Graphical Analysis

A word on notation

We will write f(x) in comments rather than f[x, a].

What it is.

Graphical analysis is a simple technique for seeing how trajectories evolve over many iterations. It exploits the fact that the output of the last iterate y is the input to the next x, so that we can use the line y = x to guide us from one iteration to the next. The fixed points are where the line y = x crosses f(x). Furthermore, as you will show in an exercise, a fixed point x0 of the map f(x) is attracting (attracts nearby points) if |f'(x0)| < 1 and repelling if |f'(x0)| > 1. Since the line y = x has slope one and crosses the fixed point at x0, it is trivial to check the stability by inspection with graphical analysis. We provide 2 examples below. The first shows a stable fixed point for f(x). Note that f(x) has smaller slope than the line y = x. The second shows an attracting period-2 point. Again note how the function (and its composition) crosses the line y = x.

Example 1

Example 2