In the theory of dynamical systems, a Poincaré map is the intersection of a trajectory of something which moves periodically (or quasi-periodically , or chaotically), in a space of at least three dimensions, with a hyperplane of one fewer dimension. More precisely, one observes the return of the trajectory to the hyperplane which starts at a given point of it. The name is for Henri Poincaré.
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 on a plane looks like a tangled mess, while the Poincaré map shows the structure more clearly.
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