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 the first question from yesterday’s list: the first split 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.
We also have seen some progress on question 3, finding a 5-cycle at a = 3.906, a 7-cycle at a = 3.702, and our very first 3-cycle at a = 3.83. A follow-up question we could ask here: Can a cycle of any length be found for a > 3.6?
We still haven’t answered question 2: whether or not both points of the 2-cycle at a = 3.4 split simultaneously to form the four cycle found at a = 3.5, or if, for the briefest of moments, one point splits before the other and we can find another 3-cycle somewhere between 3.4 < a < 3.5. Your homework this break is to investigate this some more.