Quantitative universality for a class of nonlinear transformations
Abstract
A large class of recursion relations x _{n + 1} = λ f(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 maximumbar x. Withf(bar x)  f(x) ̃  {x  bar x} ^z (for {x  bar x}  sufficiently small), z > 1, the universal details depend only upon z. In particular, the local structure of highorder stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratio α (α = 2.5029078750957... for z = 2). This structure is determined by a universal function g ^{*}( x), where the 2^{n}th iterate of f, f ^{(n)}, converges locally to α ^{n} g ^{*}( α ^{n} x) for large n. For the class of f's considered, there exists a λ _{n} such that a 2^{n}point stable limit cycle includingbar x exists; λ _{∞}  λ _{n} R δ ^{n} ( δ = 4.669201609103... for z = 2). The numbers α and δ have been computationally determined for a range of z through their definitions, for a variety of f's for each z. We present a recursive mechanism that explains these results by determining g ^{*} as the fixedpoint (function) of a transformation on the class of f's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.
 Publication:

Journal of Statistical Physics
 Pub Date:
 July 1978
 DOI:
 10.1007/BF01020332
 Bibcode:
 1978JSP....19...25F
 Keywords:

 Recurrence;
 bifurcation;
 limit cycles;
 attractor;
 universality;
 scaling;
 population dynamics