E Todesco and G Turchetti 1995 J. Phys. A: Math. Gen. 28 2325 doi:10.1088/0305-4470/28/8/023
Nonresonant limit for sequences of resonant orbits: the case of holomorphic maps
E Todesco and G Turchetti
Show affiliationsThe approximation of a nonresonant orbit with a sequence of resonant orbits is considered for the holomorphic maps of the complex plane. The problem is motivated by Hamiltonian dynamics (Greene's conjecture) and we consider a complexified Hamiltonian map in the region (far from the section of real dynamics) where it can be reduced to a holomorphic map of a single complex variable. For a sequence of maps in normal form with linear resonant frequencies, the limit to a linear map with nonresonant diophantine frequency has a simple interpretation: the flower-like resonant orbits become circles due to the increase of the number of petals and the freezing of radial motion. A similar non-trivial result is proved for small perturbations of the normal forms by investigating the behaviour of the conjugation functions.
- PACS
- MSC
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37J40 Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (See also 53D20)
- Subjects
- Dates
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Issue 8 (21 April 1995)
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Nonresonant limit for sequences of resonant orbits: the case of holomorphic maps
E Todesco and G Turchetti 1995 J. Phys. A: Math. Gen. 28 2325
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