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Journal of Statistical Physics

, Volume 19, Issue 1, pp 25–52 | Cite as

Quantitative universality for a class of nonlinear transformations

  • Mitchell J. Feigenbaum
Articles

Abstract

A large class of recursion relationsxn + 1 = λf(xn) 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−ng*(αnx) for largen. For the class off's considered, there exists aλn such that a 2n-point stable limit cycle including\(\bar x\) exists;λλn 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.

Key words

Recurrence bifurcation limit cycles attractor universality scaling population dynamics 

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Copyright information

© Plenum Publishing Corporation 1978

Authors and Affiliations

  • Mitchell J. Feigenbaum
    • 1
  1. 1.Theoretical DivisionLos Alamos Scientific LaboratoryLos Alamos

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