Chaos and sensitivity to initial conditions

Deterministic systems whose nearby trajectories separate exponentially, $|\delta(t)| \approx |\delta_0|e^{\lambda t}$ with $\lambda > 0$. Prediction is possible only up to a horizon of about $\frac1\lambda\ln\frac{\text{tolerance}}{|\delta_0|}$: better measurements buy time only logarithmically.

Laplace imagined that knowing the present exactly would reveal the future. Poincaré (three bodies) and Lorenz (weather, 1963) showed that even simple deterministic equations amplify any uncertainty so fast that long-term prediction is impossible in practice.

Lorenz restarted a simulation from a printout rounded to 3 decimals instead of 6; after a few simulated weeks the two forecasts had nothing in common. Stretching and folding — like kneading dough — makes neighbouring points separate exponentially while staying in a bounded region.

Weather services run ensembles (dozens of forecasts from perturbed initial states) and report probabilities instead of a single answer. Chaotic simulations are not reproducible bit-for-bit across compilers or GPUs: a different rounding order is a tiny perturbation that grows. And chaos sets hard limits on what any model — learned or physical — can predict.