Some foundational issues in multiple scale theory

Abstract

“Optimist” and “pessimist” positions are defined, with regard to the use of multiple scale methods for initial value problems in dynamical systems, and some of the intuitions underlying each position are converted into theorems. It is shown that in principle multiple scale approximations exist which are valid for arbitrarily long expanding intervals, but that complex dynamics (such as chaos) can make it impossible in practice to compute these approximations beyond a certain "separation time" which is determined by the dynamics. In such situations one cannot extend the time of validity by increasing the number of time scales used.

Keywords:

• scales
• matching