Canonical Melnikov theory for diffeomorphisms

Details Canonical Melnikov theory for diffeomorphisms Canonical Melnikov theory for diffeomorphisms

2007-06-18

arxiv.org/abs/0706.2515v1

Nonlinear Sciences

Chaotic Dynamics

laTeX, 31 pages, 3 figures

Scientific paper

10.1088/0951-7715/21/3/007

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.

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