  The multiphoton ionization of hydrogen atoms in a strong elliptically
polarized microwave field exhibits complex features that are not observed for
ionization in circular and linear polarized fields. Experimental data reveal
high sensitivity of ionization dynamics to the small changes of the field
polarization. The multidimensional nature of the problem makes widely used
diagnostics of dynamics, such as Poincar\'{e} surfaces of section, impractical.
We analyze the phase space dynamics using finite time stability analysis
rendered by the fast Lyapunov Indicators technique. The concept of
zero--velocity surface is used to initialize the calculations and visualize the
dynamics. Our analysis provides stability maps calculated for the initial
energy at the maximum and below the saddle of the zero-velocity surface. We
estimate qualitatively the dependence of ionization thresholds on the
parameters of the applied field, such as polarization and scaled amplitude.
