• DOI:10.1007/BF01020332
• Corpus ID: 124498882

Quantitative universality for a class of nonlinear transformations

@article{Feigenbaum1978QuantitativeUF,
  title={Quantitative universality for a class of nonlinear transformations},
  author={Mitchell J. Feigenbaum},
  journal={Journal of Statistical Physics},
  year={1978},
  volume={19},
  pages={25-52}
}

• M. Feigenbaum
• Published 1 July 1978
• Mathematics
• Journal of Statistical Physics

AbstractA 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;λ∞ −λn 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.