While the possible dynamics of linear systems are rather restricted, even very simple nonlinear systems can have very complex dynamical behaviour. Nonlinearity is closely associated with chaos, even though not all nonlinear mappings produce chaotic dynamics [49].

A dynamical system is chaotic if its future behaviour is very sensitive to the initial conditions. In such systems two orbits starting close to each other will rather soon behave very differently. This makes any long term prediction of the evolution of the system impossible. Even for a deterministic system that is perfectly known, the initial conditions would have to be known arbitrarily precisely, which is of course impossible in practice. There is, however, great variation in how far ahead different systems can be predicted.

A classical example of a chaotic system is the weather conditions in the atmosphere of the Earth. It is said that a butterfly flapping its wings in the Caribbean can cause a great storm in Europe a few weeks later. Whether this is actually true or not, it gives a good example on the futility of trying to model a chaotic system and predict its evolution for long periods of time.

Even though long term prediction of a chaotic system is impossible, it is still often possible to predict statistical features of its behaviour. While modelling weather has proven troublesome, modelling the general long term behaviour, the climate, is possible. It is also possible to find certain invariant features of chaotic systems that provide qualitative description of the system.