Chaos is a striking example of emergence. Deterministic equations of motions lead to completely unpredictable over time. Randomness has emerged from these deterministic laws.
From (Crutchfield 1994):
Where in the determinism did the randomness come from? The answer is that the effective dynamic, which maps from initial conditions to states at a later time, becomes so complicated that an observer can neither measure the system accurately enough nor compute with sufficient power to predict the future behavior when given an initial condition. The emergence of disorder here is the product of both the complicated behavior of nonlinear dynamical systems and the limitations of the observer.
- James P. Crutchfield. . "The Calculi of Emergence: Computation, Dynamics and Induction". Physica D: Nonlinear Phenomena 75 (1):11–54. DOI.