Quantitative universality for a class of nonlinear transformations. (English) Zbl 0509.58037

In this interesting paper, the asymptotic behavior of a class of recursion relations \((*)\) \(x_{n+1}=\lambda f(x_n)\) is studied from a quantitative viewpoint. The functions \(f\) which are considered have a unique differentiable maximum \(\overline x\) near which \((**)\) \(f(\overline x)-f(x)\sim|\overline x-x|^z\) for some \(z>1\). The exponent \(z\) determines the quantitative bifurcation and stable limit cycle behavior of \((*)\) for all \(f\) satisfying \((**)\). Included in the class described by \((**)\) is a class of functions studied earlier [N. Metropolis et al., J. Comb. Theory, Ser. A 15, 25–44 (1973; Zbl 0259.26003)]. Included in the relation \((*)\) are models of population behavior. Numerical results suggest that corresponding to \(z\), there exist generic constants relating the bifurcation points and the local behavior of the iterates \(y^{(2n)}\), \(n=0,1,\cdots\), where \(g^{(0)}(x)=\overline xf(x)\). The iterates \(g^{(2n)}\) become increasingly oscillatory and have increasingly many fixed points. (Note that it is possible to rescale so that \(f(\overline x)=1\), and when \(\lambda=\overline x\), \(\lambda f(\overline x)=\overline xf(\overline x)=\overline x\).) It is conjectured that \(\{g^{(2n)}\}\) converges to a universal function \(g^\ast\) which characterizes the behavior of \((*)\) for the class \((**)\). The present paper is primarily heuristic in nature. A rigorous discussion is to be given in a sequel.


37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37G99 Local and nonlocal bifurcation theory for dynamical systems


Zbl 0259.26003
Full Text: DOI

Online Encyclopedia of Integer Sequences:

Decimal expansion of Feigenbaum’s constant 0.399535...


[1] N. Metropolis, M. L. Stein, and P. R. Stein, On Finite Limit Sets for Transformations on the Unit Interval,J. Combinatorial Theory 15(1):25 (1973). · Zbl 0259.26003 · doi:10.1016/0097-3165(73)90033-2
[2] M. Feigenbaum, The Formal Development of Recursive Universality, Los Alamos preprint LA-UR-78-1155.
[3] B. Derrida, A. Gervois, Y. Pomeau, Iterations of Endomorphisms on the Real Axis and Representations of Numbers, Saclay preprint (1977). · Zbl 0416.28012
[4] J. Guckenheimer,Inventiones Math. 39:165 (1977). · Zbl 0354.58013 · doi:10.1007/BF01390107
[5] R. H. May,Nature 261:459 (1976). · Zbl 1369.37088 · doi:10.1038/261459a0
[6] J. Milnor, W. Thurston, Warwick Dynamical Systems Conference,Lecture Notes in Mathematics, Springer Verlag (1974).
[7] P. Stefan,Comm. Math. Phys. 54:237 (1977). · Zbl 0354.54027 · doi:10.1007/BF01614086
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.