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. Strange Attractors

3. “r” Values – Rate of Growth Population Size Stabilizes to one size Oscillates between two sizes Oscillates among four sizes Oscillates among unlimited sizes, with fractal windows of stability Point Attractor Limit Cycle Attractors Strange Attractors

4. Properties of Complex Evolutionary Systems To Attract To cause to draw near or adhere by physical force: Magnetic poles are attracted to their opposites. Universality Potential energy Kinetic energy Equilibrium Point Attractor To draw by a physical force causing or tending to cause to approach, adhere, or unite; pull (opposed to repel): The gravitational force of the earth attracts smaller bodies to it.

5. Real Space and Phase Space Attractors In mathematics, an attractor is a region of phase space that "attracts" all nearby points as time passes. That is, the changing values have a trajectory which moves across the phase space toward the attractor, like a ball rolling down a hilly landscape toward the valley it is attracted to. Phase space is simply an X-Y or X-Y-Z graph on which the history of the system through time is seen as the changing coordinates proscribed by the changing variables of the system.

6. Limit Cycle Attractor Real Space and Phase Space

7. Point Attractor Real Space and Phase Space

8. Time Series Diagram Time Series Diagrams Time Series and Phase Space Diagrams Phase Space Diagrams PointLimit Cycle Strange

9. r = 3.8 Strange Attractor

10. Properties of Complex Evolutionary Systems Strange Attractors with Examples Lorenz Strange Attractor Universality A strange attractor is one in which we can see recognizable shapes in phase space, but the system never follows exactly the same trajectory through the phase space. In this Lorenz or butterfly strange attractor the system irregularly switches from one side to the other. A fountain is a strange attractor

12. Chaotic Pendulum Limit Cycle Attractors in Phase Space

13. Chaotic Pendulum Strange Attractors in Phase Space

14. http://www.stsci.edu/~lbradley/seminar/images/lorenz3d.gif Examples of strange attractors

16. http://www.thphys.uni-heidelberg.de/~gasenzer/pendelplots/pppoi_q=4_g=1.5.jpeg Examples of strange attractors

17. Properties of Complex Evolutionary Systems Strange Attractors - Turbulence Universality

18. Learning Outcomes 14. Strange Attractors Chaos/complex systems have behaviors that may superficially appear random, but have recognizable larger scale patterns.