  We study the invasion of an unstable state by a propagating front in a
peculiar but generic situation where the invasion process exhibits a remnant
instability. Here, remnant instability refers to the fact that the spatially
constant invaded state is linearly unstable in any exponentially weighted space
in a frame moving with the linear invasion speed. Our main result is the
nonlinear asymptotic stability of the selected invasion front for a
prototypical model coupling spatio-temporal oscillations and monotone dynamics.
We establish stability through a decomposition of the perturbation into two
pieces: one that is bounded in the weighted space and a second that is
unbounded in the weighted space but which converges uniformly to zero in the
unweighted space at an exponential rate. Interestingly, long-time numerical
simulations reveal an apparent instability in some cases. We exhibit how this
instability is caused by round-off errors that introduce linear resonant
coupling of otherwise non-resonant linear modes, and we determine the
accelerated invasion speed.
