First integrals and invariants of systems of ODEs

TL;DR

This paper develops an algorithmic approach using computational algebra to identify generators of monomial and polynomial first integrals in autonomous ODE systems, especially with complex eigenvalues, linking invariants to normal forms.

Contribution

It introduces a novel computational method for finding algebraic invariants of ODE systems with complex eigenvalues, enhancing understanding of their structure and normal forms.

Findings

Algorithm successfully identifies generators of first integrals.

Method applies to systems with algebraic complex eigenvalues.

Provides insights into the algebraic structure of invariants.

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.