Lyapunov unstable elliptic equilibria

Fayad, Bassam

doi:10.1090/jams/997

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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