what is the physical interpretation of strongly stability in differential equations ?

Question

The solution $x(t)$ of $X'= F(t,x)$ is said to be strongly stable if, for each $\varepsilon >0,$ there exists a $\delta= \delta(\varepsilon)>0$ such that, for any solution $\bar{x(t)}= x(t,t_0,\bar{x_0})$ of $X'= F(t,x),$ the inequalities $t_1\ge t_0$ and $\|\bar{x(t_1)}-x(t_1)\|\leq \delta$ imply $\|\bar{x(t)}-x(t)\|\leq \varepsilon$ for all $t\ge t_0.$

Answer 1

We don't make "physical interpretations" of an abstract mathematical notion. In any event, it simply means that solutions are close for all time $t\ge t_0$ in case they are known to be sufficiently close for some time in the same interval.

PS. It is a bit peculiar to have "for all $t\ge t_0$" instead of "for all $t\ge t_1$", at the end. But the interpretation would be the same.