**Up:** *EMAT33100: Nonlinear Dynamics & Chaos*

# One-Dimensional Dynamical Systems

The study of one-dimensional discrete dynamical systems gives a
new interpretation to the investigation of functions defined on the
real line, namely the concept of iteration. Explorations are done
using the package Chaos.mpl in Maple. It is possible to use Java applets instead of Maple for
this course. These Java applets
are specified below.

## Table of Contents:

## Java Applets

John McDonald made the
Java applet Iterates versus
time for the Logistic map. You can set the number of iterates
being shown and use sliders to vary the parameter
.

There are other Java applets available on the web that let you
explore the Logistic map. The applet
Iteration
of the Logistic Map of Andy Burbanks has
sliders for changing the parameter
and the starting
point *x*_{0}.

The second applet
The
Period Doubling Route to Chaos is written by Michael Cross. It
accepts typed input, and it graphs the first, second and fourth
iterates (and so on) of the logistic map.
Complete
instructions are given at the site, but you can also
start with the abstracted
instructions that show you how to do everything needed for this
lab.

An almost complete set is provided at Boston University by Bob Devaney's group:

- Nonlinear
Web lets you do cobwebbing and investigate graphs of higher
iterates for several different dynamical systems.
- Orbit
Diagram can compute bifurcation diagrams of these systems and also
lets you zoom in and investigate particular regions of the parameter
space or phase space in detail.

## References

One-Dimensional Dynamical Systems was designed using

**Up:** *EMAT33100: Nonlinear Dynamics & Chaos*

Copyright © 2001 by
Hinke Osinga

Last modified: Wed Nov 12 10:56:03 2003