Poincaré map
From Free net encyclopedia
In mathematics, in the theory of dynamical systems, a Poincaré map or Poincaré section 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 transversal hypersurface 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 Poincaré section refers to the hyperplane, and the Poincaré map refers to the map of points in the hyperplane induced by the intersections. It is named after 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.
See also
References
- Nicholas B. TUFILLARO, Poincaré Map, (1997)
- Shivakumar JOLAD, Poincare Map and its application to 'Spinning Magnet' problem, (2005)