Solutions of simple nonlinear systems can behave in extremely complicated ways. This observation, and the subsequent mathematical treatment of ‘chaos’, is one of the most exciting recent developments of mathematics. Loosely speaking, a chaotic solution is aperiodic but bounded and nearby solutions separate rapidly in time. This latter property, called sensitive dependence upon initial conditions, can be thought of as a loss of memory of the system of the past history of any solution. It implies that long term predictions of the system are almost impossible despite the deterministic nature of the equations. Historically the possibility of aperiodic solutions with complicated geometric structure was known to both Poincaré and Birkhoff in the late nineteenth and early twentieth centuries, but it was not until computer simulation of differential equations became feasible that the subject really took off. This is probably because it is extremely difficult to get any intuitive feel for how a system behaves simply by looking at the equations. The computer allows one to see the type of result one might try to prove and motivates the development of conjectures and theorems.

Two important papers appeared in the 1960s, one on the applied side of the subject and one on the pure. In 1963, Lorenz published a paper called Deterministic Non-periodic Flows in which he described the numerical results he had obtained by integrating a simple third order system of ordinary differential equations on a computer (this was not the first such paper, but it has become the most influential).