1. 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)

2. Converting a Time Series Diagram Into a Bifurcation Diagram

3. This axis was a time series, but becomes . . .
Population
Size
= X
This axis was a time series, but becomes . . .
r Value
r = 2.9
r = 2.7
X = .655
X = .629

4. It would be reasonable for population sizes in the gap to fall between those for r = 2.7 and r = 2.9
And for population size to drop as ‘r’ drops
r = 2.9
r = 2.7
X = .655
X = .629

5. split
split
split
r = 3.3
r = 3.5
X = .48 & .82
X = .50, .87, .38, .82

6.
r = 3.8

7. Very Complex Behavior Very Simple Behavior Bifurcation Diagram
Modeling an Evolutionary System
Bifurcation Diagram
Population Size
3rd
Bifurcation
A Bifurcation is a change in basic behavior of a system
2nd
Bifurcation
1st
Bifurcation
Very
Complex
Behavior
Very Simple
Behavior
“r” Values – Rate of Growth

8. Bifurcation Diagram Modeling an Evolutionary System Population Size
3rd
Bifurcation
Why are some systems stable, reliable, predictable, and others are not?
2nd
Bifurcation
1st
Bifurcation
Refrigerators
Computers
Cars
Airplanes
Weather
Stock Market
Human Behavior
Outcome of sport events
Weather
Stock Market
Human Behavior
Outcome of sport events
Because we engineer human-made systems to operate at low ‘r’
“r” Values – Rate of Growth

9. Learning Outcomes 5. Bifurcations 6. Instability increases with ‘r’
Bifurcations are a change in the behavior of the system, the entire range of behaviors or a particular system can be shown in one bifurcation diagram.
6. Instability increases with ‘r’
The harder the system is pushed, the higher the ‘r’ value the more unstable and unpredictable its behavior becomes. This is seen in the bifurcation cascade: settling to one population value, splitting to 2, then 4, then 8, etc. population values, and finally visiting so many population values a pattern cannot be seen.