  We investigate theoretically the dynamics of a Josephson junction in the
framework of the RSJ model. We consider a junction that hosts two supercurrrent
contributions: a $2\pi$- and a $4\pi$-periodic in phase, with intensities
$I_{2\pi}$ and $I_{4\pi}$ respectively. We study the size of the Shapiro steps
as a function of the ratio of the intensity of the mentioned contributions,
i.e. $I_{4\pi}/I_{2\pi}$. We provide detailed explanations where to expect
clear signatures of the presence of the $4\pi$-periodic contribution as a
function of the external parameters: the intensity AC-bias $I_\text{ac}$ and
frequency $\omega_\text{ac}$. On the one hand, in the low AC-intensity regime
(where $I_\text{ac}$ is much smaller than the critical current, $I_\text{c}$),
we find that the non-linear dynamics of the junction allows the observation of
only even Shapiro steps even in the unfavorable situation where
$I_{4\pi}/I_{2\pi}\ll 1$. On the other hand, in the opposite limit
($I_\text{ac}\gg I_\text{c}$), even and odd Shapiro steps are present.
Nevertheless, even in this regime, we find signatures of the
$4\pi$-supercurrent in the beating pattern of the even step sizes as a function
of $I_\text{ac}$.
