The theory of dynamical systems is the basic mathematical tool for analysing time series. This section presents a brief introduction to the basic concepts. For a more extensive treatment, see for example [1].

The general form for an autonomous discrete-time dynamical system is
the *map*

where and is a

For a general autonomous differential equation

we define the

where is the unique solution of Equation (2.2) with the initial condition , evaluated at time . The function in Equation (2.2) is called the

Setting , where , gives an autonomous discrete-time dynamical system like in Equation (2.1). The discrete system defined in this way samples the values of the continuous system at constant intervals . Thus it is a discretisation of the continuous system.