Strongly formal Weierstrass non-integrability for polynomial differential systems in $\mathbb{C}^2$

Abstract

Recently it has been given a criterion for determining the weakly formal Weierstrass non-integrability of polynomial differential systems in $\mathbb{C}^2$. Here we extend this criterion for determining the strongly formal Weierstrass non-integrability which includes the weakly formal Weierstrass non-integrability of polynomial differential systems in $\mathbb{C}^2$. The criterion is based on the solutions of the form $y=f(x)$ with $f(x) \in \mathbb{C}[[x]]$ of the differential system whose integrability we are studying. The results are applied to a differential system that contains the famous force-free Duffing and the Duffing–van der Pol oscillators

Keywords

• liouville integrability
• weierstrass integrability
• polynomial differential systems