Criterion for the Existence of Two-Point Oscillatory Solution of a Perturbed System with a Relay

We study an $n$-dimensional system of ordinary differential equations with a constant matrix in the linear part, a discontinuous hysteresis-type nonlinearity, and a continuous bounded perturbation function in the nonlinear part. The nonlinearity is described by a characteristic of the on-off nonideal relay. The matrix of the system has real simple nonzero eigenvalues. We study oscillatory solutions with two switching points in the phase space of the system and an arbitrary period of return to each of these points. We consider the system in the original and canonical forms. The Cauchy problem is solved with initial and boundary conditions at the switching points. For the canonical system with nonzero vector feedback, the vector of units in the case of nonlinearity, and a perturbation function of general form, we prove a criterion for the existence and uniqueness of a solution with an arbitrary return period. Moreover, in the case of a periodic perturbation function, a necessary and sufficient condition for the existence of a unique periodic solution with a given period is obtained. We present an example of the existence of a solution for a three-dimensional system.