Piecewise constant modes of operation of fully coupled networks and their limit integro-differential systems

Fully connected networks of oscillators and their limit systems of integro-differential equations with periodic boundary conditions are considered. It is assumed that the connection is weak, i.e. the coefficient at the integral term is small. In the problem of stability of the zero equilibrium state, the simplest critical cases of loss of stability are distinguished. In these situations, quasi-normal forms are constructed, which are integro-differential equations for which several continuous families of piecewise constant two-step solutions are analytically determined, and their stability is studied. The existence of piecewise constant solutions with more than one discontinuity point is shown. A numerical experiment illustrating the analytical constructions is performed.