Quantitative universality for a class of nonlinear transformations |
| |
Authors: | Mitchell J Feigenbaum |
| |
Institution: | (1) Theoretical Division, Los Alamos Scientific Laboratory, Los Alamos, New Mexico |
| |
Abstract: | A large class of recursion relationsx
n + 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
. 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 ratio ( = 2.5029078750957... forz = 2). This structure is determined by a universal functiong
*(x), where the 2nth iterate off,f
(n), converges locally to
–n
g
*(
n
x) for largen. For the class off's considered, there exists a
n such that a 2n-point stable limit cycle including
exists;
–
n R~
–n ( = 4.669201609103... forz = 2). The numbers and 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.Research performed under the auspices of the U.S. Energy Research and Development Administration. |
| |
Keywords: | Recurrence bifurcation limit cycles attractor universality scaling population dynamics |
本文献已被 SpringerLink 等数据库收录! |
|