We consider an approach to the classical center--focus problem for plane ODE systems with polynomial right hand sides. For the wide class of these systems the necessary and sufficient center conditions are expressed through the variational equations of high order. In contrast to the existing methods, this paper deals not with obtaining the center conditions as such, but it is devoted primarily to the computation of explicit asymptotics of the Poincar'e mapping, which allows to investigate the cycle generation with degeneracy of arbitrary co-dimension.

Mathematical problems and theory of numerical methods