Briggs, Keith A precise calculation of the Feigenbaum constants. (English) Zbl 0744.58053 Math. Comput. 57, No. 195, 435-439 (1991). Let \(f_{\mu,z}(x)=1-\mu| x|^ z\). If for a particular parameter value there exists a stable \(p\)-cycle, then as \(\mu\) is increased the period will be observed to double, so that a stable \(2p\)- cycle appears. Denote the critical \(\mu\) value at which the \(2^ j\) cycle first appears by \(\mu_ j\) and let \(d_ j\) be the value of the nearest cycle element to 0 in \(2^ j\) cycle. The author calculates to high precision the Feigenbaum constants \[ \delta_ z:=\lim_{j\to \infty}(\mu_ j-\mu_{j-1})/(\mu_{j+1}-\mu_ j)\hbox{ and } \lim_{j\to \infty}d_ j/d_{j+1} \] associated with period doubling bifurcations for maps \(f_{\mu,z}\) for integers \(2\leq z\leq 12\). Multiple-precision floating-point techniques are used to find a solution of Feigenbaum’s functional equation, and hence the constants. Reviewer: M.C.Zdun (Kraków) Cited in 5 Documents MSC: 37G99 Local and nonlocal bifurcation theory for dynamical systems 37B99 Topological dynamics 39B12 Iteration theory, iterative and composite equations 26A18 Iteration of real functions in one variable 11Y60 Evaluation of number-theoretic constants Keywords:Feigenbaum constants; Feigenbaum functional equation iteration; bifurcation; stable \(p\)-cycle; asymptotically periodic sequence PDFBibTeX XMLCite \textit{K. Briggs}, Math. Comput. 57, No. 195, 435--439 (1991; Zbl 0744.58053) Full Text: DOI Online Encyclopedia of Integer Sequences: Decimal expansion of Feigenbaum bifurcation velocity. Decimal expansion of Feigenbaum reduction parameter.