An equilibrium $x^*$ is said to be stable iff for all $\epsilon>0$, there exist $\delta>0$ s.t. $|x(0)-x^*|<\delta$ $\implies$ for all $t>0$, $|x(t)-x^*|<\epsilon.$
Does anyone know if the delta in the above definition of stability depends on $t$ and $\epsilon$ or only on $\epsilon$ ?