Cobweb plots for the logistic map: a Geogebra applet

A few days ago I posted Geogebra applets illustrating discrete dynamical systems. I was using these in a differential equations lecture that I gave. In the next lecture I showed the students how to draw cobweb plots for 1-dimensional discrete dynamical systems.

A discrete dynamical system is a function f in which the range is a subset of the domain. Then, given any seed value, x_0, we can produce an orbit, x_0,x_1,x_2,\ldots, by iterating the function. That is, x_1=f(x_0), x_2=f(x_1)=f(f(x_0)), etc. The usual notation is that f^n=f\circ\cdots\circ f (composition of f with itself n times). So x_n=f^n(x_0).

A cobweb plot gives an easy way to quickly visualize an orbit of a dynamical system simply by looking at the graph y=f(x). We construct it as follows.

Draw the graphs of y=f(x) and y=x. Plug x_0 into the function. On the graph, draw a vertical line from x_0 on the x-axis up to the point (x_0,f(x_0))=(x_0,x_1). Now x_1 is a y-coordinate, but we want to plug it in as an x-value, so draw a horizontal line over to the line y=x. This will be the point (x_1,x_1). Now repeat. Draw a vertical line up to (x_1,f(x_1))=(x_1,x_2). Then draw a horizontal line over to (x_2,x_2). The x-coordinate of each vertical line is a point in the orbit.

A cobweb plot allows you to quickly spot attracting and repelling fixed points (fixed points can be found where the graphs y=f(x) and y=x cross). You can also see periodic orbits. Sometimes you can see chaotic behavior as well.

The Geogebra applet below generates cobweb plots for the family of logistic maps f(x)=kx(1-x) for different parameter values k. For small values of k every point in [0,1] is a fixed point or is attracted to a fixed point. For larger k you’ll see periodic orbits and chaotic behavior.