Commensurate cyclotron and axial oscillations (${\Omega }_z/{\Omega }_c=p/q$)

A rational ratio of axial and cyclotron frequencies makes a nondegenerate combined orbit close after an integer number of turns. For ratio $p/q$ in lowest terms it closes after $q$ transverse periods and $p$ axial periods. An integer ratio is a sufficient special case. An irrational ratio instead gives quasiperiodic motion when both component amplitudes are nonzero.