A spectrum of universality classes in period doubling and period tripling

Abstract

We have studied the functional dependence of the period-doubling and period-tripling universal bifurcation ratios α and δ on the order of the critical point z in the iterated map f;(χ) = 1 − a ¦χ¦ z , where z can be even, odd o r fractional. For the period-doubling sequence, α( z) is seen to be a monotonically decreasing function of z, and δ( z) a monotonically increasing function. For the period-tripling sequence, α( z) is still monotonically decreasing; however, δ( z) displays an interesting minimum at z = 2.