Poincaré maps are used to investigate periodic or quasi-periodic dynamical systems. Often these systems exhibit a periodic cycle or a chaotic attractor. A Poincaré section is now assumed to be a part of a plane, which is placed within the 3D phase space of the continuous dynamical system such that either the periodic orbit or the chaotic attractor intersects the Poincaré section. The Poincaré map is now defined as a discrete function , which associates consecutive intersections of a trajectory of the 3D flow with (see Fig. 5.1).
There are some important relations between a 3D flow and the
corresponding Poincaré map: A cycle
of the 3D system
which intersects the Poincaré section
in q points
(q1) is related to a periodic
point
of Poincaré map
p, i.e.,
c is a critical point of the
map
pq. Furthermore stability characteristics of the cycle are
inherited by the critical point: stable, unstable, or saddle
cycles result in stable, unstable, or saddle nodes, respectively.
Therefore many characteristics of periodic or quasi-periodic
dynamical systems can be derived from the corresponding Poincaré map.