A MELNIKOV VECTOR FOR N-DIMENSIONAL MAPPINGS

We use the Fredholm alternative to derive a Mel'nikov vector for pertu rbations of N-dimensional maps with homoclinic connections. If the unp erturbed mapping is integrable, this vector assumes a simple form, whi ch we use to determine conditions for transversal and tangential inter section between the invariant manifolds in a four-dimensional map of t he McMillan type. We also discuss conditions for non-transversal inter section which accurately predict the crossing of invariant manifolds f rom one part of 4-D space into another.