Question 1 asks us if the shadowing lemma tells us that noise-added trajectories on chaotic attractors are shadowed by true trajectories

And this is true; this is precisely the spirit of the shadowing lemma

That is, if youre on a chaotic attractor and youre bumped by a small amount of noise, then youll land on an area of the attractor that you would have gotten to anyway if youd continued along that same trajectory for infinitely long

That is, that a noise-added trajectory on a chaotic attractor is shadowed by a true trajectory

So a little bit of noise wont be a problem

However, question 2 really gets at the nuances of the shadowing lemma

It asks if the shadowing lemma tells us that a chaotic attractor is immune to any level of noise; i.e., that all noise-added trajectories, regardless of the size of the noise, will remain on the attractor

And this is false

If the noise is large enough, for example, you could bump the trajectory off of the attractor and into the basin of attraction of a different attractor

So you may actually reach a trajectory, or an attractor, that you wouldnt have normally reached, even if the trajectory was infinitely long, because the noise bumped you out of the basin of attraction

The final question asks if the shadowing lemma holds for all nonlinear systems

And this is false; the shadowing lemma really only holds for nonlinear, chaotic dynamical systems