  We study the emergence and the stability of temporal localized structures in
the output of a semiconductor laser passively mode-locked by a saturable
absorber in the long cavity regime. For large yet realistic values of the
linewidth enhancement factor, we disclose the existence of secondary dynamical
instabilities where the pulses develop regular and subsequent irregular
temporal oscil- lations. By a detailed bifurcation analysis we show that
additional solution branches that consist in multi-pulse (molecules) solutions
exist. We demonstrate that the various solution curves for the single and
multi-peak pulses can splice and intersect each other via transcritical
bifurcations, lead- ing to a complex web of solution. Our analysis is based
upon a generic model of mode-locking that consists in a time-delayed dynamical
system, but also upon a much more numerically efficient, yet approximate,
partial differential equation. We compare the results of the bifurcation
analysis of both models in order to assess up to which point the two approaches
are equivalent. We conclude our analysis by the study of the influence of group
velocity dispersion, that is only possible in the framework of the partial
differential equation model, and we show that it may have a profound impact on
the dynamics of the localized states.
