  Within the standard Lagrangian and Hamiltonian setting, we consider a
position-dependent mass (PDM) classical particle performing a damped driven
oscillatory (DDO) motion under the influence of a conservative harmonic
oscillator force field $V\left( x\right) =\frac{1}{2}\omega ^{2}Q\left(
x\right) x^{2}$ and subjected to a Rayleigh dissipative force field
$\mathcal{R}\left( x,\dot{x}\right) =\frac{1}{2}b\,m\left( x\right)
\dot{x}^{2}$ in the presence of an external periodic (non-autonomous) force
$F\left( t\right) =F_{\circ }\,\cos \left( \Omega t\right) $. Where, the
correlation between the coordinate deformation $\sqrt{Q(x)}$ and the velocity
deformation $\sqrt{m(x)}$ is governed by a point canonical transformation
$q\left( x\right) =\int \sqrt{m\left( x\right) }dx=\sqrt{% Q\left( x\right)
}x$. Two illustrative examples are used: a non-singular PDM-DDO, and a
power-law PDM-DDO models. Classical-states $\{x(t),p(t)\}$ crossings are
analysed and reported. Yet, we observed/reported that as a classical state
$\{x_{i}(t),p_{i}(t)\}$ evolves in time it may cross itself at an earlier
and/or a latter time/s.
