# How to control the angles of Kuramoto model by controlling its order parameter?

Consider Homogenous Kuramoto model in this paper. In theorem 3.1, the author derive condition on $$A$$ such that all second-order critical points of $$E(\theta)$$ are in two opposite quadrants, by saying that $$|\sin(\theta_i-\theta_r)|\leq\frac{\sqrt{2}}{2},\forall i\in[n]$$.

To derive condition on $$A$$ in order to ensure this, they use order parameter:

since $$\|r\||\sin(\theta_i-\theta_r)|=\Big|\sum_j(1-a_{ij})\sin(\theta_i-\theta_j)\Big|$$, we can control $$\|r\|$$ and $$\Big|\sum_j(1-a_{ij})\sin(\theta_i-\theta_j)\Big|$$ for controlling $$|\sin(\theta_i-\theta_r)|$$ for all $$i\in[n]$$.

Now I want to have a stronger result:

For any second-order critical point $$\theta$$, its entries $$\theta_i$$ are in the same quadrant. Thus, in addition to above condition, we need to ensure

$$\cos(\theta_i-\theta_r)>0,\forall i\in[n]$$

What I tried:

Similar to the proof in theorem 3.1,

$$\|r\|e^{-i(\theta_i-\theta_r)}=re^{i\theta_r}=\sum_je^{-i(\theta_i-\theta_j)}$$

Take the real part, we have

$$\|r\|\cos(\theta_i-\theta_r)=\sum_j\cos(\theta_i-\theta_j)$$, for all $$i\in[n]$$.

Thus we need for all $$i\in[n]$$, $$\cos(\theta_i-\theta_r)=\frac{\sum_j\cos(\theta_i-\theta_j)}{\|r\|}>0$$.

Equivalently we need $$\sum_j\cos(\theta_i-\theta_j)>0$$ for all $$i$$.

Any insight is appreciated!