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.