The essential idea of a Poincaré map is to boil down the way you represent a dynamical system. For this, the system has to have certain properties, namely to return to some region in its state space from time to time. This is fulfilled if the dynamics is periodic, but it also works with chaotic dynamics.
To give a simple example, instead of analysing the entire trajectory of a planet, you would only look at its position once a year, more precisely, whenever it intersects (with a given direction) a plane
- that is perpendicular to the plane in which the planets trajectories lie,
- that contains the central celestial body around which the planet rotates.
This plane is a Poincaré section for the orbit of this planet, as it is transversal to the flow of the system (which goes along the planet’s trajectories).
Now, if the planet’s orbit is exactly periodic with a period length corresponding to one year, our yearly recording would always yield the same result. With other words, our planet would intersect the Poincaré section at the same point every year. If the planet’s orbit is however more complicated, e.g., the Perihelion precession of Mercury, the point of intersection with the Poincaré section will slightly change each year. You can then consider a Poincaré map which describes how the intersection point for one year depends on the intersection point for the previous year.
While I have only looked at the geometrical position for this example, you can also look at other observables and probably need to, if you cannot fully deduce the position in phase space from the geometrical position. In our example, you would also need to record the impulse of the planet (or some other observable).
Now, what’s the purpose of this? If our planet’s orbit only deviates from perfect periodicity slightly, what happens during one year is just going in circles and thus “rather boring” and obfuscating the interesting things that happen on larger time scale. The latter can be observed on our Poincaré map, which shows us how the orbit slightly changes each year. Therefore it may be easier or more illustrative to just analyse the Poincaré map instead of the entire trajectory.
This is even more pronounced for billiard: Between two collisions with a boundary, the dynamics is just $\dot{x}=v$.
In particular, certain properties of your underlying dynamics translate to the Poincaré map, e.g.: If the dynamics is chaotic, so is your Poincaré map. If, in our planet example, the dynamics is periodic with a period of four years, your Poincaré map will alternate between four points.
If your dynamic is quasi-periodic with two incommensurable frequencies (for example, if one observable is $\sin(x)+\sin(\pi x)$), the intersections with your Poincaré section will all lie on a closed curve. For example, most straight trajectories on the surface of a torus correspond to a dynamics with incommensurable frequencies and will eventually come arbitrarily close to any point on the torus, i.e., they fill the torus’s surface. Thus the intersection of the trajectory with a Poincaré section that is perpendicular to the torus’s surface at all points will yield the border of a circle (and non-perpendicular Poincaré sections will yield something close to an ellipsis).
In general, the dimension of the intersections with the Poincaré section is the dimension of the attractor minus one.
Also, if you want to model an observed system in the sense of finding equations that reproduce its dynamics to some extent, you might start with modelling the Poincaré map (i.e., find an explicit formula for it).
