Geometric Singular Perturbation Analysis of the Yamada Model

The Yamada model for lasers with saturable absorber is known to display interesting dynamics on widely different time scales. Methods from geometric singular perturbation theory are used to analyze the dynamics. On the slow scale solutions drift along a slow manifold which is attracting on one side of a nonhyperbolic line and repelling at the other side. On the fast scale solutions jump from the repelling part of the slow manifold back to the attracting part. This mechanism generates periodic and homoclinic orbits of relaxation type. A detailed analysis of the complicated behavior of the homoclinic orbits in the singular limit is given. The recently developed blow-up method for systems of singularly perturbed ordinary differential equations is used to analyze the dynamics near the nonhyperbolic line where an essential part of the dynamics takes place. In the blown-up system a singular homoclinic orbit interacting with a transcritical bifurcation of equilibria is identified. A return map along the singular homoclinic orbit is studied to prove that the homoclinic orbit persists along a smooth curve in parameter space which is shown to be exponentially close to the transcritical bifurcation curve.