Algorithm for constructing asymptotics of periodic solutions in laser models with a rapidly oscillating delay

The problem of stability of the equilibrium state in a laser system with fast oscillating coefficients is considered. A system averaged over fast oscillations and with a distributed delay is constructed. Critical cases in the problem of the stability of the equilibrium state are singled out. It is shown that the threshold value of the feedback coefficient at which the equilibrium state becomes unstable increases due to rapid oscillations compared to the corresponding value in the absence of modulation. In critical cases, normal forms are constructed – equations for the slowly varying amplitude of the periodic solutions. The conditions for the existence, stability and instability of cycles are revealed.