First Integrals and Invariants of Systems of Ordinary Differential Equations

Mateja Grašič, Abdul Salam Jarrah and Valery G. Romanovski ()
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Mateja Grašič: Faculty of Natural Science and Mathematics, University of Maribor, Koroška Cesta 160, SI-2000 Maribor, Slovenia
Abdul Salam Jarrah: Department of Mathematics and Statistics, American University of Sharjah, Sharjah P.O. Box 26666, United Arab Emirates
Valery G. Romanovski: Faculty of Natural Science and Mathematics, University of Maribor, Koroška Cesta 160, SI-2000 Maribor, Slovenia

Mathematics, 2025, vol. 13, issue 21, 1-17

Abstract: We investigate the interplay between monomial first integrals, polynomial invariants of certain group action, and the Poincaré–Dulac normal forms for autonomous systems of ordinary differential equations with a diagonal matrix of the linear part. Using tools from computational algebra, we develop an algorithmic approach for identifying generators of the algebras of monomial and polynomial first integrals, which works in the general case where the matrix of the linear part includes algebraic complex eigenvalues. Our method also provides a practical tool for exploring the algebraic structure of polynomial invariants and their relation to the Poincaré-Dulac normal forms of the underlying vector fields.

Keywords: first integral; polynomial invariants; Poincaré-Dulac normal form; systems of ordinary differential equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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