Dynamical Properties of Periodic Solutions of Integro-Differential Equations

Spatially distributed integro-differential systems of equations with periodic boundary conditions are considered. In applications, such systems arise as limiting ones for some nonlinear fully coupled ensembles. The simplest critical cases of zero and purely imaginary eigenvalues in the problem of stability of the zero equilibrium state are considered.
In these two situations, quasinormal forms are constructed, for which the question of the existence of piecewise constant solutions is studied. In the case of a simple zero root, the conditions for the stability of these solutions are determined. The existence of piecewise constant solutions with more than one discontinuity point is shown. An algorithm for calculating solutions of the corresponding boundary value problem by numerical methods is presented. A numerical experiment is performed, confirming the analytical constructions.