Periodic solutions of a system of differential equations with hysteresis nonlinearity in the presence of eigenvalue zero

Abstract

We study an $n$-order system of ordinary differential equations with a nonlinearity of nonideal-relay-type with hysteresis and an external periodic perturbations. We consider the existence of solutions with periods equal or multiple to the period of the external perturbation and two points of switching within period. The problem is solved in the case where the collection of simple real eigenvalues of the matrix of the system contains an eigenvalue equal to zero. By a nonsingular transformation, the system is reduced to a canonical system of a special form that enables us to perform its analysis by analytic methods. We propose an approach to finding the points of switching for the representation point of the periodic solution and to a choice of the parameters of the nonlinearity and the feedback vector. We prove a theorem on necessary conditions for the existence of the periodic solutions of the system. Sufficient conditions for the existence of the required solutions are established. We also perform an analysis of stability of the solutions by using the point mapping and the fixed-point method. We present an example that confirms the accumulated results.