Approximations in the stability problem for linear periodic systems with aftereffect

The asymptotic stability of a linear periodic system of differential equations with aftereffect is determined by the location of the spectrum of the infinite-dimensional, compact monodromy operator. Analytical representations of such operators can be obtained only for systems of a special type. In numerical simulations, finite-dimensional approximations of the monodromy operators are used. In this paper, we examine a procedure for approximating a system of differential equations with aftereffect by systems of ordinary differential equations of large dimension proposed by N. N. Krasovskii. Finite-dimensional approximations for monodromy operators are constructed in the Hilbert space of states of a periodic system with aftereffect. We prove that increasing of the dimension of finite-dimensional approximations leads to increasing of the approximation accuracy.