Supplements for our CMJ Article

The idea for the article began with a lab exercise Jody Sorensen and I designed for the students in our class on Chaotic Dynamical Systems. Small groups of students were each assigned a range of C values and asked to find the periodic points (up to period 8) for f(x)=x^2+c by purely graphical techniques: namely, graphing f ^ n to observe where it crossed the line y=x. The slope at such a crossing determined whether to corresponding periodic point was on an attracting or repelling cycle. The students collected all their data on a large sheet of graph paper, and plotted a very detailed bifurcation diagram. This lab, the resulting plot, and the subsequent computer program for drawing bifurcation diagrams automatically, became the subject of our article.

Details. With the exception of the plots in our first page below, all the diagrams here are screen-captured images from the DOS program "orbits", which I wrote in Turbo Pascal. Despite rave reviews, I've been writing a more user-friendly Windows version in Visual Basic; it was used in making the plots in the first page below. Both programs are capable of making these diagrams at a much higher resolution than anything seen in these pictures: close to 3000x3000 pixels, for printing purposes. The detail revealed in both diagrams at such resolutions is always amazing; it's unfortunate you can't see it on the screen.
On this page, we review and compare the bifurcation and orbit diagrams.
An HTML help file for the program "orbits".
This page has orbit and bifurcation diagrams which are labeled with helpful information. (The second page of our help file, it is independently useful for the general information it presents.)
Here we present two additional figures. The first superimposes the orbit diagram on top of the bifurcation diagram. The second shows just the bifurcation diagram, with the curves colored according to the prime period of the points on the curves.
Download a copy of the "orbits" program.
View the COLOR versions of most of the figures in the CMJ article.
Visit my colleague Jody's home page.

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© 2001 by Chip Ross
Associate Professor of Mathematics
Bates College
Lewiston, ME 04240