  Classical and quantum dynamics of a harmonic oscillator in a monochromatic
wave is studied in the exact resonance and near resonance cases. This model
describes, in particular, a dynamics of a cold ion trapped in a linear ion trap
and interacting with two lasers fields with close frequencies. Analytically and
numerically a stability of the ``classical ground state'' (CGS) -- the vicinity
of the point ($x=0, p=0$) -- is analyzed. In the quantum case, the method for
studying a stability of the quantum ground state (QGS) is suggested, based on
the quasienergy representation. The dynamics depends on four parameters: the
detuning from the resonance, $\delta=\ell-\Omega/\omega$, where $\Omega$ and
$\omega$ are, respectively, the wave and the oscillator's frequencies; the
positive integer (resonance) number, $\ell$; the dimensionless Planck constant,
$h$, and the dimensionless wave amplitude, $\epsilon$. For $\delta=0$, the CGS
and the QGS are unstable for resonance numbers $\ell=1, 2$. For small
$\epsilon$, the QGS becomes more stable with increasing $\delta$ and decreasing
$h$. When $\epsilon$ increases, the influence of chaos on the stability of the
QGS is analyzed for different parameters of the model, $\ell$, $\delta$ and
$h$.
