One-Dimensional Dynamical Systems

Table of Contents:

• Part 1: Introduction
  
• Warm Up Questions
  
• Part 2: Logistic Growth Model
  
• Part 3: Iteration
  
• Part 4: Linear and Nonlinear Behaviour
  
• Part 5: Bifurcation
• Questions on One-Dimensional Dynamical Systems

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 x₀.

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

• Chaos (in Dutch), edited by A.W. Grootendorst, CWI Syllabus 41, Stichting Mathematisch Centrum, Amsterdam, 1996.
  
• An Introduction to Chaotic Dynamical Systems (second edition), by Robert L. Devaney, Addison-Wesley, 1987.
  
• notes used in Math5337, Spring 1995 by Evelyn Sander.