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,…*)

There were two important takeaways from this algebraic definition of a 2-cycle. 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 served as our proof of Question 2 posed early in our exploration of this pattern.

The second takeaway is graphing the function *y* = *f *(*f *(*x*)) can be a way of **finding new cycles**. Fixed points on the graph of *y* = *f *(*f *(*x*)) that are not common with the graph of *y = **f *(*x*) will be the parameters of our 2-cycle. By extension, graphing *y = f *(*f *(*f *(*x*))), or any number of nested iterations, will give us a tool of finding new cycles. More crucially, this also shows us **why cycles spontaneously appear from chaos**. Explore the graph here. For *a* = 3.84, the “wiggliness” of the graph of *y = f *(*f *(*f *(*x*))) is enough for the fingers of the graph to touch the line *y = x*. But for *a* < 3.84, it isn’t. The moment that *a* becomes large enough for those fingers to touch the line *y = x* is the moment that a 3-cycle is born!