We have two differential equations $\dot x_1=f_1(x_1)$ and $\dot x_2=f_2(x_2)$ which are too complex to solve but we could show by the same Lyapunov function $V$ (and its derivatives) that equilibrium points $x_1=0$ and $x_2=0$ are globally asymptotically stable (GAS).
If we know $\Vert \dot V(x_1) \Vert \le \Vert \dot V(x_2) \Vert$ $\forall x_1 = x_2$ (for example $\dot V(x_1)=-x_1^2/(1+x_1^2), \dot V(x_2)=-x_2^2$ ) and $x_1(0)=x_2(0)\gt 0$, can we say that $x_1(t_a)\ge x_2(t_a)$ $\forall t_a$ (in this inequality we take the same moment of time)?