We make use of iterative Bogoliubov transformations (IBT) and employ a procedure derived by Mundarain et al. to derive an approximate analytic expression for the evolution operator of a system consisting of a quartic oscillator coupled to a time-dependent electric field. As a result, this expression resumes the whole iterative process as a global squeezing and displacement transformation in which the compression coefficients and translation parameters are obtained explicitly, thus giving a complete analytic structure to the process that supports and complements the conventional IBT and the bilinear approximation (Echave, J., et al. J Chem Phys 1990, 92, 1188), which are mainly numerical and semiclassic approaches respectively.