## 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 in which the range is a…

## Sines and cosines (part 2)

In my previous post I asked the following question. What -values satisfy the equation ? More generally, let  and . Let be the composition of with itself times. Similarly, let be the composition of with itself times. What are the solutions to ? First, let us look at some properties of the iterates of and . is periodic with…

## Sharkovsky’s theorem

In this post I would like to share one of the most surprising, remarkable, and beautiful results in the study of discrete dynamical systems. It relates to an unusual ordering of the positive integers: First, some definitions. The basic object of study in a discrete dynamical system is the orbit. Let be a function from…