We did some more research in the logistic map and the catalog of behavior we’ve been using for the past few days. We’ve answered one question from yesterday’s list, the first one: The first bifurcation happens at precisely a = 3.0. With a proof in class, we showed that this is because for a = 3.0, the slope of the curve at the value of the fixed point (which happens to precisely equal 2/3) is exactly -1. Any value of a less than 3.0 will have a slope at its fixed point that is shallower than -1, and as a result the fixed point is an attractor. Any value of a greater than 3.0 will have a slope at its fixed point that is steeper than -1, and as a result the fixed point is a repeller. So when we notice that there’s a 2-cycle at a = 3.1, this is because the fixed point is a repeller, pushing the orbit to the 2-cycle. This 2-cycle exists for all values of 3.0 < a < 3.1.
As a clue to further revelations, I also provided the answer to question 2: In fact, the 2 elements of the 2-cycle split simultaneously, meaning that the 2-cycle splits directly to a 4-cycle (and furthermore that many students’ hunt for the elusive 3-cycle within the interval 3.4 < a < 3.5 is fruitless!).
Nobody has found a 3-cycle yet, or indeed any cycle that isn’t a power of 2 (we found an 8-cycle at a = 3.55, a 16-cycle shortly thereafter, and a 32-cycle shortly after that).
Your homework this weekend is to turn your attention to the chaos past a = 3.6 and to address question 3. Is there truly only chaos there? Or is there something else, lurking in the shadows…