Introduction

Abstract

In this book we will study equations of the following form

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

(0.1)

and

$$x \mapsto g\left( {x;\mu } \right),$$

(0.2)

with x ∈ U ⊂ ℝⁿ, t ∈ ℝ¹, and μ ∈ V ⊂ ℝ^p where U and V are open sets in ℝⁿ and ℝ^p, respectively. The overdot in (0.1) means “$\frac{d}{{dt}},$” and we view the variables μ as parameters. In the study of dynamical systems the dependent variable is often referred to as “time.” We will use this terminology from time to time also. We refer to (0.1) as a vector field or ordinary differential equation and to (0.2) as a map or difference equation. Both will be termed dynamical systems. Before discussing what we might want to know about (0.1) and (0.2), we need to establish a bit of terminology.