# The universal metric properties of nonlinear transformations

Articles

- Received:

DOI: 10.1007/BF01107909

- Cite this article as:
- Feigenbaum, M.J. J Stat Phys (1979) 21: 669. doi:10.1007/BF01107909

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## Abstract

The role of functional equations to describe the exact local structure of highly bifurcated attractors of We conjecture that ℒ possesses a unique eigenvalue in excess of 1, and show that this δ is the λ-convergence rate. The form (

*x*_{n+1}=*λf(x*_{n}) independent of a specific*f*is formally developed. A hierarchy of universal functions*g*_{r}*(x)*exists, each descriptive of the same local structure but at levels of a cluster of 2^{r}points. The hierarchy obeys*g*_{r−1}*(x)*=−*αg*_{r}(g_{r}(x/*α*), with*g*=lim_{r → ∞}g_{r}existing and obeying*g(x)*= −*αg*(g(x/*α*), an equation whose solution determines both*g*and*α*. For*r*asymptotic*g*_{r}∼ g − δ^{−r}*h*^{*}where δ > 1 and*h*are determined as the associated eigenvalue and eigenvector of the operator ℒ:$$\mathcal{L}\left[ \psi \right] = - \alpha \left[ {\psi \left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right) + g'\left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right)\psi \left( {{{ - x} \mathord{\left/ {\vphantom {{ - x} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right]$$

^{*}) is then continued to all*λ*rather than just discrete*λ*_{r}and bifurcation values*Λ*_{r}and dynamics at such*λ*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 the*g*_{r}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 to*g*_{r}'s, thereby providing some elementary approximation schemes for obtaining*λ*_{r}for a chosen*f*.### Key words

Recurrencebifurcationattractoruniversalfunctional equationsscalingconjugacyspectrum of linearized operator## Copyright information

© Plenum Publishing Corporation 1979