Journal of Statistical Physics

, Volume 21, Issue 6, pp 669–706

The universal metric properties of nonlinear transformations

• Mitchell J. Feigenbaum
Articles

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:
$$\mathcal{L}\left[ \psi \right] = - \alpha \left[ {\psi \left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right) + g'\left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right)\psi \left( {{{ - x} \mathord{\left/ {\vphantom {{ - x} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right]$$
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.

Key words

Recurrence bifurcation attractor universal functional equations scaling conjugacy spectrum of linearized operator

Preview

References

1. 1.
Mitchell J. Feigenbaum,J. Stat. Phys. 19:25 (1978).Google Scholar
2. 2.
P. Collet, J.-P. Eckmann, and O. E. Lanford III, Universal Properties of Maps on an Interval, in draft.Google Scholar
3. 3.
P. Collet and J.-P. Eckmann, Bifurcations et Groupe de Renormalisation, IHES/P/78/250.Google Scholar
4. 4.
B. Derrida, A. Gervois, and Y. Pomeau, Universal Metric Properties of Bifurcations of Endomorphisms, Saclay preprint (1977).Google Scholar
5. 5.
B. Derrida, A. Gervois, and Y. Pomeau, Iterations of Endomorphisms on the Real Axis and Representation of Numbers, Saclay preprint (1977).Google Scholar
6. 6.
N. Metropolis, M. L. Stein, and P. R. Stein,J. Combinatorial Theory 15:25 (1973).Google Scholar
7. 7.
K. Wilson and J. Kogut,Phys. Rep. 12C:75 (1974).Google Scholar