  Excitability, encountered in numerous fields from biology to neurosciences
and optics, is a general phenomenon characterized by an all-or-none response of
a system to an external perturbation. When subject to delayed feedback,
excitable systems can sustain multistable pulsing regimes, which are either
regular or irregular time sequences of pulses reappearing every delay time.
Here, we investigate an excitable microlaser subject to delayed optical
feedback and study the emergence of complex pulsing dynamics, including
periodic, quasiperiodic and irregular pulsing regimes. This work is motivated
by experimental observations showing these different types of pulsing dynamics.
A suitable mathematical model, written as a system of delay differential
equations, is investigated through an in-depth bifurcation analysis. We
demonstrate that resonance tongues play a key role in the emergence of complex
dynamics, including non-equidistant periodic pulsing solutions and chaotic
pulsing. The structure of resonance tongues is shown to depend very sensitively
on the pump parameter. Successive saddle transitions of bounding saddle-node
bifurcations constitute a merging process that results in unexpectedly large
locking regions, which subsequently disconnect from the relevant torus
bifurcation curve; the existence of such unconnected regions of periodic
pulsing is in excellent agreement with experimental observations. As we show,
the transition to unconnected resonance regions is due to a general mechanism:
the interaction of resonance tongues locally at an extremum of the rotation
number on a torus bifurcation curve. We present and illustrate the two generic
cases of disconnecting and of disappearing resonance tongues. Moreover, we show
how a maximum and a minimum of the rotation number appears naturally when two
torus bifurcation curves undergo a saddle transition (where they connect
differently).
