Qualitative theory of differential equations

Abstract

In this chapter we consider the differential equation

$${\rm{\dot x}}\, = \,{\rm{f}}\,\left( {t,{\rm{x}}} \right)$$

(1)

where

$${\rm{x}}\,{\rm{ = }}\,\left( {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}\\ \vdots \\ {{x_n}\left( t \right)} \end{array}} \right),$$

and

$${\rm{f}}\,\left( {t,\,{\rm{x}}} \right)\, = \,\left( {\begin{array}{*{20}{c}} {{f_1}\left( {t,{x_1}, \ldots ,{x_n}} \right)}\\ \vdots \\ {{f_n}\,\left( {t,{x_1}, \ldots ,{x_n}} \right)} \end{array}} \right)$$

is a nonlinear function of x ₁,…,x _n. Unfortunately, there are no known methods of solving Equation (1). This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of (1) explicitly. For example, let x ₁(t) and x ₂(t) denote the populations, at time t, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of x ₁(t) and x ₂(t) are governed by the differential equation (1). In this case, we are not really interested in the values of x ₁(t) and x ₂(t) at every time t. Rather, we are interested in the qualitative properties of x ₁(t) and x ₂(t). Specically, we wish to answer the following questions.