Painlevé analysis and normal forms theory

Nonlinear vector fields have two important types of singularities: the fixed points in phase space and the time singularities in the complex plane. The first singularities are locally analyzed via normal form theory whereas the second ones are studied by the Painlevé analysis. In this paper, normal form theory is used to describe the solutions around their complex-time singularities. To do so, a transformation mapping the local series around the singularities to the local series around a fixed point of a new system is introduced. Regular normal form theory is then used on this new system. It is shown 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, the connection between partial and complete integrability and the structure of local series around both type of singularities are established. A new proof of the convergence of the local Psi-series is given and an explicit method to prove the existence of finite time blow-up manifold in phase space is presented.