## 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.
Show Scanned Page ### MSC:

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

Zbl 0259.26003
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### Online Encyclopedia of Integer Sequences:

Decimal expansion of Feigenbaum’s constant 0.399535...

### References:

  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  M. Feigenbaum, The Formal Development of Recursive Universality, Los Alamos preprint LA-UR-78-1155.  B. Derrida, A. Gervois, Y. Pomeau, Iterations of Endomorphisms on the Real Axis and Representations of Numbers, Saclay preprint (1977). · Zbl 0416.28012  J. Guckenheimer,Inventiones Math. 39:165 (1977). · Zbl 0354.58013  R. H. May,Nature 261:459 (1976). · Zbl 1369.37088  J. Milnor, W. Thurston, Warwick Dynamical Systems Conference,Lecture Notes in Mathematics, Springer Verlag (1974).  P. Stefan,Comm. Math. Phys. 54:237 (1977). · Zbl 0354.54027
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