Golden mean renormalization for the Harper equation: The strong coupling fixed point

Abstract

A simpler derivation of Feigenbaum's renormalization group equation for the period-doubling bifurcation sequence Am.We construct a renormalization fixed point corresponding to the strong coupling limit of the golden mean Harper equation. We give an analytic expression for this fixed point, establish its existence and uniqueness, and verify properties previously seen only in numerical calculations. The spectrum of the linearization of the renormalization operator at this fixed point is also explicitly determined. This strong coupling fixed point also helps describe the onset of a strange nonchaotic attractor in quasiperiodically forced systems.

Renormalization analysis of correlation properties in a quasiperiodically forced two-level system

“…As in studies of the self-similar fluctuations of the localized eigenstates of the Harper equation [7], the birth of a strange nonchaotic attractor [9], and of the autocorrelation function of a strange nonchaotic attractor [4], this self-similarity is explained by means of a functional recurrence, the key to the understanding of which is the dynamics of a simple piecewise-linear map of the interval [11,12,13]. …”

“…In [11] we proved that there exists a fixed point of the multiplicative version of (1.1) of the type numerically found in [7]. (To do this at times we considered the additive recurrence (1.1) in our proof.)…”

Strange nonchaotic attractors in Harper maps

“…10,11,14,21,22 As we have seen, we do not use localization to prove the existence of SNA. Besides, localization may not hold in all the Harper maps considered here, because an energy a in the spectrum with nonzero Lyapunov exponent ͑for which the Harper map has a SNA͒ may not be an eigenvalue of the operator.…”

“…These two hypotheses lie at the heart of many heuristic arguments for establishing the existence of SNA in Harper maps. 10,11,14,21,22 …”

“…[5][6][7][8][9] Our goal in this paper is to prove the existence and abundance of strange nonchaotic attractors ͑SNA͒ in the family of Harper maps. This is a family of 1D quasiperiodically forced maps that many authors have suggested, based on numerical experiments and heuristic arguments, as a scenario in which SNA appear, [10][11][12][13][14][15] but proofs are scarce. This equation also arises naturally from problems in mathematical physics and it is a paradigm of a 1D quasiperiodically forced map.…”