ABSTRACT
The solution of initial value problems is the first example of a singular perturbation problem. In many problems, it is apparent that two time scales are at work, but it is not immediately obvious what the precise choice of time scales should be. The function is called an adiabatic invariant for this problem, and a representation of the solution shows both the slow amplitude variation and the frequency modulation of this oscillator. Many of the equations governing chemical reactions exhibit the feature that different reactions take place at vastly different rates. Boundary value problems often show the same features as initial value problems with multiple scales in operation, and resulting boundary layers. This chapter ends with two examples of interest in neurophysiology. These examples show that shock layers do not necessarily occur at boundaries, and that one need not have an explicit representation of the solution to understand its behavior.
