## Beautiful theorems about dynamical systems on the plane

I was reading through some papers written by my Ph.D. advisor (John Franks) from the early 1990’s and was reminded of a few beautiful results about the dynamics of planar homeomorphisms. So I thought I’d share them here. For those of you who are not familiar with the terminology, a planar homeomorphism is a bijective…

## Irrational rotations of the circle and Benford’s law

Take a collection of real-world data such as the lengths of all rivers in the world, the populations of counties in the United States, the net worths of American corporations, or the street addresses of all residents of Detroit. Strip away all the information except the leading digits. What percentage of these digits do you…

## 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…

## Three cool facts about rotations of the circle

I was playing around with GeoGebra and made this applet about one of the simplest, but most intersting dynamical systems: the rigid rotation of a circle. Let me tell you a little about this fascinating subject. Let denote a circle. For simplicity, let’s think of it as a circle with circumference 1. Let be any…

## 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…

## Fractals on NOVA

I was shocked yesterday afternoon. I turned on the television to see what was on for my kids (PBS, of course) only to see, not Bob the Builder or Word World, but the Cantor set being constructed right before my eyes. It turned out that it was a new NOVA special called Hunting the Hidden…