Bounded polynomial vector fields

Abstract

We prove that, for generic bounded polynomial vector fields in R n {{\mathbf {R}}^n} with isolated critical points, the sum of the indices at all their critical points is ( − 1 ) n {( - 1)^n} . We characterize the local phase portrait of the isolated critical points at infinity for any bounded polynomial vector field in R 2 {{\mathbf {R}}^2} . We apply this characterization to show that there are exactly seventeen different behaviours at infinity for bounded cubic polynomial vector fields in the plane.