  The coherent state representations of the group $G = W_1 \otimes G_0$ (where
$G_0 = SU(2), SU(1,1)$) are used in computer simulation of the dynamics of
single two-level atom $(G_0 = SU(2))$ interacting with a quantized photon
cavity mode - the Jaynes - Cummings model (JCM) without the rotating wave
approximation and, in general, nonlinear in photon variables). The second case
(hyperbolic Jaynes - Cummings model (HJCM), $G_0 = SU(1,1))$ corresponds to the
quantum dynamics of quadratic nonlinear coupled oscillators (the parametric
resonance on double field frequency and a three - wave parametric processes of
nonlinear optics). Quasiclassical dynamical equations for parameters of
approximately factorizable coherent states for these models are derived and
regimes of motion for "atom" and field variables are analyzed.
