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