Perturbative methods for the Painlevé test (Q2734888)

Some examples of ordinary differential equations having movable critical singularities, which the ``usual Painlevé test'' of Kowalewski and Gambier fails to detect, are considered. Algorithmic methods able to perform this detection are explaned not in full theoretical detail but on examples selected for their simplicity. The considered examples are the following NEWLINE\[NEWLINEu' +4uu'+2u^3=0, \quad u''''+3uu''-4{u'}^2=0,\quad u'''+uu''-2{u'}^2=0.NEWLINE\]NEWLINE More details can be found in [\textit{R. Conte} (ed.), The Painlevé property. One century later. CRM Series in Mathematical Physics (Springer-Verlag: New York) (1999; Zbl 0989.00036)].NEWLINENEWLINEFor the entire collection see [Zbl 0960.00047].