Non-formally integrable centers admitting an algebraic inverse integrating factor

TL;DR

This paper investigates the existence of algebraic inverse integrating factors in non-formally integrable systems, characterizing centers especially in systems with specific quasi-homogeneous lowest-degree terms.

Contribution

It introduces conditions for the existence of algebraic inverse integrating factors in non-formally integrable systems and characterizes centers in these systems, including those with a specific quasi-homogeneous term.

Findings

Identifies conditions for algebraic inverse integrating factors in non-integrable systems.

Characterizes centers in systems with lowest-degree term (-y^3, x^3)^T.

Provides criteria for the existence of centers based on inverse integrating factors.

Abstract

Westudy the existence of a class of inverse integrating factor for a family of non formally integrable systems, in general, whose lowest-degree quasi-homogeneous term is a Hamiltonian vector field. Once the existence of an inverse integrat ing factor is established, we characterize the systems having a center. Among others, we characterize the centers of the systems whose lowest-degree quasiho mogeneous term is (-y3,x3)T with an algebraic inverse integrating factor. Keywords: Nonlinear differential systems, Inverse integrating factor, Integrability problem, Degenerate center problem