  In this paper we study the singular vanishing-viscosity limit of a gradient
flow in a finite dimensional Hilbert space, focusing on the so-called delayed
loss of stability of stationary solutions. We find a class of time-dependent
energy functionals and initial conditions for which we can explicitly calculate
the first discontinuity time $t^*$ of the limit. For our class of functionals,
$t^*$ coincides with the blow-up time of the solutions of the linearized system
around the equilibrium, and is in particular strictly greater than the time
$t_c$ where strict local minimality with respect to the driving energy gets
lost. Moreover, we show that, in a right neighborhood of $t^*$, rescaled
solutions of the singularly perturbed problem converge to heteroclinic
solutions of the gradient flow. Our results complement the previous ones by
Zanini, where the situation we consider was excluded by assuming the so-called
transversality conditions, and the limit evolution consisted of strict local
minimizers of the energy up to a negligible set of times.
