  This is a complete study of the dynamics of polynomial planar vector fields
whose linear part is a multiple of the identity and whose nonlinear part is a
contracting homogeneous polynomial. The contracting nonlinearity provides the
existence of an invariant circle and allows us to obtain a classification
through a complete invariant for the dynamics, extending previous work by other
authors that was mainly concerned with the existence and number of limit
cycles. The general results are also applied to some classes of examples:
definite nonlinearities, $\ZZ_2\oplus\ZZ_2$ symmetric systems and
nonlinearities of degree 3, for which we provide complete sets of phase
portraits.
