Main Question or Discussion Point
I'm not sure I'm following the whole intro chapter 1C "Phase Singularities of Maps". Is he saying that the existence of an oscillator somehow requires a singularity?
I'm a bit confused by your reference to change in winding number. The point, I thought, is that the winding number is an integer. It's minimum value per change in the input phase must be 1 cycle and that really this thought puzzle is about the difficulty of having a map from a real to an integer while still assuming infinitesimal continuity. There is always a smaller move you can make with the input - which still must result in a minimum of 1 integer winding cycle. Or there is no non-zero change in the input that does not cause a minimum of one cycle.A change in winding number isn't a small change. Far from singularities, small changes in the input phases doesn't change the winding number of the output. When the phase space trajectories are deformed in a way that crosses the singularity, the winding number can change.
Winding numbers for trajectories are just like a loop of string wrapped several times around an infinite solid rod, or a steel ring, or such. The winding number around the rod can't change without cutting and reattaching the string, or teleporting a hank of string discontinuously through the rod, etc.
If you haven't read it, Strogatz's "Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering" is an excellent broad based introduction to this topic, and has a very good discussion of winding numbers.
The first two pages of Hatcher's "Algebraic Topology" has a pretty good explanation of retractions. Plus, the book is available free as a PDF from the author's site: https://www.math.cornell.edu/~hatcher/AT/ATpage.htmlWish I had a good grasp of what exactly "manifold retraction" means...