Assume there is a dynamical system
$$ \frac{d x(t)}{dt} = A \cdot x(t) + q(x(t)) $$
and that $A$ is stable and that $q$ is a nonlinear and very complicated function. We only know $q$ is smooth and bounded from above and below and that $0$ is the only fixpoint.
Can't I just replace $q$ with a "fake disturbance input" $d$:
$$ \frac{d x(t)}{dt} = A \cdot x(t) + d(t) $$
and show that this system is input to state stable for a bounded $d(t)$? (which is much easier)
Doesn't that also show that the system with $q$ is stable or am I missing something?