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<p>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. Finitedimensional 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.</p>

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Approximations in the Stability Problem for Linear Periodic Systems with Aftereffect

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • Yu. F. Dolgii,
  • R. I. Shevchenko

摘要

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. Finitedimensional 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.