Dynamics of the Harper map: Localized states, Cantor spectra and Strange nonchaotic attractors

Abstract

The Harper (or ``almost Mathieu'') equation plays an important role in studies of localization. Through a simple transformation, this equation can be converted into an iterative two dimensional skew--product mapping of the cylinder to itself. Localized states of the Harper system correspond to fractal attractors with nonpositive maximal Lyapunov exponent in the dynamics of the associated Harper map. We study this map and these strange nonchaotic attractors (SNAs) in detail in this paper. The spectral gaps of the Harper system have a unique labeling through a topological invariant of orbits of the Harper map. This labeling associates an integer index with each gap, and the scaling properties of the width of the gaps as a function of potential strength, $ε$ depends on the index. SNAs occur in a large region in parameter space: these regions have a tongue--like shape and end on a Cantor set on the line $ε= 1$ where the states are critically localized, and the spectrum is singular continuous. The SNAs of the Harper map are described in terms of their fractal properties, and the scaling behaviour of their power--spectra. These are created by unusual bifurcations and differ in many respects from SNAs that have hitherto been studied. The technique of studying a quantum eigenvalue problem in terms of the dynamics of an associated mapping can be applied to a number of related problems in 1~dimension. We discuss generalizations of the Harper potential as well as other quasiperiodic potentials in this context.