Incommensurate standard map

Abstract: We introduce and study the extension of the Chirikov standard map when the kick potential has two and three incommensurate spatial harmonics. This system is called the incommensurate standard map. At small kick amplitudes the dynamics is bounded by the isolating Kolmogorov-Arnold-Moser surfaces while above a certain kick strength it becomes unbounded and diffusive. The quantum evolution at small quantum kick amplitudes is somewhat similar to the case of Aubru-Andr\'e model studied in mathematics and experiments with cold atoms in a static incommensurate potential. We show that for the quantum map there is also a metal-insulator transition in space while in momentum we have localization similar to the case of 2D Anderson localization. In the case of three incommensurate frequencies of space potential the quantum evolution is characterized by the Anderson transition similar to 3D case of disordered potential. We discuss possible physical systems with such map description including dynamics of comets and dark matter in planetary systems.