Homoclinic orbits in the Maxwell-Bloch equations with a probe

The dynamics of an ensemble of two-level atoms in a single-mode resonant laser cavity with external pumping and a weak coherent probe is investigated. The system is represented as a perturbation to an integrable system, the Jaynes-Cummings model, in which there are no losses. Using an analytical perturbation technique, the Melnikov method, we show the presence of special homoclinic orbits, which persist under small perturbation from the homoclinic structure present in the integrable case on codimension-one surfaces in the parameter space. Two cases are considered: one in which we consider the equations with small relaxation parameters and no probe and the second in which we consider the small relaxation parameters and the effects of the probe. The persistence of homoclinic orbits for larger parameters is demonstrated through numerical continuation using the software package AUTO. The breakup of these homoclinic orbits is believed to be a source of chaos in the laser system.