A codimension three bifurcation for the laser with saturable absorber

We study the five mode equations which are used to model the dynamical behavior of a laser with saturable absorber in the mean field limit and exact resonance. We show that in this system a codimension three bifurcation exists where a tricritical point of the stationary solution encounters a double zero eigenvalue. A center manifold reduction is performed to fix the three-dimensional submanifold in parameter space where this degeneracy occurs. The associated Takens-normal form is given. By unfolding the normal form we obtain all structurally stable phase portraits near this bifurcation point and display them in the form of bifurcation diagrams with the laser pumping rate as a distinguished bifurcation parameter. These diagrams allow a unifying analytical and geometrical description of many different numerical solutions of the equations describing a laser with absorber. In particular, they yield the connection of the small amplitude periodic solutions with passive Q-switching and suggest new bifurcation processes, which one can expect to occur for physical parameters near the critical submanifold. The existence of a codimension four bifurcation is indicated.