Poincaré map

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In mathematics, particularly in dynamical systems, a Poincaré map or Poincaré section, named after Henri Poincaré, is the intersection of a trajectory which moves periodically (or quasi-periodically, or chaotically), in a space of at least three dimensions, with a transversal hypersurface of one fewer dimension. More precisely, one considers a trajectory with initial conditions on the hyperplane and observes the point at which this trajectory returns to the hyperplane. The Poincaré section refers to the hyperplane, and the Poincaré map refers to the map of points in the hyperplane induced by the intersections.

It differs from a recurrence plot in that space, not time, determines when to plot a point. For instance, the locus of the moon when the earth is at perihelion is a recurrence plot; the locus of the moon when it passes through the plane perpendicular to the earth's orbit and passing through the sun and the earth at perihelion is a Poincaré map. It was used by Michel Hénon to study the motion of stars in a galaxy, because the path of a star projected onto a plane looks like a tangled mess, while the Poincaré map shows the structure more clearly.