  In this paper, we aim to study non-convex minimization problems via
second-order (in-time) dynamics, including a non-vanishing viscous damping and
a geometric Hessian-driven damping. Second-order systems that only rely on a
viscous damping may suffer from oscillation problems towards the minima, while
the inclusion of a Hessian-driven damping term is known to reduce this effect
without explicit construction of the Hessian in practice. There are essentially
two ways to introduce the Hessian-driven damping term: explicitly or
implicitly. For each setting, we provide conditions on the damping coefficients
to ensure convergence of the gradient towards zero. Moreover, if the objective
function is definable, we show global convergence of the trajectory towards a
critical point as well as convergence rates. Besides, in the autonomous case,
if the objective function is Morse, we conclude that the trajectory converges
to a local minimum of the objective for almost all initializations. We also
study algorithmic schemes for both dynamics and prove all the previous
properties in the discrete setting under proper choice of the step-size.
