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<p>The natural concepts of Perron and upper-limit stability of the zero solution of a&#xa0;differential system are defined, as well as their numerous varieties: from global to particular stability or instability and analogues of the same properties, extending not to all, but to almost all perturbed solutions. Their logical connections with the corresponding Lyapunov concepts and with each other, with the signs of the Perron and Lyapunov exponents and with special indicators are investigated. Their specific features for one-dimensional, autonomous and linear systems are studied. In particular, the independence of most of these properties from the phase domain of the system has been proven. A&#xa0;complete coincidence of research possibilities by the first approximation of stability and asymptotic stability of all three types was discovered. A&#xa0;similar coincidence has been established for partial and particular stability by the first approximation, and in the one-dimensional case for all of the listed types of stability, as well as for all types of instability. Bibliography: 46 titles.</p>

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ON PERRON, LYAPUNOV, AND UPPER-LIMIT STABILITY PROPERTIES OF DIFFERENTIAL SYSTEMS

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • I. N. Sergeev

摘要

The natural concepts of Perron and upper-limit stability of the zero solution of a differential system are defined, as well as their numerous varieties: from global to particular stability or instability and analogues of the same properties, extending not to all, but to almost all perturbed solutions. Their logical connections with the corresponding Lyapunov concepts and with each other, with the signs of the Perron and Lyapunov exponents and with special indicators are investigated. Their specific features for one-dimensional, autonomous and linear systems are studied. In particular, the independence of most of these properties from the phase domain of the system has been proven. A complete coincidence of research possibilities by the first approximation of stability and asymptotic stability of all three types was discovered. A similar coincidence has been established for partial and particular stability by the first approximation, and in the one-dimensional case for all of the listed types of stability, as well as for all types of instability. Bibliography: 46 titles.