Quasi-Poincaré compactifications and blow-up solutions of ordinary differential equations

Kaname Matsue

Published 2016-11-19Version 1

We construct a generalized compactification of Euclidean spaces as well as dynamical systems on them for studying blow-up solutions of ordinary differential equations. The basic idea is based on the quasi-homogeneous desingularization (blowing-up) of singularities. We apply this desingularization to infinity so that divergent solutions including grow-up and blow-up solutions for differential equations whose asymptotic form is not necessarily homogeneous can be treated on compact spaces. As a prototype, we define the quasi-homogeneous version of Poincar\'e compactifications and discuss fundamental properties which play important roles to study blow-up solutions of inhomogeneous vector fields. As a demonstration of applicability, we consider singular shock waves consisting of several pieces of blow-up solutions and bounded solutions near infinity. We see that the quasi-Poincar\'e compactification well describes singular profiles of singular shocks observed in preceding studies.