Two-point oscillatory solutions to system with relay hysteresis and nonperiodic external disturbance

Abstract

We study an n-dimensional system of ordinary differential equations with a constant matrix, a relay-type nonlinearity, and an external disturbance in the right-hand side. We consider a nonideal relay characteristic. The external disturbance is described by the product of an exponential function and a sine function with an initial phase as a parameter. We assume the matrix of the linear part and the vector at the relay characteristic such that, by a nonsingular transformation, the system is reduced to the form with the diagonal matrix and the vector being opposite to the unit vector. We establish a necessary and sufficient condition for the existence of two-point oscillatory solutions, i.e., the solutions with two fixed points on the hyperplanes of the relay switching in phase space. Also, we give the sufficient conditions under which such solutions do not exist. We provide a supporting example, which demonstrates how to apply the obtained results.