Consider two solutions(or trajectories), say $x_1(t)$ and $x_2(t)$, of a system of differential equaions. That is, $$ x_1'(t)=x_2'(t)=f(x,t), t\ge0. $$
Also, $\|x_2(0)-x_1(0)\|<\epsilon$ for some small constant $\epsilon$. So, $x_1(t)$ and $x_2(t)$ can be regarded as two trajectories starting with two close initial conditions in the system . I would like to ask is there any theorem to show that under what conditions, we can say $x_1(t)$ and $x_2(t)$ are very close if the perturbation in the initial condition is small. For example, is any theorem to say something like this?
For every $\delta>0$, there exists $\epsilon>0$ such that if $\|x_2(0)-x_1(0)\|<\epsilon$, then we have $\|x_2(t)-x_1(t)\|<\delta, \forall t\ge0$.
I know there is Lyapunov stability to show the stability of the equilibrium point. However, I want the something stronger than that. It is about the "stability of the trajectories". It would be greatly appreciated if you could help with this question.