  Semiconductor lasers exhibit a wealth of dynamics, from emission of a
constant beam of light, to periodic oscillations and excitability. Self-pulsing
regimes, where the laser periodically releases a short pulse of light, are
particularly interesting for many applications, from material science to
telecommunications. Self-pulsing regimes need to produce pulses very regularly
and, as such, they are also known to be particularly sensitive to
perturbations, such as noise or light injection.
  We investigate the effect of delayed optical feedback on the dynamics of a
self-pulsing semiconductor laser with saturable absorber (SLSA). More
precisely, we consider the Yamada model with delay -- a system of three
delay-differential equations (DDEs) for two slow and one fast variable -- which
has been shown to reproduce accurately self-pulsing features as observed in
SLSA experimentally. This model is also of broader interest because it is quite
closely related to mathematical models of other self-pulsing systems, such as
excitable spiking neurons.
  We perform a numerical bifurcation analysis of the Yamada model with delay,
where we consider both the feedback delay, the feedback strength and the
strength of pumping as bifurcation parameters. We find a rapidly increasing
complexity of the system dynamics when the feedback delay is increased from
zero. In particular, there are new feedback-induced dynamics: stable
quasi-periodic oscillations on tori, as well as a large degree of
multistability, with up to five pulse-like stable periodic solutions with
different amplitudes and repetition rates. An attractor map in the plane of
perturbations on the gain and intensity reveals a Cantor set-like, intermingled
structure of the different basins of attraction. This suggests that, in
practice, the multistable laser is extremely sensitive to small perturbations.
