On One Type of Oscillatory Solutions of a Nonautonomous System with Relay Hysteresis

We consider an $n$-dimensional system of first-order ordinary differential equations with a constant matrix having real, simple, and nonzero eigenvalues, with a discontinuous nonlinearity of two-position relay type with positive hysteresis and a continuous bounded perturbation function. We study continuous two-point oscillatory solutions with a certain return period of the representative point to the switching hyperplane in the state space. When solving the Cauchy problem with initial condition at the switching point, we use the fitting method. We construct a system of transcendental equations for the switching times and points. We prove a criterion for the existence and uniqueness of a solution with some fixed return period. For a system in the canonical form with diagonal matrix and with feedback vector of a special form, we obtain conditions for the solvability of a system of transcendental equations for the first switching time for a given return period and formulas for the switching points. For a three-dimensional system, we give a numerical example to illustrate the theoretical results.