About Poincaré maps

  

Figure 5.1: An illustration of the Poincaré map definition.

A Poincaré section is used to construct a (n-1)-dimensional discrete dynamical system, i.e., a Poincaré map, of a continuous flow given in n dimensions. This reduced system of n-1 dimensions inherits many properties, e.g., periodicity or quasi-periodicity, of the original system. We will concentrate in the following on the case of n being equal to three.

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  $\mathcal{S}$ 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  $\mathbf{p}:\mathcal{S}\to\mathcal{S}$, which associates consecutive intersections of a trajectory of the 3D flow with  $\mathcal{S}$ (see Fig. 5.1).

There are some important relations between a 3D flow and the corresponding Poincaré map: A cycle  $\mathcal{C}$ of the 3D system which intersects the Poincaré section  $\mathcal{S}$ in q points (q$\ge$1) is related to a periodic point  $\mathbf{c}=\mathcal{C}\!\cap\!\mathcal{S}=\mathbf{p}^q(\mathbf{c})$ of Poincaré map  p, i.e., c is a critical point of the map  p^q. 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.

mailto:helwig@cg.tuwien.ac.at.