On the coercivity of continuously differentiable vector fields

TL;DR

This paper establishes a characterization of coercivity for continuously differentiable vector fields based on their conservative parts, providing conditions for the coexistence of equilibrium states.

Contribution

It proves that a vector field's coercivity is equivalent to its conservative part's coercivity and offers conditions for equilibrium coexistence.

Findings

Coercivity of vector fields is characterized by their conservative parts.

Provides criteria for the coexistence of equilibrium states.

Establishes a theoretical link between vector field coercivity and conservative components.

Abstract

Given an arbitrary fixed continuously differentiable vector field on $\mathbb{R}^n$, we prove that this vector field is coercive if and only if its conservative part is coercive. We apply this result in order to provide sufficient conditions to guarantee the co-existence of equilibrium states of a continuously differentiable vector field and its conservative part.