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**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.

### MSC:

37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

### Citations:

Zbl 0259.26003
Full Text:
DOI

### References:

[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 |

[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 |

[5] | R. H. May,Nature 261:459 (1976). · Zbl 1369.37088 |

[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 |

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