IMPLEMENTATION, OF THE PAINLEVE TEST FOR ORDINARY DIFFERENTIAL-SYSTEMS

By definition, the Painleve test is the set of all techniques which en ables one to generate necessary (although generally not sufficient) co nditions for the integrability of ordinary (or partial) non-linear dif ferential systems, in the sense of Painleve. In conclusive cases, the results of Painleve local singularity analysis provide ale most fundam ental knowledge related with the integrability of such systems. Moreov er, the qualitative behaviour of dynamical systems may also be probed in terms of their analytic structure. The Painleve test may then provi de, in some cases, quite pertinent indications in favour of possibly c haotic dynamical behaviours in the general solution of such non-linear systems. The implementation of the Painleve test in the algebraic pro gramming language REDUCE is presented in the case of systems of non-li near ordinary differential equations which are polynomials in the depe ndent variables and their derivatives. The implementation includes a r outine for the automatic search for movable singularity families, and a second routine for the process of the Painleve test itself, either w ith the help of the classical (and widely used) Kowalevskaya and Gambl er method or within the perturbative framework developed by Conte, For dy and Pickering.