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Nonanalytic First Integrals of Analytic Systems of Differential Equations in a Neighborhood of Stable Equilibria

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Abstract

In even-dimensional phase spaces, we give examples of analytic systems of differential equations that have isolated equilibria and admit nonanalytic first integrals. These integrals are positive definite in a neighborhood of the equilibria, which proves the stability of the equilibria (on the entire time axis). However, such systems of differential equations do not admit nontrivial first integrals in the form of formal power series at all. In particular, the Lyapunov stability of equilibria of analytic systems does not imply their formal stability. In the case of an odd-dimensional phase space, all isolated equilibria are apparently unstable.

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Authors and Affiliations

  1. Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, 119991, Russia

    V. V. Kozlov

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  1. V. V. Kozlov
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Correspondence to V. V. Kozlov.

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Translated by V. Potapchouck

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Kozlov, V.V. Nonanalytic First Integrals of Analytic Systems of Differential Equations in a Neighborhood of Stable Equilibria. Diff Equat 59, 862–865 (2023). https://doi.org/10.1134/S0012266123060137

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  • DOI: https://doi.org/10.1134/S0012266123060137

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