# The universal metric properties of nonlinear transformations

Ontology type: schema:ScholarlyArticle

### Article Info

DATE

1979-12

AUTHORS ABSTRACT

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. More... »

PAGES

669-706

### References to SciGraph publications

• 1978-07. Quantitative universality for a class of nonlinear transformations in JOURNAL OF STATISTICAL PHYSICS
• ### Journal

TITLE

Journal of Statistical Physics

ISSUE

6

VOLUME

21

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf01107909

DOI

http://dx.doi.org/10.1007/bf01107909

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1048384192

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