Canonical Melnikov theory for diffeomorphisms

We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection. We show how our definition reproduces the classical methods of Poincar\'{e} and Melnikov and specializes to methods previously used for exact symplectic and volume-preserving maps. We use the method to detect the transverse intersection of stable and unstable manifolds and relate this intersection to the set of zeros of the Melnikov displacement.Comment: laTeX, 31 pages, 3 figure

Quadratic Volume Preserving Maps

We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Henon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume preserving maps in three space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.Comment: Ams LaTeX file with 4 figures (figure 2 is gif, the others are ps