Darboux integrability of polynomial differential systems in $\mathbb R^3$

Abstract

In this article we study the Darboux integrability of the polynomial differential systems \[ \dot x = y-x^2, \quad \dot y=z-x, \quad \dot z= -d-ax -by-cz. \] This system comes from the study of a Hopf bifurcation in slow-fast systems with two slow variables and one fast variable. The tools used here for studying the Darboux integrability can be applied to arbitrary polynomial differential systems in $\mathbb R^3$.