We use the Fredholm alternative to derive a Mel'nikov vector for perturbations of N-dimensional maps with homoclinic connections. If the unperturbed mapping is integrable, this vector assumes a simple form, which we use to determine conditions for transversal and tangential intersection between the invariant manifolds in a four-dimensional map of the McMillan type. We also discuss conditions for non-transversal intersection which accurately predict the crossing of invariant manifolds from one part of 4-D space into another.
ASJC Scopus subject areas
- Physics and Astronomy(all)