Analysis of conservative systems by approximating area-preserving point mapping

The reduction of the study of continuous dynamical systems to the study of point mappings was first effected by Poincaré, and later more generally by Birkhoff, in the conservative case. However, the analytical expressions of these mappings are, in most cases, rather complicated or even unknown. This paper deals with the construction of area- preserving polynomial point mappings in the plane osculating the actual one (also area preserving) to a certain degree. These mappings can be used for an approximate study of the behaviour of solutions in a neighbourhood of the origin taken as a fixed point. To illustrate this approach, some numerical examples are given.