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### Model: Feigenbaum Ratio: Change Comes Faster at a Constant Ratio

Bifurcation Diagram - Part 3 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:** Mathematically, Feigenbaum demonstrated that period doubling comes at a constant rate given by the delta constant: 4.66921166091029..., and that this constant arises in any dynamical system that approaches chaotic behavior via period-doubling bifurcations: fluid-flow turbulence, electronic oscillators, chemical reactions, and even the Mandelbrot set.

**Presentation:** In introductory presentations we do not go into the mathematics of this, but while we have a bifurcation diagram up it is easy to demonstrate that each bifurcation comes faster than the previous one. In

Power Point we just draw a series of vertical lines at each bifurcation and note that they come faster and faster (Figure 5).

**Anticipated Learning Outcome: **
10. All complex systems accelerate their rate of change—bifurcations—at the same rate. This is a universality property; if this rate of change in behavior of a system is detected, the system is a complex system.

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