Strange Attractors

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

4. Universality Properties of Complex Evolutionary Systems To Attract To cause to draw near or adhere by physical force: Magnetic poles are attracted to their opposites. 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. Potential energy Kinetic energy Equilibrium Point Attractor

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. Real Space and Phase Space Limit Cycle Attractor

7. Real Space and Phase Space Point Attractor

8. Time Series and Phase Space Diagrams Time Series Diagram Time Series Diagrams Point Limit Cycle Limit Cycle Strange Phase Space Diagrams

9. Strange Attractor r = 3.8

10. Universality Properties of Complex Evolutionary Systems Strange Attractors with Examples 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 Lorenz Strange Attractor

12. Chaotic Pendulum Limit Cycle Attractors in Phase Space

13. Chaotic Pendulum Strange Attractors in Phase Space

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

15. Examples of strange attractors

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

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

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