For Question 1, the Farmer/Sidorowich filtering strategy leverages the stable and unstable manifold structure of the attractor to remove noise from a chaotic trajectory

And this is true

Remember, it moves a cluster of points forward in time to stretch it along the unstable manifold, and backward in time to stretch it along the stable manifold

And looks at the intersection of what was left of this noise ball, forward and backward in time, stretched along the stable and unstable manifolds

Giving you a more precise location of where the point actually lies

For Question 2, low-pass filtering of chaotic systems is a bad idea because it can remove signal, not just noise, and this is true

Remember that chaotic trajectories can be thought of as having infinite periods

That means all frequencies are present in a chaotic trajectory

This means, if you did a low-pass filter, you would effectively be removing a good portion of the actual signal, not just the noise

So doing low-pass filtering of a chaotic time series is generally a very bad idea

For Question 3, were interested in the following topology-based approaches for identifying noisy points in a trajectory from a dynamical system

Part a asks, if the forward images of two nearby points are not close, one of those two points may have been perturbed by noise, and this is true

If the forward images of two nearby points are not close, this would violate the continuity properties of the dynamical systems were looking at

For part b, if the attractor contains isolated points, they may be the result of noise, and this is also true

Attractors  at least the attractors were looking at in this course  are connected sets

If you have points that are isolated  that is, theyre not connected to the rest of the set  this may mean that that point is simply perturbed by noise, or thats a noisy point