ISSN:
1572-9613
Keywords:
Recurrence
;
bifurcation
;
limit cycles
;
attractor
;
universality
;
scaling
;
population dynamics
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract A large class of recursion relationsx n + 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 −n g *(α n x) for largen. For the class off's considered, there exists aλ n such that a 2n-point stable limit cycle including $$\bar x$$ exists;λ ∞ −λ n 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01020332
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