Asymptotics and Perturbation

Abstract

Approximations of some kind are essential in the computation of the solutions of differential equations representing the system dynamics. Asymptotic solutions constitute a large class of such approximations which have been widely used in many areas such as the analysis of complex systems. They have been used, for example, for preliminary design purposes or for computing the response of a dynamic system, such as a linear or nonlinear (constant or variable) system. They are computationally efficient as evidenced by the classical examples in Chap. 2. They have the highly desirable property that the salient aspects of the system dynamics are preserved and can be calculated rapidly. This is generally because, a complex phenomenon is represented in terms of simpler descriptions as it is impossible, in general, to derive exact representations of the phenomena under study. The mathematical models are usually in the form of nonlinear or nonautonomous differential equations. Even simpler models such as linear time-invariant (LTI) systems, in certain situations present difficulties. This is the case, for instance, when the system has widely separated eigenvalues, or equivalently, a mixture of motions at different rates. Such systems are often called stiff systems. For many complex mathematical models, the only recourse is through approximations.