0022-4715 1572-9613 Springer Nature Journal of Statistical Physics dimensions_id pub.1048384192 6 structure levels hierarchy https://doi.org/10.1007/bf01107909 1979-12-01 equations role point functional equation https://scigraph.springernature.com/explorer/license/ descriptive 669-706 local structure 2022-05-20T07:16 The universal metric properties of nonlinear transformations clusters article same local structure attractors The role of functional equations to describe the exact local structure of highly bifurcated attractors ofxn+1 =λf(xn) independent of a specificf is formally developed. A hierarchy of universal functionsgr(x) exists, each descriptive of the same local structure but at levels of a cluster of 2r points. The hierarchy obeysgr−1(x)=−αgr(gr(x/α), withg=limr → ∞ gr existing and obeyingg(x) = −αg(g(x/α), an equation whose solution determines bothg andα. Forr asymptoticgr ∼ g − δ−rh* where δ > 1 andh are determined as the associated eigenvalue and eigenvector of the operator ℒ:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{L}\left[ \psi \right] = - \alpha \left[ {\psi \left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right) + g'\left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right)\psi \left( {{{ - x} \mathord{\left/ {\vphantom {{ - x} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right]$$ \end{document} We conjecture that ℒ possesses a unique eigenvalue in excess of 1, and show that this δ is the λ-convergence rate. The form (*) is then continued to allλ rather than just discreteλr and bifurcation valuesΛr and dynamics at suchλ is determined. These results hold for the high bifurcations of any fundamental cycle. We proceed to analyze the approach to the asymptotic regime and show, granted ℒ's spectral conjecture, the stability of thegr limit of highly iterated λf's, thus establishing our theory in a local sense. We show in the course of this that highly iterated λf's are conjugate togr's, thereby providing some elementary approximation schemes for obtainingλr for a chosenf. en 1979-12 articles false doi 10.1007/bf01107909 Mathematical Sciences Feigenbaum Mitchell J. Pure Mathematics Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico Springer Nature - SN SciGraph project 21