A Graphical Analysis

It is useful in general to have a geometrical picture, called a state space or phase space plot, which shows how the next iterate $x_{n+1}$ depends on the current value $x_n$. This is done as follows. First draw the curve $y=f(x)$ to represent the right hand side of Eq.(3.1) . On the same graph plot the line $y=x$ to represent the iteration process $x_{n+1}=f(x_n)$. Fig(3.1a) illustrates the method for the case $f(x)=x^2$. Starting with any $x_0$, for example $x_0=0.8$, the next value $x_1=f(x_0)$ is obtained by drawing a vertical line from $x_0=0.8$ to the curve $y=x^2$. Since $x_1=f(x_0)$ is now needed as the next input to the iterative map, graphically this is achieved by moving horizontally to the $y=x$ line and then moving vertically to the 'square' curve to give $x_2$. The procedure is repeated to produce a generic 'cobweb' diagram. For this example Fig(3.1a) shows a step-wise approach to the fixed point at the origin.

It is easy to see from this diagram that for the example considered, the origin is an attractive (stable) fixed point with a basin of attraction $[0,1)$, while as Fig(3.1b) shows, the point $x^{*}=1$ is an unstable fixed point.

Since the fixed points above correspond to a situation where the future values of $x$ at each time step remain the same, they are called period-one fixed points. This terminology is introduced as we will later generalise the concept of fixed points. Since period-one fixed points correspond to solutions of $x_{n+1} = x_n$, graphically these can be found from the intersection of the mapping function $y=f(x)$ with the staright line $y=x$. For the case $f(x)=x^2$, the two intersections are at the origin and at $x=1$ as we already know.