On the approach of Holmes and Sanders to the Melnikov procedure in the method of averaging

In 1963 Melnikov has offered a computational approach to the problem of finding transversal heteroclinic points for certain plane periodic systems. Recently Holmes has proposed a similar scheme for a different type of equations which are related to the method of averaging. In the sequence Sanders has pointed out that the reasoning of Holmes is not conclusive. In this note we review the problem based on a different generalization of the Melnikov scheme.