Painlevé analysis and normal forms theory (Q5940204)

In the study of nonlinear vector fields, there are only two types of singularities. The first singularities are locally analysed via normal form theory, whereas the second ones are studied by the Painlevé analysis.NEWLINENEWLINENEWLINEHere, the author uses normal form theory to describe the solutions around their complex-time singularities. To do so, he introduces a transformation mapping the local series around the singularities to the local series around a fixed point of a new system. Then he uses regular normal form theory in this new system. He shows that a vector field has the Painlevé property only if the associated system is locally linearizable around its fixed points, a problem analogous to the classical problem of the centre. Moreover, he establishes a connection between partial and complete integrability and the structure of local series around both types of singularities. He further gives a new proof for the convergence of the local Psi-series and presents an explicit method to prove the existence of a finite time blow-up manifold in phase space.NEWLINENEWLINENEWLINEThe paper appears to be a good addition to the subject.