Suppose you have a vector field $V$ on an even dimensional sphere $S^{2n}$ and you consider the flow induced by the vector field i.e. $$\dot{x} = V(x).$$ By topological constraints, it must have at least two equilibria. Suppose that we also know that the equilibria for this vector fields are not centers i.e. if you look at the eigenvalues of the Jacobian of this vector field, they all have non-zero real part.
Question: Can we conclude that at least one of these equilibria is a sink, and another one a source? i.e. that one is asymptotically stable and the other completely unstable?
My thoughts: If we assume the vector field is given by the gradient of a function, i.e. $$V = - \nabla f,$$ then this is surely so since a smooth function on a sphere always attains a maximum and a minimum and these are clearly a sink and source respectively.
If we remove the conditions about no centers, then this is surely false. For example, consider $S^2$ and a rotation which fixes the north and south pole. This clearly has no sink nor source, and only centers. All other possible counter examples I could think of failed due to not satisfying this condition.
Any ideas?