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Poincaré section and Poincaré map

A Poincaré section is a surface in phase space that cuts across the flow of a dynamical system. It is a carefully chosen (in general, curved) surface in the phase space that is crossed by almost all orbits. It is a tool developed by Poincaré for visualization of the flow in more than two dimensions. The Poincaré section has one dimension less than the phase space and the Poincaré map transforms the Poincaré section onto itself by relating two consecutive intersection points, say and . We note that only those intersection points count, which come from the same side of the section. The Poincaré map is invertible because one gets from by following the orbit backwards. A Poincaré map turns a continuous-time dynamical system into a discrete-time one. If the Poincaré section is carefully chosen no information is lost concerning the qualitative behaviour of the dynamics. For example, if the system is being attracted to a limit cycle, one observes dots converging to a fixed point in the Poincaré section.

Subsections

- Poincaré maps in CL/SMALL>_MATCONT