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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.