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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
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