On periodic solutions and attractors for the Maxwell--Bloch equations

We consider the Maxwell-Bloch system which is a finite-dimensional approximation of the coupled nonlinear Maxwell-Schrödinger equations. The approximation consists of one-mode Maxwell field coupled to two-level molecule. We construct time-periodic solutions to the factordynamics which is due to the symmetry gauge group. For the corresponding solutions to the Maxwell–Bloch system, the Maxwell field, current and the population inversion are time-periodic, while the wave function acquires a unit factor in the period. The proofs rely on high-amplitude asymptotics of the Maxwell field and a suitable extension of the Lefschetz theorem on fixed points and the Euler characteristic for noncompact manifolds. We also prove the existence of the global compact attractor.