Polynomial Differential Systems in R3 Having Invariant Weighted Homogeneous Surfaces

Abstract

In this paper we give the normal form of all polynomial differential systems in R3 having a weighted homogeneous surface f= 0 as an invariant algebraic surface and characterize among these systems those having a Darboux invariant constructed uniquely using this invariant surface. Using the obtained results we give some examples of stratified vector fields, when f= 0 is a singular surface. We also apply the obtained results to study the Vallis system, which is related to the so-called El Niño atmospheric phenomenon, when it has a cone as an invariant algebraic surface, performing a dynamical analysis of the flow of this system restricted to the invariant cone and providing a stratification for this singular surface.