In many practical problems, it is desirable to find an equilibrium. For example, equilibria are important in transportation engineering.
Many urban areas suffer from traffic congestion. Intuitively, it may seem that a road expansion (e.g., the opening of a new road) should always improve the traffic conditions. However, in reality, a new road can actually worsen traffic congestion. It is therefore extremely important that before we start a road expansion project, we first predict the effect of this project on traffic congestion.
When a new road is built, some traffic moves to this road to avoid congestion on the other roads; this causes congestion on the new road, which, in its turn, leads drivers to go back to their previous routes, etc. What we want to estimate is the resulting equilibrium.
In many problems -- e.g., in many transportation problems -- natural iterations do not converge. It turns out that the convergence of the corresponding fixed point iterations can be improved if we consider these iterations as an approximation to the appropriate dynamical system.