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 in which the range is a subset of the domain. Then, given any seed value, , we can produce an orbit, , by iterating the function. That is, , , etc. The usual notation is that (composition of with itself times). So .
A cobweb plot gives an easy way to quickly visualize an orbit of a dynamical system simply by looking at the graph . We construct it as follows.
Draw the graphs of and . Plug into the function. On the graph, draw a vertical line from on the -axis up to the point . Now is a -coordinate, but we want to plug it in as an -value, so draw a horizontal line over to the line . This will be the point . Now repeat. Draw a vertical line up to . Then draw a horizontal line over to . The -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 and 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 for different parameter values . For small values of every point in is a fixed point or is attracted to a fixed point. For larger you’ll see periodic orbits and chaotic behavior.