On the integrability of a planar system of Ode’s near a degenerate stationary point

Abstract

We consider an autonomous system of ordinary differential equations, which is solved with respect to derivatives. To study the local integrability of the system near a degenerate stationary point, we use an approach based on power geometry and the computation of the resonant normal form. For a concrete planar 5-parameter system, we found a complete set of necessary conditions on the parameters of the system for which the system is locally integrable near a degenerate stationary point. These sets of parameters consist of 4 two-parameter subsets in this 5-parameter space. For 3 such subsets, we found sufficient conditions for local integrability by independent methods. Since these methods are constructive, we obtain first integrals of the system. Thus, for these 3 subsets of parameters, the system is globally integrable. For the fourth subset, at the moment we have only approximations of local integrals in the form of truncated power series in the parameters of the system. We could not yet manage to sum them up to finite functions. Bibliography: 8 titles.