Monodromy, how to choose u?

[freemin7]

The article on monodromy does not provide any guidance how to choose such an u. Questions like how many dimensions it should have and how one should integrate it into the system could be addressed or a reference provided.

[PBrdng]

In general, there is no "correct" way of choosing parameters. For instance, you could set some of the coefficients of your systems of parameters and some not, or all of them. This requires some trial-and-error before it really works well, unfortunately.

[freemin7]

Are there some general principles one should be aware of? Like linear independence conditions, rules how the number of u must scale with the number of variables. How the order of the u should behave relative to the equation it occurs in?

[PBrdng]

Only rules of thumbs: when I do it, I always try first few u-variables and see how it goes (because fewer us = smaller linear system to solve internally). The order of u and linear independence mostly don't matter. But when I say this, it is not a hard fact. In fact, monodromy is not well-understood at this point. Many open research questions.