Finding of periodic solution of the Mathieu equation with the delay

Abstract

The work suggests an approach for finding periodic solution of the nonlinear delayed differential Mathieu equations applied in the theory of oscillatory processes. The application of the numerical-analytical method to finding periodic solutions of this equation is known. This idea includes reducing the equation to the system of the first order. The article proposes the use of the previously developed method for finding periodic solutions of nonlinear second-order ordinary differential equations, also used for equations with delay, without being reduced to a system. In this case, the Green's function is constructed for a self-adjoint differential operator of the second derivative, defined on functions that satisfy periodic boundary conditions. The necessary and sufficient conditions for the existence of a periodic solution of the Mathieu equation are given. The solution itself is found by the method of successive approximations. The estimates for the method's rate of convergence were obtained.