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?
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.
Wish I had a good grasp of what exactly "manifold retraction" means...