Let $A(x)$ be a symmetric negative semi-definite matrix which depends continuously on the parameter $x\in\mathbb{R}^{d}$. We consider the differential equation $$\dot{x} = (I-xx^*)A(x)x$$ on the unit sphere. Is there anything known about the convergence behaviour of the trajectories for $t\to\infty$?

If the matrix is constant it converges to a stable equilibrium.