Harper equation, the dissipative standard map and strange nonchaotic attractors: Relationship between an eigenvalue problem and iterated maps

The almost periodic eigenvalue problem described by the Harper equation is connected to other classes of quasiperiodic behavior; the dissipative dynamics on critical invariant tori and quasiperiodically driven maps. Firstly, the strong coupling limit of the supercritical Harper equation and the strong dissipation limit of the critical standard map play equivalent role in describing the universal characteristics of these systems. Secondly, a simple transformation is used to relate the Harper equation to a quasiperiodically forced one-dimensional map. In this case, the localized eigenstates of the supercritical Harper equation correspond to strange but nonchaotic attractors of the driven map. Furthermore, the existence of localization in the eigenvalue problem is associated with the appearance of homoclinic points in the corresponding map.