We explored cycles a bit more in class today, noting that if {*p,q*} are the two values of a 2-cycle, then by definition *f *(*p*) = *q* and *f *(*q*) = *p*. Put another way, *f *(*f *(*p*)) = *p* and *f *(*f *(*q*)) = *q.* What’s important to note, though, is that the reverse is **not true**. If *f *(*f *(*n*)) = *n*, then *n* could be a member of a two cycle (*n,m,n,m,n,m,…*) **or** it could just be a fixed point: (*n,n,n,n,n,n,…*)

One important takeaway from this is that the first is that the local slope (Instantaneous Rate of Change) to all points of a 2-cycle, or indeed any cycle, is always the same. As a result, all points of a cycle will pass an IRC of -1 simultaneously, meaning all points of a cycle bifurcate simultaneously. This (and the Chain Rule from calculus) served as our proof of Question 2 posed early in our exploration of this pattern.

The implication of this is clear: after *a* = 3.0, the pattern of the orbit of the logistic function bifurcates to a two cycle. By *a* = 3.5 (more specifically at *a* = 3.449490…), it has bifurcated again to a four-cycle. Soon after (at *a* = 3.544090…), it bifurcates yet again to an eight-cycle, then to a 16-cycle (at *a* = 3.564407…), a 32-cycle (at *a* = 3.568759…), and so on. These bifurcations happen over increasingly shorter intervals for values of *a*. The fascinating thing is that the ratio between these intervals approaches a constant value, approximately 4.669202…, known as Feigenbaum’s Constant (this link goes to a Numberphile video about this idea that we watched a portion of in class today; the picture of orbit destinations that we have been drawing is known as the Feigenbaum Plot, which we will look at in detail tomorrow).