Dynamics is the study of the movement through time of variables such as heartbeat, temperature, species population, voltage, production, employment, prices and so forth.

This is often achieved by means of equations linking the values of variables at different, uniformly spaced instants of time, i.e., difference equations, or by systems relating the values of variables to their time derivatives, i.e., ordinary differential equations. Dynamical phenomena can also be investigated by other types of mathematical representations, such as partial differential equations, lattice maps or cellular automata. In this book, however, we shall concentrate on the study of systems of difference and differential equations and their dynamical behaviour.

In the following chapters we shall occasionally use models drawn from economics to illustrate the main concepts and methods. However, in general, the mathematical properties of equations will be discussed independently of their applications.

A static problem

To provide a first, broad idea of the problems posed by dynamic vis-à-vis static analysis, we shall now introduce an elementary model that could be labelled as ‘supply-demand-price interaction in a single market’. Our model considers the quantities supplied and demanded of a single good, defined as functions of a single variable, its price, p. In economic parlance, this would be called partial analysis since the effect of prices and quantities determined in the markets of all other goods is neglected. It is assumed that the demand function D(p) is decreasing in p (the lower the price, the greater the amount that people wish to buy), while the supply function S(p) is increasing in p (the higher the price, the greater the amount that people wish to supply).