Inverse Approach In The Study Of Ordinary Differential Equations

TL;DR

This paper extends classical results to construct differential systems in higher dimensions with specified integrals, analyzing their integrability and algebraic limit cycles, especially focusing on polynomial cases.

Contribution

It generalizes Eruguin's 1952 results to higher dimensions and constructs polynomial systems with specific invariants, exploring their integrability and algebraic properties.

Findings

Constructed a non-Darboux integrable polynomial system with a given algebraic invariant.

Analyzed the system's Darboux integrability, Poincare's problem, and Hilbert's 16th problem.

Provided insights into algebraic limit cycles in polynomial differential systems.

Abstract

We extend the Eruguin result exposed in the paper "Construction of the whole set of ordinary differential equations with a given integral curve" published in 1952 and construct a differential system in $\Bbb{R}^N$ which admits a given set of the partial integrals, in particular we study the case when theses functions are polynomials. We construct a non-Darboux integrable planar polynomial system of degree $n$ with one invariant irreducible algebraic curve $g(x,y)=0$. For this system we analyze the Darboux integrability, Poincare's problem and 16th's Hilbert problem for algebraic limit cycles.