Kepler map
We present a consecutive derivation of mapping equations of motion for the one-dimensional classical hydrogenic atom in a monochromatic field of any frequency. We analyze this map in the case of high and low relative frequency of the field and transition from regular to chaotic behavior. We show that the map at aphelion is suitable for investigation of transition to chaotic behavior also in the low frequency field and even for adiabatic ionization when the strength of the external field is comparable with the Coulomb field. Moreover, the approximate analytical criterion (taking into account the electron's energy increase by the influence of the field) yields a threshold field strength quite close to the numerical results. We reveal that transition from adiabatic to chaotic ionization takes place when the ratio of the field frequency to the electron Kepler frequency approximately equals 0.1. For the dynamics and ionization in a very low frequency field the Kepler map can be converted to a differential equation and solved analytically. The threshold field of the adiabatic ionization obtained from the map is only 1.5% lower than the exact field strength of static field ionization.