  We consider the Yamada model for an excitable or self-pulsating laser with
saturable absorber, and study the effects of delayed optical self-feedback in
the excitable case. More specifically, we are concerned with the generation of
stable periodic pulse trains via repeated self-excitation after passage through
the delayed feedback loop, as well as their bifurcations. We show that onset
and termination of such pulse trains correspond to the simultaneous bifurcation
of countably many fold periodic orbits with infinite period in this delay
differential equation. We employ numerical continuation and the concept of
reappearance of periodic solutions to show that these bifurcations coincide
with codimension-two points along families of connecting orbits and fold
periodic orbits in a related advanced differential equation. These points
include heteroclinic connections between steady states, as well as homoclinic
bifurcations with non-hyperbolic equilibria. Tracking these codimension-two
points in parameter space reveals the critical parameter values for the
existence of periodic pulse trains. We use the recently developed theory of
temporal dissipative solitons to infer necessary conditions for the stability
of such pulse trains.
