On necessary conditions for existence of periodic solutions in a dynamic system with discontinuous nonlinearity and an external periodic influence

The system of ordinary differential equations with discontinuous nonlinearity of a non-ideal relay type and an external continuous periodic influence in the right-hand side is considered in the Euclidean space. Necessary conditions for existence of periodic solutions with given properties in problems of the specified class imposed on the coefficients of the system are obtained by means of accurate analytical methods. An approach for finding moments of time and switching points of the image point of the required solution is suggested in the case when the period of the solution is multiple to the period of the function describing the external perturbation.