In the following notes, how to linearize a state equation is described. The part I don't understand is why you can just remove the $\delta$ like that. I think the state equation should be:
\begin{align} \delta\dot x(t) = A(t)\delta x(t) + B(t)\delta u(t) \end{align}
Since the linearization only gives us how much change there is given the change in inputs.
I guess, here it is just a redesignation, as $\delta x(t)\rightarrow x(t)$. As you left only terms with $\delta$, so there is no reason to keep $\delta$ for all the terms. $x(t)$ and $u(t)$ are still excitations over $x_0(t)$ and $u_0(t)$.