  In this paper, we first analyze a parametric oscillator with both mass and
frequency time-dependent. We show that the evolution operator can be obtained
from the evolution operator of another parametric oscillator with a constant
mass and time-dependent frequency followed by a time transformation
$t\rightarrow\int_0^t dt'\,1/m(t')$. Then we proceed by investigating the
quantum dynamics of a parametric oscillator with unit mass and time-dependent
frequency in a Kerr medium under the influence of a time-dependent force along
the motion of the oscillator. The quantum dynamics of the time-dependent
oscillator is analyzed from both analytical and numerical points of view in two
main regimes: (i) small Kerr parameter $\chi$, and (ii) small confinement
parameter $k$. In the following, to investigate the characteristics and
statistical properties of the generated states, we calculate the
autocorrelation function, the Mandel $Q$ parameter, and the Husimi
$Q$-function.
