  We present a set of numerical simulations of stellar explosions induced by
relativistic jets emanating from a central engine sitting at the center of
compact, dying stars. We explore a wide range of durations of the central
engine activity, two candidate stellar progenitors, and two possible values of
the total energy release. We find that even if the jets are narrowly
collimated, their interaction with the star unbinds the stellar material,
producing a stellar explosion. We also find that the outcome of the explosion
can be very different depending on the duration of the engine activity. Only
the longest-lasting engines result in successful gamma-ray bursts. Engines that
power jets only for a short time result in relativistic supernova explosions,
akin to observed engine-driven SNe such as SN2009bb. Engines with intermediate
durations produce weak gamma-ray bursts, with properties similar to nearby
bursts such as GRB 980425. Finally, we find that the engines with the shortest
durations, if they exist in nature, produce stellar explosions that lack
sizable amounts of relativistic ejecta and are therefore dynamically
indistinguishable from ordinary core-collapse supernovae.
