Critical States and Fractal Attractors in Fractal Tongues: Localization in the Harper map

Abstract

Localized states of Harper's equation correspond to strange nonchaotic attractors (SNAs) in the related Harper mapping. In parameter space, these fractal attractors with nonpositive Lyapunov exponents occur in fractally organized tongue--like regions which emanate from the Cantor set of eigenvalues on the critical line $\epsilon =1$. A topological invariant characterizes wavefunctions corresponding to energies in the gaps in the spectrum. This permits a unique integer labeling of the gaps and also determines their scaling properties as a function of potential strength.

11 Pages, 5 figures(three eps format and two in gif format)

References in corpus (3)

• Strange Nonchaotic Attractors
• Collision and symmetry-breaking in the transition to strange nonchaotic attractors
• On the singular spectrum of the Almost Mathieu operator. Arithmetics and Cantor spectra of integrable models

Cited by in corpus (5)

• How chaotic are strange nonchaotic attractors
• Strange Nonchaotic Attractors in Harper Maps
• Torus Fractalization and Intermittency
• Local symmetry theory of resonator structures for the real-space control of edge states in binary aperiodic chains
• Comment: "Critical states and fractal attractors in fractal tongues: Localization in the Harper map" [Phys. Rev. E64 (2001) 045204]