Dynamics of a singularly perturbed system of two differential equations with delay

We study a two-dimensional singularly perturbed system with delay, which is a simplification of models used in laser physics. We analyze several cases of a small parameter multiplying the derivative in the first equation and investigate the behavior of solutions in a neighborhood of a stationary point when the system parameters pass through bifurcation values. Methods for local asymptotic analysis are used to construct special nonlinear equations describing the structure of solutions and the asymptotic approximation of solutions of the original problem.