4 ideas
In nonlinear systems, infinitesimally small differences in starting conditions amplify exponentially over time, making long-term prediction impossible even when the underlying equations are fully deterministic. A rounding error in the thousandth decimal place eventually dominates the outcome, so the system is predictable in principle but unpredictable in practice.
A meteorologist re-ran a weather model entering 0.506 instead of the stored 0.506127, expecting a near-identical result, but the two runs diverged completely within simulated weeks. This accidental discovery revealed that deterministic weather equations could never yield reliable long-range forecasts, coining the popular image of the butterfly effect.
Chaotic systems never repeat exactly yet remain confined to a bounded region of state space whose shape — an attractor of infinite, fractal complexity — they trace forever without crossing themselves. This means disorder has hidden geometric order: the system is unpredictable in detail but constrained to a definite pattern.
Feigenbaum found that wildly different systems approaching chaos through period-doubling share identical numerical constants governing the transition, regardless of their specific physics. This means the route to chaos is a universal property independent of the medium — dripping faucets, populations, and fluids obey the same scaling laws.