Analytics of period doubling

Approximate analytic solutions for periodic orbits of the quadratic mapxrx(1–x) are developed using algebraic methods. These solutions form the basis of an exact algorithm which predicts the quantitative order of periodic points characteristic of the Feigenbaum scenario. The algorithm holds for any one dimensional unimodal map. A general procedure is developed which permits calculation of period doubling parameters for large period orbits from those of low period to any desired degree of accuracy. Explicit equations are given through second order.