Nonlinear Rotations on a 3-Dimensional Lattice

Abstract

The study of resonances of Hamiltonian systems with a divided phase space is a well-established area of research which attracted the interests of many researchers over the years. We study such resonances over a discrete space focusing on the rotational motion of some orbits in the 3-Dimensional lattice, ?3. In this study, we construct the discrete standard map from 2-Dimensional to 3-Dimensional lattices. Then, we identify and categorize any transformation that could give periodicity for the 3-Dimensional
lattice on a specific map. The study aims to determine the points that give the periodicity and at the same time investigate the behavior of the points for the nonlinear stable orbits over a discrete space.  For some choice of parameters ???? ?, our findings showed that the orbit of the 3-Dimensional map is periodic depending on the initial conditions. Some arbitrary initial conditions may be periodic in a 3-Dimensional lattice.
Keywords: Hamiltonian systems, nonlinear rotations, space discretization, arithmetic dynamics