In this chapter, we turn to the analysis of non-linear difference equations. Since most applications in economics assume a stationary environment, that is, an environment that does not change over time, the focus of our study will be on autonomous equations. Such equations typically admit constant solutions – so-called fixed points – or periodic solutions, and the first step of the analysis of an autonomous difference equation is often the identification of these simple types of solutions. Therefore, we collect in section 3.1 a number of results about fixed points and periodic points.

As a next step, we turn to the investigation of the dynamics locally around the fixed points and the periodic points, respectively. This is greatly facilitated by local linearization techniques and the Hartman–Grobman theorem, which we present in section 3.2. In section 3.3, we introduce the important notion of the stability of a fixed point or a periodic point and we derive stability criteria. Some of these criteria are based on local linearization techniques, whereas others involve Lyapunov functions.

As we shall see in section 3.4, the appropriate definition of stability for an economic problem depends on how many of the system variables are pre-determined and how many are jump variables. This will lead us to the concept of saddle point stability. Finally, in section 3.5 we demonstrate that in the case of ‘too much stability’, a notion that we will formally define, purely deterministic economic models can admit stochastic solutions if we properly take into account the influence that expectations have on the behaviour of economic agents.

Invariant sets, fixed points, and periodic points

In this section, we consider autonomous difference equations of the form

xt+1= f(xt), (3.1)

where the law of motion f : X → X is a given function and the system domain X ⊆ ℝn is a non-empty set.