Rule 19 of 36 · Chapter III — Force & Motion
Small causes can have large effects
Why this rule exists
In nonlinear systems, tiny differences in starting conditions grow exponentially, so two nearly identical states diverge into utterly different futures. This is deterministic chaos: the equations are exact, yet long-term prediction is impossible because any measurement error, however small, is amplified without limit. The effect appears wherever feedback compounds — weather, turbulence, three gravitating bodies — and sets a horizon beyond which forecasting fails.
In practice
Ask whether your system is linear or nonlinear before trusting a long forecast. If errors compound, quote a prediction horizon rather than a single trajectory, and describe the system statistically — averages, attractors, distributions — instead of chasing one exact path. You reduce sensitivity by damping the feedback or shortening the interval you predict over.
When it doesn't apply
Sensitive dependence is not universal. Linear or strongly damped systems wash out small perturbations and stay predictable; stable equilibria pull nearby states back. Chaos requires nonlinearity and enough freedom for trajectories to stretch and fold.