Problems with different time scales

In this section we discuss a very simple problem. Consider the scalar initial value problem Here ε > 0 is a small constant and a = a 1 + i a 2 , a 1 , a 2 real, is a complex number with | a | = 1. We can write down the solution of (1.1) explicity. It is where is the forced solution and is a solution of the homogeneous equation y S varies on the time scale ‘1’ while y F varies on the much faster scale 1/ε. We say that y S , y F vary on the slow and fast scale, respectively. We use also the phrase: y S and y F are the slow and the fast part of the solution, respectively.