As is well known, there is a compason principle for scalar differential equations $\dot x(t)=a(x(t))$ and $\dot y(t)=b(y(t))$ with $x(t_0)=y(t_0)$ and $a(\cdot)\leq b(\cdot)$, with the result being that $x(t)\leq y(t), \forall t\geq t_0$.

Is there a similar version of comparison principle for vector differential euqations $\dot x=F(x)$ and $\dot y=G(y)$? More specifically, if $\|F(x)\|\leq \|G(x)\|$ and $x(t_0)=y(t_0)$, what information can be obtained regarding the relationship of the two differential equations?

As is well known, there is a comparison principle for scalar differential equations $\dot x(t)=a(x(t))$ and $\dot y(t)=b(y(t))$ with $x(t_0)=y(t_0)$ and $a(\cdot)\leq b(\cdot)$, with the result being that $x(t)\leq y(t), \forall t\geq t_0$.

Is there a similar version of comparison principle for vector differential euqations $\dot x=F(x)$ and $\dot y=G(y)$? More specifically, if $\|F(x)\|\leq \|G(x)\|$ and $x(t_0)=y(t_0)$, what information can be obtained regarding the relationship of the two differential equations?