In this paper classical nonlinear dynamics approaches are used to study the intense field stabilization (IFS) for ID model atom according to differential equation (DE) and map approximation (MA). The exact stability boundary and region of 1-period fixed points in the parameter space of field strength and frequency are obtained, and the results of MA are in excellent agreement with those of DE. The electrons dynamical behavior in phase space is investigated and the IFS mainly occurring in high frequency regime is confirmed. The stabilization phenomenon connected with the stable fixed points is also examined. In particular it is shown that this phenomenon depends on tightly the KAM tori. It can be predicted that the stabilization probability is just the fraction of initial electrons confined by the KAM tori.