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Abstract <p>This paper investigates the construction of an equivalent quasi-Hamiltonian system for a linear differential equation with periodic coefficients that contains only even-order derivatives and depends on a small parameter. We develop an approach based on transforming the unperturbed equation into an equivalent Hamiltonian system in normal form. It is shown that that the corresponding linear non-degenerate transformation provides a solution to the main problem. The obtained results are extended to an analogous problem for a linear periodic system with a small parameter, where the unperturbed system is defined by a Frobenius matrix. We discuss applications of these results to stability theory problems for solutions of differential equations in critical cases.</p>

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Hamiltonian Structures of Periodic Differential Equations with Even-Order Derivatives

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • M. G. Yumagulov,
  • L. S. Ibragimova,
  • T. S. Oripov

摘要

Abstract

This paper investigates the construction of an equivalent quasi-Hamiltonian system for a linear differential equation with periodic coefficients that contains only even-order derivatives and depends on a small parameter. We develop an approach based on transforming the unperturbed equation into an equivalent Hamiltonian system in normal form. It is shown that that the corresponding linear non-degenerate transformation provides a solution to the main problem. The obtained results are extended to an analogous problem for a linear periodic system with a small parameter, where the unperturbed system is defined by a Frobenius matrix. We discuss applications of these results to stability theory problems for solutions of differential equations in critical cases.