Hopf bifurcation theory for an oscillator subject to a weak feedback but a
large delay is investigated for a specific laser system. The problem is mot
ivated by semiconductor laser instabilities which are initiated by undesira
ble optical feedbacks. Most of these instabilities are starting from a sing
le Hopf bifurcation. Because of the large delay, a delayed amplitude appear
s in the slow time bifurcation equation which generates new bifurcations to
periodic and quasi-periodic states. We determine analytical expressions fo
r all branches of periodic solutions and show the emergence of secondary bi
furcation points from double Hopf bifurcation points. We study numerically
different cases of bistability between steady, periodic, and quasi-periodic
regimes. Finally, the validity of the Hopf bifurcation approximation is in
vestigated numerically by comparing the bifurcation diagrams of the origina
l laser equations and the slow time amplitude equation.