Periodicity of a discrete standard map on 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. One can study such resonances over a discrete space. This paper focuses on the rotational motion of the orbits in the 3-dimensional lattice, ℤ3. We construct discrete maps from 2 dimensional to 3-dimensional. We consider the transformations that give periodicity to the 3-dimensional space. On a specific value of initial conditions and parameters alpha and beta, the orbit of the 3-dimensional map is periodic while the rest are non-periodic. Some observations on the reduction from a 3 dimensional map to a 2-dimensional by using principal component analysis (PCA) are investigated analytically in this paper.