I will cite the following definition (from an O.D.E textbook) of the stability of a solution:
Definition. Let $f:[0,\infty)\times\Omega\to\mathbb{R}^n$ be continuous and locally Lipschitz on $\Omega\subseteq\mathbb{R}^n$. Then a solution $\phi:[0,\infty)\to\mathbb{R}^n$ of the system:
$$ x'(t)=f(t,x(t)), \forall\ t\in [0,\infty) $$
is called stable if for every $a\geq 0$ there exists $\mu(a)>0$ such that for every $\xi\in\Omega$ with $||\xi-\phi(a)||<\mu(a)$, the unique saturated solution $x$, of the problem
$$ \begin{cases} x'(t)=f(t,x(t)), \forall\ t\in [0,\infty) \\ x(a)=\xi\end{cases} $$
is defined on $[a,+\infty)$ and for every $\epsilon>0$,there exists $\delta(\epsilon,a)\in (0,\mu(a)]$ such that for each $\xi\in\Omega$ with $||\xi-\phi(a)||<\delta(\epsilon,a)$ we have that:
$$||x(t)-\phi(t)||<\epsilon, \ \forall t\in [a,\infty).$$
My question is: Why can't we define the stability of a solution on a bounded interval $[a,b]$?? Why we must consider only systems on $[a,+\infty)$?