Implementation of the Painlevé test for ordinary differential systems

AbstractBy definition, the Painlevé test is the set of all techniques which enables one to generate necessary (although generally not sufficient) conditions for the integrability of ordinary (or partial) non-linear differential systems, in the sense of Painlevé. In conclusive cases, the results of Painlevé local singularity analysis provide the most fundamental knowledge related with the integrability of such systems. Moreover, the qualitative behaviour of dynamical systems may also be probed in terms of their analytic structure. The Painlevé test may then provide, in some cases, quite pertinent indications in favour of possibly chaotic dynamical behaviours in the general solution of such non-linear systems.The implementation of the Painlevé test in the algebraic programming language reduce is presented in the case of systems of non-linear ordinary differential equations which are polynomials in the dependent variables and their derivatives. The implementation includes a routine for the automatic search for movable singularity families, and a second routine for the process of the Painlevé test itself, either with the help of the classical (and widely used) Kowalevskaya and Gambier method or within the perturbative framework developed by Conte, Fordy and Pickering