Symmetry structure of accelerator modes in the standard map

The regular motions in nonlinear Hamiltonian dynamical systems have been best studied in terms of the symmetry analysis based on the involution decomposition. As for the standard map, because of its intrinsic periodicity, there appear various orders of the accelerator modes, a special type of regular motions. Extension of the involution decomposition taking account of the periodicity makes it possible to investigate the symmetry structure of the accelerator modes. Explicit analysis has been carried out for the period-2, the period-3 and the period-4 accelerator modes. The present approach proves to be useful for other mappings with proper periodicity.