ABSTRACT
In this chapter we present the asymptotic methods of averaging and singular perturbation. Suppose we are given the state equation x˙ = f (t, x, ε), where ε is a “small” positive parameter, and, under certain conditions, the equation has an exact solution x(t, ε). Equations of this type are encountered in many applications. The goal of an asymptotic method is to obtain an approximate solution x˜(t, ε) such that the approximation error x(t, ε)− x˜(t, ε) is small, in some norm, for small ε and the approximate solution x˜(t, ε) is expressed in terms of equations simpler than the original equation. The practical significance of asymptotic methods is in revealing underlying multiple timescale structures inherent in many practical problems. Quite often the solution of the state equation exhibits the phenomenon that some variables move in time faster than other variables, leading to the classification of variables as “slow” and “fast.” Both the averaging and singular perturbationmethods deal with the interaction of slow and fast variables.
We start by a brief description of the perturbation method that seeks an approximate solution as a finite Taylor expansion of the exact solution. Then, we introduce the averaging method in its simplest form, which is sometimes called “periodic averaging” since the right-hand side function is periodic in time. Next, we introduce the singular perturbation model and give its two timescale properties. Finally, we show how to improve the accuracy of the reduced model of a singularly perturbed system.
