The Pioncaré map takes the point x {\displaystyle x} in the Poincaré section S {\displaystyle S} from x {\displaystyle x} to x ′ {\displaystyle x'} and from x ′ {\displaystyle x'} to x ″ {\displaystyle x''} .
The Pioncaré map takes the point in the Poincaré section from to and from to .
The Pioncaré map takes the point 
