Exercises in Iterational Asymptotics II

TL;DR

This work advances the theory of iterational asymptotics for nonlinear recurrences, focusing on oscillatory and monotone convergence near attractive fixed points. It employs a brute-force matching-coefficient approach to derive high-order asymptotic expansions when standard algorithms fail due to missing terms in the local nonlinearity, and it computes precise constants that govern subleading behavior. Key results include explicit asymptotics for the logistic-type map around the fixed point $x=2/3$, constants $C_{\mathrm{o}}$ and $C_{\mathrm{e}}$ for the even/odd subsequences, the constant $C$ in the monotone $y_k$ sequence for various polynomial truncations, and product-form representations yielding linear convergence rates for the cosine and logistic maps. The findings illuminate how small structural gaps in the nonlinear terms affect asymptotic corrections and demonstrate how high-precision constants can be extracted via brute-force coefficient matching, with potential implications for related iterated systems and fixed-point analyses.

Abstract

The nonlinear recurrences we consider here include the functions $3x(1-x)$ and $\cos(x)$, which possess attractive fixed points $2/3$ and $0.739...$ (Dottie's number). Detailed asymptotics for oscillatory convergence are found, starting with a 1960 paper by Wolfgang Thron. Another function, $x/(1+x\ln(1+x))$, gives rise to a sequence with monotonic convergence to $0$ but requires substantial work to calculate its associated constant $C$.