Quantitative universality for a class of nonlinear transformations

Abstract

A large class of recursion relationsx _{n + 1} = λf(x_n) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximum$\bar x$. With$f(\bar x) - f(x) \sim \left| {x - \bar x} \right|^z (for\left| {x - \bar x} \right|$ sufficiently small),z > 1, the universal details depend only uponz. In particular, the local structure of high-order stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratioα (α = 2.5029078750957... forz = 2). This structure is determined by a universal functiong ^*(x), where the 2ⁿth iterate off,f ⁽ⁿ⁾, converges locally toα ⁻ⁿ g ^*(α ⁿ x) for largen. For the class off's considered, there exists aλ _n such that a 2ⁿ-point stable limit cycle including$\bar x$ exists;λ _∞ −λ _n R~δ ⁻ⁿ (δ = 4.669201609103... forz = 2). The numbersα andδ have been computationally determined for a range ofz through their definitions, for a variety off's for eachz. We present a recursive mechanism that explains these results by determiningg ^* as the fixed-point (function) of a transformation on the class off's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.