The universal metric properties of nonlinear transformations

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(g_r(x/α), withg=lim_{r → ∞} g_r 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.