Analytics of period doubling
For mappings \(F_ 0(x)=rx(1-x)\) of the interval [0,1] (0\(\leq r\leq 4)\) it is well known that there are parameter values \(r_ 2<r_ 4<...<r_{2^ k}<...\to r_{\infty}\) such that for \(r\in (r_ p,r_{2p}]\) there is a cycle of \(p=2^ k\) periodic points \(a_ n^{(p)}(r)\) \((n=1,...,p)\). Here an approximative formula for the values \(a_ n^{(p)}(r)\) is presented which consists of k summands each of which allows two values. This is used for an approximate determination of \(r_ p\) (including \(r_{\infty})\) and the parameters \(r^*_ p\subset (r_ p,r_{2p})\) at which the cycle becomes superstable (i.e. \((dF_ 0^ p/dx)(a_ n^{(p)}(r^*_ p))=0).\) Further remarks concern, among others, the relation of the order in the cycle \(\{a_ n^{(p)}(r)\}_ n\) given by the iteration of \(F_ 0\) and the numerical order.
scientific article; zbMATH DE number 4096420