The Classification of Time Invariants (First Integrals) of Multinomial Systems of O.D.E.s and the Surprising Link Between Algebraic and Logarithmic Time Invariants, Dictated by the Method of Arrays

TL;DR

This paper explores the classification of time invariants (first integrals) in multinomial ODE systems, revealing a connection between algebraic and logarithmic invariants through the method of arrays, and provides practical methods for their determination.

Contribution

It introduces a systematic approach using the method of arrays to classify and compute first integrals of multinomial ODE systems, highlighting a surprising link between algebraic and logarithmic invariants.

Findings

First integrals can be obtained by solving linear systems.

The response of first integrals to system changes is characterized.

An easy method for finding specific classes of first integrals is provided.

Abstract

For a large class of systems of o.d.e.'s which have first integrals, the method of arrays yields the following results: i) The first integrals $I$ can be found by solving systems of linear equations. ii) How the first integral $I$ responds to changes in the system $S$. iii) An easy way for finding the first integral for a special class of first integrals if they exist.