question
My question regarding the argument of the proof of theorem 3.1 in this paper.
In the proof the Lasalle's invariance principle is used. From what I learned, radially unboundedness must hold for $V$ in order for global stability. However in the proof, I did not see whether and why radially unboundedness of $V(\vec\theta):=-\sum a_{ij}\cos(\theta_i-\theta_j)$ hold. It seems obvious that $-\sum a_{ij}\cos(\theta_i-\theta_j)$ is not radially unbounded (Take $\vec\theta=(\theta_0,\cdots,\theta_0)$ and $\theta_0 \rightarrow \infty$, then $-\sum a_{ij}\cos(\theta_i-\theta_j)=\sum a_{ij}$ which does not goes to infinity).
If it is indeed not radially unbounded, then how does the global stability obtained in the paper? Is it because The system being defined on a compact manifold $\mathbb{T}^n$ stated in the proof?
Could someone help? Many thanks in advance!