  In this work, we approach the problem of finding the zeros of a continuous
and monotone operator through a second-order dynamical system with a damping
term of the form $1/t^{r}$, where $r\in [0, 1]$. The system features the time
derivative of the operator evaluated along the trajectory, which is a
Hessian-driven type damping term when the governing operator comes from a
potential. Also entering the system is a time rescaling parameter $\beta(t)$
which satisfies a certain growth condition. We derive
$o\left(\frac{1}{t^{2r}\beta(t)}\right)$ convergence rates for the norm of the
operator evaluated along the generated trajectories as well as for a gap
function which serves as a measure of optimality for the associated variational
inequality. The parameter $r$ enters the growth condition for $\beta(t)$: when
$r < 1$, the damping $1/t^{r}$ approaches zero at a slower speed than
Nesterov's $1/t$ damping; in this case, we are allowed to choose $\beta(t)$ to
be an exponential function, thus having linear convergence rates for the
involved quantities. We also show weak convergence of the trajectories towards
zeros of the governing operator. Through a particular choice for the operator,
we establish a connection with the problem of minimizing a smooth and convex
function with linear constraints. The convergence rates we derived in the
operator case are inherited by the objective function evaluated at the
trajectories and for the feasibility gap. We also prove weak convergence of the
trajectories towards primal-dual solutions of the problem.
  A discretization of the dynamical system yields an implicit algorithm that
exhibits analogous convergence properties to its continuous counterpart.
  We complement our theoretical findings with two numerical experiments.
