Teaching Chaos and Complex Evolutionary Systems Theories at the Introductory Level

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Model: Bifurcation Diagram -
Generating the Diagram

(Part 1 of 4 parts)

Strategies and Rubrics for Teaching Chaos and Complex Systems Theories as Elaborating, Self-Organizing, and Fractionating Evolutionary Systems,

Fichter, Lynn S., Pyle, E.J., and Whitmeyer, S.J., 2010, Journal of Geoscience Education (in press)
Description: It is difficult to sense the behavior and properties of a system through many values of r with a dozen or dozen and a half time series diagrams, so we convert them into a bifurcation diagram (Power Point). The horizontal time series axis of the graph is replaced with r values with the population sizes plotted at successively higher values of r; the vertical axis remains population size. Figure 2 was created using the DOS Xnextbif.exe program to generate the data, which was then imported into Grapher to create the bifurcation diagram (any program that generates a graph will work, but we have found that spread sheet programs do not generate very good graphs). (Lab experiment with instructions on how to generate the bifurcation diagram.) Bifurcation is also called period doubling.

Presentation: A major point here is for students to understand different graphic representations, and in this case how the transformation is made from a time series diagram to a bifurcation diagram. This diagram is used multiple times to demonstrate other properties of chaos/complex systems. With Power Point animations (link below) we make this transformation in class, and it takes 5 minutes or less.

Anticipated Learning Outcome:
    5. Bifurcations are a change in the behavior of the system and the entire range of behaviors can be summarized in one diagram. Eric Newman, at his MyPhysicsLab web site has an applet of a chaotic pendulum. In just a couple of minutes we can demonstrate that bifurcations to complexity occur in a system that, going back to Galileo, is often taken as one of the best examples of a classical system (Lab experiment for exploring the chaotic pendulum linked below.)
    6. The harder a system is pushed, the higher the r value, the more unstable and unpredictable its behavior becomes.
         Unlike classical systems which can be described as simple, predictable, and with gradual change that goes to equilibrium, complex systems are ambiguous, unpredictable, and undergo sudden changes (bifurcations). Equilibrium means the system is dead.

Figure 2 - Bifurcation Diagram


- a DOS program that generates the bifurcation data file

Lab Exercise: X next Bifurcation

- an exercise used in GEOL 200 - Evolutionary Systems that explains how to generate the data file and convert it into a bifurcation diagram using Grapher.

Lab Exercise: The Driven Chaotic Pendulum

Power Point: Generating a Bifurcation Diagram

- a Power Point series of slides showing how the bifurcation diagram is generated from a series of time series diagrams.

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