## Geogebra applet for families of discrete dynamical systems

As I mentioned recently, I taught the last two weeks of my colleague’s differential equations course. The topic was discrete dynamical systems. I posted links to a few Geogebra applets that I made, namely, applets for illustrating one-dimensional dynamical systems and an applet to generate cobweb plots for the logistic map. This is my third and…

## Applet for discrete dynamical systems

I’m teaching the last two weeks of one of my colleagues’ Differential Equations course. I’m leading the class through a chapter on discrete dynamical systems. In preparation for the first lecture I created a couple of java applets using Geogebra. I thought others might be interested in them, so I’m linking to them here. The…

## 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…