Consider an n-dimensional deterministic dynamical system (x^_)^. = f^_(x) and let S be an n - 1-dimensional surface of section that is traverse to the flow, i.e., all trajectories starting from S flow through it and are not parallel to it. Then a Poincaré map P is a mapping from S to itself obtained by following trajectories from one intersection of the surface S to the next. Poincaré maps are useful when studying swirling flows near periodic solutions in dynamical systems.

surface of section

Henri Poincaré