Quantitative universality for a class of nonlinear transformations

@article{Feigenbaum1978QuantitativeUF,
  title={Quantitative universality for a class of nonlinear transformations},
  author={M. Feigenbaum},
  journal={Journal of Statistical Physics},
  year={1978},
  volume={19},
  pages={25-52}
}
  • M. Feigenbaum
  • Published 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… Expand
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References

SHOWING 1-10 OF 10 REFERENCES
On the bifurcation of maps of the interval
  • 123
  • PDF
On Finite Limit Sets for Transformations on the Unit Interval
  • 487
  • PDF
Simple mathematical models with very complicated dynamics
  • R. May
  • Computer Science, Medicine
  • Nature
  • 1976
  • 5,431
  • PDF
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