differential equations - Dimension of Poincaré Map

I am used to seeing bi-dimensional Poincaré maps, as the ones shown in this post:

Poincaré maps and interpretation

In that example, one manages to draw a bi-dimensional map because the number of degrees of freedom is quite limited.

My question is: does it make sense to draw bi-dimensional Poincaré maps also for higher-dimensional dynamical systems (e.g. a 5-body problems)? If yes, which is a good recipe to decide the section plane?

Or, instead, the Poincaré map is of higher dimensionality?

Answer

A Poincaré map can be used to obtain a representation of the original, $N$-dimensional dynamics in any number of dimensions $n

That said, a 2-D map of a system with 4 or more dimensions is seldom used, because of the difficulty in interpreting the resulting figures -- unless, and that's an important exception, the original system trajectories are actually contained in a lower dimensional manifold (i.e., the system effectively has fewer dimensions). Also, if $n>3$, the Poincaré map of course won't be of much help for visualization purposes.

As for how to define the section, the more you know about the system (symmetries, bounds, invariant sets, etc.), the better the guess you can make. Apart from that, it's usually a matter of trial and error, where experience and luck will be helpful.