Mathematically, chaos refers to a very specific kind of unpredictability: deterministic behavior that is very sensitive to its initial conditions. In other words, infinitesimal variations in initial conditions for a chaotic dynamic system lead to large variations in behavior.
Chaotic systems consequently appear disordered and random. However, they are actually deterministic systems governed by physical or mathematical laws, and so are completely predictable given perfect knowledge of the initial conditions. In other words, a chaotic system will always exhibit the same behavior when seeded with the same initial conditions – there is no inherent randomness in this regard. However, such perfect knowledge is never attainable in real life; slight errors are intrinsic to any physical measurement. In a chaotic system, these slight errors will give rise to results which differ wildly from the correct result. A commonly used example is weather forecasting, which is only possible up to about a week ahead, despite theoretically being perfectly possible at any level (ignoring the effects of the uncertainty principle).
Chaotic systems display sensitivity to initial conditions. Tiny initial differences can result, over time, in completely different results.
Edward Lorenz and Henri PoincarÃ© were early pioneers of chaos theory, and James Gleickâ€™s 1987 book Chaos: Making a New Science helped to popularize the field.