Lyapunov unstable elliptic equilibria
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- by Bassam Fayad
- J. Amer. Math. Soc. 36 (2023), 81-106
- DOI: https://doi.org/10.1090/jams/997
- Published electronically: January 27, 2022
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Abstract:
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 $\mathbb {R}^{2d}$, $d\geq 4$, 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 $\mathbb {R}^4$, 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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Bibliographic Information
- Bassam Fayad
- Affiliation: Institut de Mathématiques de Jussieu–Paris Rive Gauche (IMJ-PRG), French National Centre for Scientific Research (CNRS), 58-56, Avenue de France, Paris, France
- MR Author ID: 675142
- Received by editor(s): July 2, 2020
- Received by editor(s) in revised form: May 20, 2021, September 1, 2021, and October 18, 2021
- Published electronically: January 27, 2022
- © Copyright 2022 American Mathematical Society
- Journal: J. Amer. Math. Soc. 36 (2023), 81-106
- MSC (2020): Primary 37J06, 37J11, 37J12, 37J25, 37J30
- DOI: https://doi.org/10.1090/jams/997
- MathSciNet review: 4495839