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A new diffusion mechanism from the neighborhood of elliptic equilibria for Hamiltonian flows in three or more degrees of freedom is introduced. We thus obtain explicit real entire Hamiltonians on R 2 d \mathbb {R}^{2d} , d ≥ 4 d\geq 4 , that have a Lyapunov unstable elliptic equilibrium with an arbitrary chosen frequency vector whose coordinates are not all of the same sign. For non-resonant frequency vectors, our examples all have divergent Birkhoff normal form at the equilibrium. On R 4 \mathbb {R}^4 , we give explicit examples of real entire Hamiltonians having an equilibrium with an arbitrary chosen non-resonant frequency vector and a divergent Birkhoff normal form.
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Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation
Abstract Consider an analytic Hamiltonian system near its analytic invariant torus $$\mathcal T_0$$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $$\mathcal T_0$$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation—not just a formal power series—bringing the Hamiltonian into its Birkhoff normal form.
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- Award ID(s):
- 1800241
- PAR ID:
- 10358888
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 42
- Issue:
- 3
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 1166 to 1187
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2021.71
