see: math.stackexchange.com/questions/94599/…

Just noting here the "swimming against the tide" is also name of a memoir about VI Arnold, so his work could probably be apt here.

Yes because the basin of attraction should be smooth in 2D autonomous systems

Yes, wikipedia has more info: en.wikipedia.org/wiki/Dennis_Sullivan and refers to the en.wikipedia.org/wiki/Uniformization_theorem

Add SIAM Math analysis to the list

Synchrony can understood as stability of a possibly time-varying "equilibrium", such as a periodic orbit. I recommend Nonlinear Dynamics book by Strogatz

Depending upon the problem, the states represent energy or probabilities. In Either case, you want them to be integrable, hence the restriction.

The geometry is in the strain, not stress. Newtonian fluid assumption that stress is simply proportional to strain gives this relation for stress. The derivation of strain is in many textbooks. Basically you need to compute shear strain ( angle) as a function of velocity gradienta

looks like a contrived problem:the solution was the written down first and then the problem was created