4.1 Introduction

A consistent theme running throughout this text is the nonexistence of explicit exact general solutions for an arbitrary linear or nonlinear differential equation. Even for the few instances where such expressions exist, their functional forms may be such that the explicit forms do not provide much useful information on the details of the solutions. Often, what is required is a detailed understanding of the qualitative properties of the solutions along with good analytical approximations to the solutions. The main goal of this chapter is to introduce some techniques that allow, in part, these two issues to be resolved. Our focus is on qualitative methods for one- and two-dimensional dynamical systems. While these systems are restricted to just a single and two coupled, first-order differential equations, they are general enough to cover the mathematical modeling requirements of a broad and interesting variety of phenomena in the natural and engineering sciences.

We begin with a study of one-dimensional autonomous systems. They are modeled by a single first-order differential equation having the form

We then define, for such equations, the concepts of fixed-points, linear stability, and global stability. Section 4.3 shows how these techniques can be applied to a number of particular differential equations having the form given by Eq. (4.1.1). We also demonstrate that these general methods can, in special cases, be also applied to first-order nonautonomous equations.

Section 4.4 discusses two-dimensional dynamical systems. Such systems are modeled by a pair of coupled, first-order differential equations

The study of these...