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.



Convergence of dynamical system on the sphere
I think you can get any flow on the unit sphere in this way. Take $A(x)$ to be the rank-one matrix $$A(x) = -(x-b)(x^\*-b^\*)$$ with $b = b(x)$ tangent to the sphere (that is, orthogonal to $x$). Then $x^\*b=b^\*x=0$ and $x^\*x = 1$, so that $$\begin{aligned}(I - xx^\*)A(x)x & = xx^\*(x-b)(x^\*-b^\*)x-(x-b)(x^\*-b^\*)x \\ & = x - (x-b) = b .\end{aligned}$$ Therefore, the differential equation under discussion takes form $$\dot{x} = b(x).$$


