Almost all the interesting dynamic problems in science and engineering are characterized by the presence of more than one significant scale, i.e. there is a variety of degrees of freedom with very different time scales. Among numerous important examples we can mention protein folding and climate. While the time scale of vibration of covalent bonds is O(10–15 s), the folding time for proteins may be of the order of seconds. Also in the case of climate, the characteristic times of the involved processes vary from days (for the atmosphere) to O(103 yr) (for the deep ocean and ice shields). In such a situation one says that the system has a multiscale character (E and Engquist 2003).

The necessity of treating the “slow dynamics” in terms of effective equations is both practical (even modern supercomputers are not able to simulate all the relevant scales involved in certain difficult problems) and conceptual: effective equations are able to catch some general features and to reveal dominant ingredients which can remain hidden in the detailed description. The study of multiscale problems has a long history in science (in particular in mathematics): an early important example is the averaging method in mechanics (Arnold 1976).