This chapter prepares the ground for much of this book in several ways. On one hand, it provides the simplest examples of dynamical behavior, with the first hints as to how more complicated behavior can arise. On the other hand, it provides some important tools and concepts that we will need frequently. There are two kinds of dynamical systems we present here as “simple”. There are linear maps, whose simplicity lies in the possibility of breaking them down into components that one can study separately. Contracting maps are simple because everything moves toward a single point. We introduce linear maps briefly here and concentrate on a preview of their utility for studying nonlinear dynamical systems. Linear maps are studied systematically in Chapter 3. We present the facts about contracting maps that will be used throughout this course. Applications pervade this book and are featured prominently in Chapter 9.

LINEAR MAPS AND LINEARIZATION

Scalar Linear Maps

The primitive discrete-time population model xi+1 = f(xi) = kxi (with k > 0) introduced in Section 1.2.9.1 has simple dynamics: Starting with any x0 ≠ 0, the sequence (xi)i∈ℕ diverges if k > 1 and goes to 0 if k < 1. Part of the simplicity is that the asymptotic behavior is independent of the initial condition; scaling x0 by a factor a scales all xi by the same factor.