Articles

Journal of Statistical Physics

, Volume 21, Issue 6, pp 669-706

The universal metric properties of nonlinear transformations

  • Mitchell J. FeigenbaumAffiliated withTheoretical Division, Los Alamos Scientific Laboratory, University of California

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Abstract

The role of functional equations to describe the exact local structure of highly bifurcated attractors ofx n+1 =λf(x n ) independent of a specificf is formally developed. A hierarchy of universal functionsg r (x) exists, each descriptive of the same local structure but at levels of a cluster of 2 r points. The hierarchy obeysg r−1 (x)=−αg r(gr(x/α), withg=limr → ∞ gr existing and obeyingg(x) = −αg(g(x/α), an equation whose solution determines bothg andα. Forr asymptoticg r ∼ g − δ−r h * 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 discreteλ r and bifurcation valuesΛ r 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 theg 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 tog r 's, thereby providing some elementary approximation schemes for obtainingλ r for a chosenf.

Key words

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