Dynamical Systems and Iterative Maps

Whereas a real dynamical system, such as the motion of the planets, is described by differential equations and continuous time, it is often convenient to consider simpler mathematical models, called difference equations, where the system evolves through a set of discrete time steps. The simplest models of difference equations are the iterative maps because the future value of some variable at time $t=(n+1)$ depends only on its value at the present time $t=n$ (here $n$ is an integer). An iterative map is of the form

$$x_{n+1} = f(x_n) ,$$ (3.1)

where $f(x)$ is called the mapping function. Thus starting with some initial value $x_0$ of the variable, its next value $x_1$ is obtained by evaluating $f(x_0)$, which itself becomes the input to evaluate $x_2$ and so on. Thus the time-evolution of the discrete variable $x$ is obtained by repeated iteration of the mapping function. Note that this mathematical iteration corresponds in physical terms to a feedback process.