Mathematics > Dynamical Systems
[Submitted on 30 Jul 2025
]
Title: First integrals and invariants of systems of ODEs
Title: 一阶积分和常微分方程组的不变量
Abstract: We investigate the interplay between monomial first integrals, polynomial invariants of certain group action, and the Poincar\'{e}-Dulac normal forms for autonomous systems of ODEs with 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 Poincar\'{e}-Dulac normal forms of the underlying vector fields.
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