### Poincaré section

(pattern formed by a the locations an orbiting body passes through a plane)

An example of a **Poincaré section** is all the points through
which an orbiting body passes through a plane intersecting the
path of the orbits, specifically passing in one direction but not the other.
The associated **Poincaré map** is a function mapping a chosen
point of the Poincaré section to the next point, and each
one to the next (but see below). Poincaré sections provide
a means of identifying long-term trends of an orbit that doesn't
simply repeat with each orbit.

While the above example gives a two-dimensional "summary" of a
three-dimensional process, in general, a *Poincaré section*
can be created for any number of dimensions, producing a summary
in one less dimensions. The type of problem is one where a point
in whatever dimensional space "moves" as time passes, i.e., the
dimensional values change over time.

The orbit example above actually is flawed: an orbit might intersect
a plane at a point, and later intersect it at the same point yet
moving in some other direction, so, by definition, a function
wouldn't be able to take the role of its associated Poincaré
Map. Thus, a Poincaré section to analyze an orbit might
more likely be done on a 6-dimensional space that also includes
three dimensions of velocity (or momentum), reducing it to 5-dimensional
summary.

(*mathematics,dynamics*)
http://en.wikipedia.org/wiki/Poincare_map

http://dspace.mit.edu/bitstream/handle/1721.1/84612/12-006j-fall-2006/contents/lecture-notes/lecnotes8.pdf

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