Interactive Chaos Tutorial gently introduces the user to discrete time dynamical systems and their possible modes of long-term behavior by investigating the logistic equation in the context of a locust population model. Interactive computational tools are provided at appropriate places. They feature both numeric and graphical output.
Type of Material:
Students could use this tutorial independently, or an instructor could build a classroom lesson around it.
The tutorial requires the Director Shockwave Plugin.
Identify Major Learning Goals:
The user should develop an enhanced understanding of the mathematical modeling process and of the meaning of chaotic behavior in a deterministic dynamical system.
Target Student Population:
High school or college math students.
Prerequisite Knowledge or Skills:
Users should understand basic algebra.
Evaluation and Observation
Overall, the presentation of the logistic equation x(n+1)=r*x(n)*(1-x(n)) in the context of a locust population model is very nice. The author points out the necessity of carefully defining assumptions in the modeling process, and steps the student through the development of chaotic behavior as r increases to four.
Three computational features are included that allow the user to study the effects of varying r, the effects of varying x(0), and the meaning of a bifurcation diagram. These are easy to use, but that latter two would likely need to be explained more fully than is done on the site in order to be effective.
There is some inconsistency in terminology that could confuse the user. The term ?birth rate? is used to describe both the logistic parameter r and the quantity r*(1-x(n)). Later, r is called the ?corn crop?. Although the corn crop is related to r, they are not the same thing.
Two links in the table of contents are backwards. The link to ?Exploring Effects of Changes in Initial Conditions? takes the user to the page for ?Exploring Effects of Changes in Corn Crop?, and vice versa.
Potential Effectiveness as a Teaching Tool
The treatment of the impact of varying the logistic parameter r is strong. Students will develop a clear understanding of this concept when using this tutorial with or without instructor guidance.
The features designed to explore bifurcation diagrams and the impact of varying the initial condition would need more explanation than is available on the site to be effective.
Ease of Use for Both Students and Faculty
Site navigation and the layout of the computational features are completely intuitive.