Quantitative universality for a class of nonlinear transformations

@article{Feigenbaum1978QuantitativeUF,
  title={Quantitative universality for a class of nonlinear transformations},
  author={Mitchell J. Feigenbaum},
  journal={Journal of Statistical Physics},
  year={1978},
  volume={19},
  pages={25-52}
}
  • M. Feigenbaum
  • Published 1 July 1978
  • Mathematics
  • Journal of Statistical Physics
AbstractA 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 $$\bar x$$ . 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… 
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References

SHOWING 1-10 OF 10 REFERENCES
On the bifurcation of maps of the interval
In the last few years there has been considerable interest in the asymptotic behavior of maps of the interval into itself under iteration. Some of this interest has come from the theory of dynamical
A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line
Two theorems are proved—the first and the more important of them due to Šarkovskii—providing complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT
On Finite Limit Sets for Transformations on the Unit Interval
TLDR
An infinite sequence of finite or denumerable limit sets is found for a class of many-to-one transformations of the unit interval into itself and the structure and order of occurrence is universal for the class.
Simple mathematical models with very complicated dynamics
  • R. May
  • Computer Science, Medicine
    Nature
  • 1976
TLDR
This is an interpretive review of first-order difference equations, which can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations.
Pomeau, Iterations of Endomorphisms on the Real Axis and Representations of Numbers, Saclay
  • 1977
Comm. Math. Phys
  • Comm. Math. Phys
  • 1977
Iterations of Endomorphisms on the Real Axis and Representations of Numbers
  • Iterations of Endomorphisms on the Real Axis and Representations of Numbers
  • 1977
For A 0 < h < 1, hf(x) has two fixed points [x* = 0, and some other x* ~ (x, 1)] both of which are repellant
  • For A 0 < h < 1, hf(x) has two fixed points [x* = 0, and some other x* ~ (x, 1)] both of which are repellant
In the interval N about 9~ such that [f'(x)[ < 1,fis concave down- ward
  • In the interval N about 9~ such that [f'(x)[ < 1,fis concave down- ward
The Formal Development of Recursive Universality, Los Alamos preprint LA-UR-78-1155
  • The Formal Development of Recursive Universality, Los Alamos preprint LA-UR-78-1155