Poincaré map

Summary

The Poincaré map, or first return map, belongs to a periodic orbit and a cross-section of it. It follows nearby solutions once around the orbit until they hit the cross-section again. It replaces a continuous-time flow in dimension $n$ by a discrete map in dimension $n - 1$. The periodic orbit becomes a fixed point of the map, and the stability of the orbit is the stability of this fixed point. The idea is due to Henri Poincaré and is a basic tool in the study of periodic orbits, bifurcations and chaos.