Poincaré compactification for n-dimensional piecewise polynomial vector fields : Theory and applications

Poincaré compactification is very important to investigate the dynamics of vector fields in the neighborhood of the infinity, which is the main concern on the escape of particles to infinity in celestial mechanics, astrophysics, astronomy and some branches of chemistry. Since then Poincaré compactification has been extended into various cases, such as: n-dimensional polynomial vector fields, Hamiltonian vector fields, quasi-homogeneous vector fields, rational vector fields, etc. In recent years, the piecewise smooth vector fields describing situations with discontinuities such as switching, decisions, impacts etc., have been attracted more and more attention. It is worth to notice that Poincaré compactification has been extended successfully to piecewise polynomial vector fields in 2-dimensional and 3-dimensional cases, and there are also works on n-dimensional Lipschitz continuous vector fields. The main goal of present paper is to extend the Poincaré compactification to n-dimensional piecewise polynomial vector fields which are usually discontinuous, this is a missing point in the existent literature. Thus we can investigate the dynamics near the infinity of n-dimensional piecewise polynomial vector fields. As an application we study the global phase portraits for a class of 3-dimensional piecewise linear differential systems.