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 f requency. We analyze this map in the case of high and low relative frequenc y of the field and transition from regular to chaotic behavior. We show tha t the map at aphelion is suitable for investigation of transition to chaoti c behavior also in the low frequency held and even for adiabatic ionization when the strength of the external held is comparable with the Coulomb held . Moreover, the approximate analytical criterion (taking into account the e lectron's energy increase by the influence of the field) yields a threshold field strength quite close to the numerical results. We reveal that transi tion 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 held the Kepler ma p 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 held strength of static field ionization.