Natural Boundaries of Normalizing Transformations
The complex time singularities of Hamilton’s equations have been investigated by Chang et al.1 in order to distinguish integrable from nonintegrable systems. The analysis of the singularities for the transformation which brings a complexified hamiltonian in normal form2 extends to hamiltonian maps; bounds are provided by KAM techniques, but the exact location of the singularities and their nature remain unknown. Some heuristic information is provided by perturbation theory3,4, but at best only the main singularities can be identified. Given an area preserving map near an elliptic point, the singularities in the action plane are poles, at any finite order; the Padé Approximants (P.A.) are adequate to detect them, as shown in5 and to provide the analytic continuation of the perturbation series beyond the resonances.
KeywordsFinite Order Natural Boundary Complex Frequency Heuristic Information Homologic Equation
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