Given a dynamical system described by the equations (for i=1,...,N) $$\frac{d y_i}{dt} = P_i - by_i + K \underset{i \neq j}{\sum_{i=1}^N} \sin(x_i-x_j)$$ $$\frac{d x_i}{dt} = y_i$$ Say that I have found a stable point in the $R^{2n}$ dimensional space $((x_1,y_1),(x_2,y_2),...,(x_n,y_n))$ such that $\frac{dx_i}{dt} = \frac{dy_i}{dt}=0$ for all $i =1,...,N$.
How do I check numerically, if this point is stable or unstable? (Preferably without checking the eigenvalues of the jacobian matrix)