Fractals and Chaos Recap for 11/28

We continued our discussion of the logistic map from yesterday by iterating the function for various values of the growth parameter a. We observed that if a is too low, the population will die out and the destination of the orbit is zero. Once a passes 1, the population will eventually settle at some proportion of the maximum population; for example at a = 2.6, the orbit of iterations settles on a value of approximately 0.6154 (precisely, this is 8/13 of the possible maximum population). For a = 3.2, we observed a two-cycle of {0.5130,0.7995}, suggesting that the population here will year by year fluctuate between 51% and 80% of its possible maximum population. For a = 3.6, we observed two bands of chaos bound within (0.32,0.6) and (0.79,0.9), suggesting that the population never dies out, but never settles at a stable value (or set of values).

Tonight, please continue to explore values of a and the destinations of orbits within this graph. Use the apps at MathInsight.org to help, as well as this Logistic Function Cobweb Diagram I made in Desmos. Keep track of your observations on the Catalog of Behavior you got in class. By tomorrow, you should at least have an observation for every tenth value of a between 2.2 and 4.0.

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