  We consider self-pulsing in lasers with a gain section and an absorber
section via a mechanism known as Q-switching, as described mathematically by
the Yamada ordinary differential equation model for the gain, the absorber and
the laser intensity. More specifically, we are interested in the case that gain
and absorber decay on different time scales. We present the overall bifurcation
structure by showing how the two-parameter bifurcation diagram in the plane of
pump strength versus decay rate of the gain changes with the ratio between the
two decay rates. In total, there are ten cases BI to BX of qualitatively
different two-parameter bifurcation diagrams, which we present with an
explanation of the transitions between them. Moroever, we show for each of the
associated eleven cases of structurally stable phase portraits (in open regions
of the parameter space) a three-dimensional representation of the organisation
of phase space by the two-dimensional manifolds of saddle equilibria and saddle
periodic orbits.
  The overall bifurcation structure constitutes a comprehensive picture of the
exact nature of the observable dynamics, including multi-stability and
excitability, which we expect to be of relevance for experimental work on
Q-switching lasers with different kinds of saturable absorbers.
