Polynomial differential systems in \(\mathbb R^3\) having invariant weighted homogeneous surfaces (Q1743471)

Invariant algebraic surfaces are quite important in understanding global dynamics of many dynamical systems. Weighted homogeneous surfaces, which are defined as \((x, y, z) \in \mathbb{R}^3\) such that \(f (x, y, z) = 0\), where \(f: \mathbb{R}^3\rightarrow \mathbb{R}\) is a weighted homogeneous polynomial, are an important class of surfaces contained in the Euclidean space \(\mathbb{R}^3\). The authors give the normal form of all polynomial differential systems in \(\mathbb{R}^3\) 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. The obtained results are applied to study the Vallis system with an invariant cone, and a dynamical analysis of the flow of this system restricted to the invariant cone is presented.