Poincaré surface of section (also referred to as Poincaré section or Poincaré map) is powerful technique extracting fundamental properties from flows in the form of a discrete maps. A trajectory or set of trajectories are sampled periodically, by looking at successive intersections with a plane in the phase space.

Many physical, chemical, and biological processes in nature are described by a set of coupled first-order autonomous differential equations, or autonomous flows. A widely used technique in the study of these systems is the Poincaré surface of section (also referred as Poincaré section or Poincaré map) technique. On a Poincaré surface of section, the dynamics can be described by a discrete map whose phase-space dimension is one less than that of the original continuous flow. This sectioning technique thus provides a natural link between continuous flows and discrete maps. With a tremendous facilitation in analysis, numerical computation, and visualization, maps also capture many fundamental dynamical properties of flows. These advantages have made the Pioncaré surface of section technique one of the most popular analysis tools in nonlinear dynamics and chaos.

A surface of section is generated by looking at successive intersections of a trajectory or a set of trajectories with a plane in the phase space. Typically, the plane is spanned by a coordinate axis and the canonically conjugate momentum axis. We will see that surfaces of section made in this way have nice properties.

In [1]:

```
%matplotlib inline
import utils
utils.plot_poincare_surface()
```