fixed. Even a chaotic deterministic model behaves this way. Chaos means that slightly different initial conditions can eventually produce very different trajectories—not that repeated runs from the identical initial condition are random.
previous realizationcurrent realization
A stochastic model contains a random term as part of the model itself. Therefore, even when every run begins from exactly the same initial state, different realizations can evolve differently because they experience different random inputs.
Deterministicdx/dt = f(x)
Stochasticdx = f(x)dt + σdWt
The term dWt represents new random forcing entering during the evolution.
Chaos is sensitivity, not randomness. A deterministic chaotic system amplifies differences that are already present. A stochastic system can introduce new differences as it evolves.
Stochastic terms can represent unresolved processes statistically. Their use does not imply that the underlying physical world is literally random.
An exact rule determines a unique trajectory only after a starting point is specified; a real forecast begins from an estimate rather than the exact state.
It can be stretched, rotated, and compressed, and its most rapidly growing direction can change with the state.
Distributions, regime probabilities, averages, and forced responses may remain informative, while slowly evolving variables can retain useful memory.
It is meaningful only when it represents the uncertainties that matter and is evaluated against outcomes.
Lorenz-63 omits spatial structure, moisture, external forcing, ocean coupling, unresolved scales, observational networks, and model error.
Precision of calculation is not the same as certainty of conclusion.