Fractals & Chaos Recap for 11/29

We did some more work collecting information for the Catalog of Behavior for the Logistic Map function we introduced on Monday, getting a clearer idea on the changes in behavior as a increases towards 4.0. The patterns we observed raised a few questions:

  1. Somewhere within the interval 3.0 < a < 3.1, the single fixed point attractor “bifurcates” (i.e., splits) into a 2 cycle. Where exactly does this happen?
  2. Later, the 2-cycle becomes a 4-cycle. Do all points of the cycle bifurcate simultaneously, or can one split before another? In other words, is there a 3-cycle within the interval 3.4 < a < 3.5, or does the 2-cycle split directly to a 4-cycle?
  3. Once chaos appears past a = 3.6, is that it? Or is there anything more to be found there?

Continue your explorations of this function to try and answer these questions. Use the applets posted yesterday. I have a special prize for the first person who can find a 3-cycle…

 

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.

Fractals & Chaos Recap for 11/27

We took some time today to finish looking at the functions in question 5 of the Iterated Functions sheet, exploring different cases of what happens with the slope of a curve immediately at its fixed point is precisely 1.

From there, we moved to a different family of functions defined by the graph y = ax(1-x). This function is called a logistic map, and it represents the growth of a capped population where x represents the ratio of the current population to a maximum sustainable population (as defined by the carrying capacity), and a is known as a fecundity rate. Naturally, both x and y are bound within the interval (0,1), as a population that reaches 0 is extinct and a population that reaches 1 will necessarily exceed its carrying capacity and become doomed. Your homework tonight is to consider this equation and those limits and to derive the boundaries for a.