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Quantitative universality for a class of nonlinear transformations

Quantitative universality for a class of nonlinear transformations,10.1007/BF01020332,Journal of Statistical Physics,Mitchell J. Feigenbaum

Quantitative universality for a class of nonlinear transformations   (Citations: 565)
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A large class of recursion relationsxn + 1 = ?f(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 ratioa (a = 2.5029078750957... forz = 2). This structure is determined by a universal functiong*(x), where the 2nth iterate off,f(n), converges locally toa-ng*(anx) for largen. For the class off's considered, there exists a?n such that a 2n-point stable limit cycle including $$\bar x$$ exists;?8 -?n R~d-n (d = 4.669201609103... forz = 2). The numbersa andd 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.
Journal: Journal of Statistical Physics - J STATIST PHYS , vol. 19, no. 1, pp. 25-52, 1978
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