Quantitative universality for a class of nonlinear transformations View Full Text


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Article Info

DATE

1978-07

AUTHORS

Mitchell J. Feigenbaum

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. With 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 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. More... »

PAGES

25-52

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf01020332

DOI

http://dx.doi.org/10.1007/bf01020332

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1023847984


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