A new algorithm for the computation of the Lyapunov constants for some degenerated critical points

The center problem for degenerated monodromic critical points is far to be solved in general. In this paper we give a procedure to solve it for a particular perturbation of critical points which dominant part near the critical point is x2n-1∂/∂x + y2m-1∂/∂y. For these critical points the problem is solved by writing its associated differential equation in the generalized polar coordinates introduced by Lyapunov and by developing a new method of computation of the so called generalized Lyapunov contants. This method is based into the transformation of the differential equation into the perturbation of a Hamiltonian system. Finally, the method is applied to solve the center and the stability problem for a particular family of differential equations.