Let $v$ be continuous and nowhere-vanishing vector field tangent to the $n$-sphere $\mathbb{S}^n$ (hence $n$ is odd, w.r.t the Hairy-Ball Theorem). Let $x$ be a trajectory on $\mathbb{S}^n$, defined for $t\geq 0$, and solution of $\dot{x}=v(x)$.

Is this system ergodic or are there any other asymptotic possibilities (such that periodicity)?