Cobweb plots

A cobweb plot is a simple and efficient tool to visualize functional iteration. The basic idea is simple: Suppose we graph the function \(f\) together with the line \(y=x\); something like so:

In this example, the two graphs intersect; that point of intersection is a fixed point. Now, suppose we're on the line at the point \((x_i,x_i)\). If we move vertically to the graph of the function, we preserve the \(x\) coordinate but change the \(y\) coordinate to \(f(x_i)\). Thus, we arrive at the point \((x_i,f(x_i)) = (x_i,x_{i+1})\). If we then move horizontally back to the line \(y=x\) we now preserve the \(y\) coordinate but change the \(x\) coordinate so that the two are the same. Thus, we arrive at the point \((x_{i+1},x_{i+1})\).

In summary: The process of moving vertically from a point on the line \(y=x\) to the graph of \(f\) and back to the line is a geometric representation of one application of the function \(f\).

Repeated application of this process represents repeated application of \(f\), i.e. iteration. Here's what that process looks like for a logistic function.

It turns out that the process is quite sensitive to the slope of the function at the point of intersection. Here's a slightly steeper logistic function.

You can explore this sensitivity with our cobweb tool.