Stability analysis of equilibrium states pabolic equation with stretch operator and delay in a nonlinear feedback functional

The paper considers the dynamics and stability of homogeneous equilibrium states of a mathematical model of a nonlinear generator of optical radiation with a stretching operator for the spatial coordinates of a light wave and a time delay in the feedback loop. The mathematical model is an initial-boundary value problem for a parabolic type equation in a circle with a spatial argument stretching operator and a time delay in a nonlinear feedback functional. We study the dynamics of homogeneous equilibrium states and their stability depending on the parameters of the initial-boundary value problem. In the plane of the main control parameters (gain and time delay), using the method of D-partitions, the regions of stability (instability) of homogeneous equilibrium states are constructed. Possible mechanisms of loss of stability by homogeneous equilibrium states are studied. The possibility of oscillatory loss of stability is shown, which is caused by the passage through the imaginary axis of the complex plane when changing the parameters of the initial-boundary value problem of one or two pairs of complex conjugate points of the spectrum of the characteristic pencil of operators, between which resonance relations are possible. The possibility of loss of stability by several equilibrium states at the same time is shown.

Keywords: spatially inhomogeneous waves, bifurcation, optical information transmission systems.