5.1 Introduction

The stability analysis of equilibrium points of an autonomous first-order system

is an important topic in the qualitative theory of ordinary differential equations (ODE). Here x = x(t) ∈ ℝn is a vector valued unknown function of the independent variable t ∈ I, an interval in ℝ, and f: ℝn → ℝn is a given vector valued function, which is assumed to be a C1 or more smooth function. This assumption ensures the uniqueness (local or global) of a solution of the system (5.1.1) with a prescribed initial value at an initial time. The positive integer n is referred to as the dimension of the system.

The system (5.1.1) is called an autonomous system because the right-side function f does not depend on t explicitly. When f depends on t explicitly as well, the system is referred to as non-autonomous. For example, the equation x′ = x + t (1D or one-dimensional equation) is non-autonomous.

In some situations, we do assume more smoothness on f, so that global existence of a solution is guaranteed; this means existence for all t. Even when a solution does not exist for all t, we can still do the phase space analysis by considering the maximum interval of existence of the solution in question. However, uniqueness plays a crucial role. In what follows, we introduce many concepts, definitions and list many results that are useful in solving the exercises. For proofs and other details, we refer to Ref.[46] or any other book with similar contents.