Héctor E Lomelí et al 2008 Nonlinearity 21 485 doi:10.1088/0951-7715/21/3/007
Canonical Melnikov theory for diffeomorphisms
Héctor E Lomelí1, James D Meiss2 and Rafael Ramírez-Ros3
Show affiliationsRecommended by D Treschev
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 Poincaré and Melnikov and specializes in 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.
- PACS
- MSC
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37C55 Periodic and quasiperiodic flows and diffeomorphisms
- Subjects
- Dates
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Issue 3 (March 2008)
Received 14 June 2007, in final form 25 January 2008
Published 18 February 2008
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Canonical Melnikov theory for diffeomorphisms
Héctor E Lomelí et al 2008 Nonlinearity 21 485
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