Hopf bifurcation subject to a large delay in a laser system

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.