The logistic function f(x) = r x (1 - x) is well known in dynamical systems theory. So is the famous bifurcation diagram for it.
I recently came upon an article in Mathematics Magazine from 1996 showing that the 3-cycle first appears at r = 1 + sqrt(8). (So the left edge of the 3-cycle "window" is 1 + sqrt(8).) In fact there are two separate articles in that issue (April 1996) giving elementary proofs. I especially like the second, which boils down to a system of two polynomial equations in two variables. Eliminating one yields a quadratic equation, and it is easy to see then that the smallest possible r is 1 + sqrt(8).
It strikes me that this method could be generalized to n-cycles. For n= 5, 7, 11, etc we would get a system of n polynomial equations. Finding the resultant should solve it, though the resultant will probably not be quadratic.
Has this been done? Is anyone aware of an approach like this?
Harmonic structure of one-dimensional quadratic maps G. Pastor, M. Romera, and F. Montoya Instituto de Fı ́sica Aplicada, Consejo Superior de Investigaciones Cientı ́ficas, Serrano 144, 28006 Madrid, Spain ͑Received 14 November 1996
Look also for other papers by Pastor and Romera
HTH